hypothesis data in statistics

Statistics Made Easy

Introduction to Hypothesis Testing

A statistical hypothesis is an assumption about a population parameter .

For example, we may assume that the mean height of a male in the U.S. is 70 inches.

The assumption about the height is the statistical hypothesis and the true mean height of a male in the U.S. is the population parameter .

A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical hypothesis.

The Two Types of Statistical Hypotheses

To test whether a statistical hypothesis about a population parameter is true, we obtain a random sample from the population and perform a hypothesis test on the sample data.

There are two types of statistical hypotheses:

The null hypothesis , denoted as H 0 , is the hypothesis that the sample data occurs purely from chance.

The alternative hypothesis , denoted as H 1 or H a , is the hypothesis that the sample data is influenced by some non-random cause.

Hypothesis Tests

A hypothesis test consists of five steps:

1. State the hypotheses.

State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false.

2. Determine a significance level to use for the hypothesis.

Decide on a significance level. Common choices are .01, .05, and .1.

3. Find the test statistic.

Find the test statistic and the corresponding p-value. Often we are analyzing a population mean or proportion and the general formula to find the test statistic is: (sample statistic – population parameter) / (standard deviation of statistic)

4. Reject or fail to reject the null hypothesis.

Using the test statistic or the p-value, determine if you can reject or fail to reject the null hypothesis based on the significance level.

The p-value tells us the strength of evidence in support of a null hypothesis. If the p-value is less than the significance level, we reject the null hypothesis.

5. Interpret the results.

Interpret the results of the hypothesis test in the context of the question being asked.

The Two Types of Decision Errors

There are two types of decision errors that one can make when doing a hypothesis test:

Type I error: You reject the null hypothesis when it is actually true. The probability of committing a Type I error is equal to the significance level, often called alpha , and denoted as α.

Type II error: You fail to reject the null hypothesis when it is actually false. The probability of committing a Type II error is called the Power of the test or Beta , denoted as β.

One-Tailed and Two-Tailed Tests

A statistical hypothesis can be one-tailed or two-tailed.

A one-tailed hypothesis involves making a “greater than” or “less than ” statement.

For example, suppose we assume the mean height of a male in the U.S. is greater than or equal to 70 inches. The null hypothesis would be H0: µ ≥ 70 inches and the alternative hypothesis would be Ha: µ < 70 inches.

A two-tailed hypothesis involves making an “equal to” or “not equal to” statement.

For example, suppose we assume the mean height of a male in the U.S. is equal to 70 inches. The null hypothesis would be H0: µ = 70 inches and the alternative hypothesis would be Ha: µ ≠ 70 inches.

Note: The “equal” sign is always included in the null hypothesis, whether it is =, ≥, or ≤.

Related: What is a Directional Hypothesis?

Types of Hypothesis Tests

There are many different types of hypothesis tests you can perform depending on the type of data you’re working with and the goal of your analysis.

The following tutorials provide an explanation of the most common types of hypothesis tests:

Introduction to the One Sample t-test Introduction to the Two Sample t-test Introduction to the Paired Samples t-test Introduction to the One Proportion Z-Test Introduction to the Two Proportion Z-Test

Featured Posts

Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike. My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

Hypothesis Testing – A Deep Dive into Hypothesis Testing, The Backbone of Statistical Inference

September 21, 2023

Explore the intricacies of hypothesis testing, a cornerstone of statistical analysis. Dive into methods, interpretations, and applications for making data-driven decisions.

In this Blog post we will learn:

What is Hypothesis Testing?
Steps in Hypothesis Testing 2.1. Set up Hypotheses: Null and Alternative 2.2. Choose a Significance Level (α) 2.3. Calculate a test statistic and P-Value 2.4. Make a Decision
Example : Testing a new drug.
Example in python

1. What is Hypothesis Testing?

In simple terms, hypothesis testing is a method used to make decisions or inferences about population parameters based on sample data. Imagine being handed a dice and asked if it’s biased. By rolling it a few times and analyzing the outcomes, you’d be engaging in the essence of hypothesis testing.

Think of hypothesis testing as the scientific method of the statistics world. Suppose you hear claims like “This new drug works wonders!” or “Our new website design boosts sales.” How do you know if these statements hold water? Enter hypothesis testing.

2. Steps in Hypothesis Testing

Set up Hypotheses : Begin with a null hypothesis (H0) and an alternative hypothesis (Ha).
Choose a Significance Level (α) : Typically 0.05, this is the probability of rejecting the null hypothesis when it’s actually true. Think of it as the chance of accusing an innocent person.
Calculate Test statistic and P-Value : Gather evidence (data) and calculate a test statistic.
p-value : This is the probability of observing the data, given that the null hypothesis is true. A small p-value (typically ≤ 0.05) suggests the data is inconsistent with the null hypothesis.
Decision Rule : If the p-value is less than or equal to α, you reject the null hypothesis in favor of the alternative.

2.1. Set up Hypotheses: Null and Alternative

Before diving into testing, we must formulate hypotheses. The null hypothesis (H0) represents the default assumption, while the alternative hypothesis (H1) challenges it.

For instance, in drug testing, H0 : “The new drug is no better than the existing one,” H1 : “The new drug is superior .”

2.2. Choose a Significance Level (α)

When You collect and analyze data to test H0 and H1 hypotheses. Based on your analysis, you decide whether to reject the null hypothesis in favor of the alternative, or fail to reject / Accept the null hypothesis.

The significance level, often denoted by $α$, represents the probability of rejecting the null hypothesis when it is actually true.

In other words, it’s the risk you’re willing to take of making a Type I error (false positive).

Type I Error (False Positive) :

Symbolized by the Greek letter alpha (α).
Occurs when you incorrectly reject a true null hypothesis . In other words, you conclude that there is an effect or difference when, in reality, there isn’t.
The probability of making a Type I error is denoted by the significance level of a test. Commonly, tests are conducted at the 0.05 significance level , which means there’s a 5% chance of making a Type I error .
Commonly used significance levels are 0.01, 0.05, and 0.10, but the choice depends on the context of the study and the level of risk one is willing to accept.

Example : If a drug is not effective (truth), but a clinical trial incorrectly concludes that it is effective (based on the sample data), then a Type I error has occurred.

Type II Error (False Negative) :

Symbolized by the Greek letter beta (β).
Occurs when you accept a false null hypothesis . This means you conclude there is no effect or difference when, in reality, there is.
The probability of making a Type II error is denoted by β. The power of a test (1 – β) represents the probability of correctly rejecting a false null hypothesis.

Example : If a drug is effective (truth), but a clinical trial incorrectly concludes that it is not effective (based on the sample data), then a Type II error has occurred.

Balancing the Errors :

In practice, there’s a trade-off between Type I and Type II errors. Reducing the risk of one typically increases the risk of the other. For example, if you want to decrease the probability of a Type I error (by setting a lower significance level), you might increase the probability of a Type II error unless you compensate by collecting more data or making other adjustments.

It’s essential to understand the consequences of both types of errors in any given context. In some situations, a Type I error might be more severe, while in others, a Type II error might be of greater concern. This understanding guides researchers in designing their experiments and choosing appropriate significance levels.

2.3. Calculate a test statistic and P-Value

Test statistic : A test statistic is a single number that helps us understand how far our sample data is from what we’d expect under a null hypothesis (a basic assumption we’re trying to test against). Generally, the larger the test statistic, the more evidence we have against our null hypothesis. It helps us decide whether the differences we observe in our data are due to random chance or if there’s an actual effect.

P-value : The P-value tells us how likely we would get our observed results (or something more extreme) if the null hypothesis were true. It’s a value between 0 and 1. – A smaller P-value (typically below 0.05) means that the observation is rare under the null hypothesis, so we might reject the null hypothesis. – A larger P-value suggests that what we observed could easily happen by random chance, so we might not reject the null hypothesis.

2.4. Make a Decision

Relationship between $α$ and P-Value

When conducting a hypothesis test:

We then calculate the p-value from our sample data and the test statistic.

Finally, we compare the p-value to our chosen $α$:

If $p−value≤α$: We reject the null hypothesis in favor of the alternative hypothesis. The result is said to be statistically significant.
If $p−value>α$: We fail to reject the null hypothesis. There isn’t enough statistical evidence to support the alternative hypothesis.

3. Example : Testing a new drug.

Imagine we are investigating whether a new drug is effective at treating headaches faster than drug B.

Setting Up the Experiment : You gather 100 people who suffer from headaches. Half of them (50 people) are given the new drug (let’s call this the ‘Drug Group’), and the other half are given a sugar pill, which doesn’t contain any medication.

Set up Hypotheses : Before starting, you make a prediction:
Null Hypothesis (H0): The new drug has no effect. Any difference in healing time between the two groups is just due to random chance.
Alternative Hypothesis (H1): The new drug does have an effect. The difference in healing time between the two groups is significant and not just by chance.

Calculate Test statistic and P-Value : After the experiment, you analyze the data. The “test statistic” is a number that helps you understand the difference between the two groups in terms of standard units.

For instance, let’s say:

The average healing time in the Drug Group is 2 hours.
The average healing time in the Placebo Group is 3 hours.

The test statistic helps you understand how significant this 1-hour difference is. If the groups are large and the spread of healing times in each group is small, then this difference might be significant. But if there’s a huge variation in healing times, the 1-hour difference might not be so special.

Imagine the P-value as answering this question: “If the new drug had NO real effect, what’s the probability that I’d see a difference as extreme (or more extreme) as the one I found, just by random chance?”

For instance:

P-value of 0.01 means there’s a 1% chance that the observed difference (or a more extreme difference) would occur if the drug had no effect. That’s pretty rare, so we might consider the drug effective.
P-value of 0.5 means there’s a 50% chance you’d see this difference just by chance. That’s pretty high, so we might not be convinced the drug is doing much.
If the P-value is less than ($α$) 0.05: the results are “statistically significant,” and they might reject the null hypothesis , believing the new drug has an effect.
If the P-value is greater than ($α$) 0.05: the results are not statistically significant, and they don’t reject the null hypothesis , remaining unsure if the drug has a genuine effect.

4. Example in python

For simplicity, let’s say we’re using a t-test (common for comparing means). Let’s dive into Python:

Making a Decision : “The results are statistically significant! p-value < 0.05 , The drug seems to have an effect!” If not, we’d say, “Looks like the drug isn’t as miraculous as we thought.”

5. Conclusion

Hypothesis testing is an indispensable tool in data science, allowing us to make data-driven decisions with confidence. By understanding its principles, conducting tests properly, and considering real-world applications, you can harness the power of hypothesis testing to unlock valuable insights from your data.

Correlation – connecting the dots, the role of correlation in data analysis, sampling and sampling distributions – a comprehensive guide on sampling and sampling distributions, law of large numbers – a deep dive into the world of statistics, central limit theorem – a deep dive into central limit theorem and its significance in statistics, skewness and kurtosis – peaks and tails, understanding data through skewness and kurtosis”, similar articles, complete introduction to linear regression in r, how to implement common statistical significance tests and find the p value, logistic regression – a complete tutorial with examples in r.

Subscribe to Machine Learning Plus for high value data science content

Machine Learning A-Z™: Hands-On Python & R In Data Science

Free sample videos:.

User Preferences

Content preview.

Arcu felis bibendum ut tristique et egestas quis:

Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris
Duis aute irure dolor in reprehenderit in voluptate
Excepteur sint occaecat cupidatat non proident

Keyboard Shortcuts

Hypothesis testing.

Key Topics:

Basic approach
Null and alternative hypothesis
Decision making and the p -value
Z-test & Nonparametric alternative

Basic approach to hypothesis testing

State a model describing the relationship between the explanatory variables and the outcome variable(s) in the population and the nature of the variability. State all of your assumptions .
Specify the null and alternative hypotheses in terms of the parameters of the model.
Invent a test statistic that will tend to be different under the null and alternative hypotheses.
Using the assumptions of step 1, find the theoretical sampling distribution of the statistic under the null hypothesis of step 2. Ideally the form of the sampling distribution should be one of the “standard distributions”(e.g. normal, t , binomial..)
Calculate a p -value , as the area under the sampling distribution more extreme than your statistic. Depends on the form of the alternative hypothesis.
Choose your acceptable type 1 error rate (alpha) and apply the decision rule : reject the null hypothesis if the p-value is less than alpha, otherwise do not reject.
$\frac{\bar{X}-\mu_0}{\sigma / \sqrt{n}}$
general form is: (estimate - value we are testing)/(st.dev of the estimate)
z-statistic follows N(0,1) distribution
2 × the area above |z|, area above z,or area below z, or
compare the statistic to a critical value, |z| ≥ z α/2 , z ≥ z α , or z ≤ - z α
Choose the acceptable level of Alpha = 0.05, we conclude …. ?

Making the Decision

It is either likely or unlikely that we would collect the evidence we did given the initial assumption. (Note: “likely” or “unlikely” is measured by calculating a probability!)

If it is likely , then we “ do not reject ” our initial assumption. There is not enough evidence to do otherwise.

If it is unlikely , then:

either our initial assumption is correct and we experienced an unusual event or,
our initial assumption is incorrect

In statistics, if it is unlikely, we decide to “ reject ” our initial assumption.

Example: Criminal Trial Analogy

First, state 2 hypotheses, the null hypothesis (“H 0 ”) and the alternative hypothesis (“H A ”)

H 0 : Defendant is not guilty.
H A : Defendant is guilty.

Usually the H 0 is a statement of “no effect”, or “no change”, or “chance only” about a population parameter.

While the H A , depending on the situation, is that there is a difference, trend, effect, or a relationship with respect to a population parameter.

It can one-sided and two-sided.
In two-sided we only care there is a difference, but not the direction of it. In one-sided we care about a particular direction of the relationship. We want to know if the value is strictly larger or smaller.

Then, collect evidence, such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, handwriting samples, etc. (In statistics, the data are the evidence.)

Next, you make your initial assumption.

Defendant is innocent until proven guilty.

In statistics, we always assume the null hypothesis is true .

Then, make a decision based on the available evidence.

If there is sufficient evidence (“beyond a reasonable doubt”), reject the null hypothesis . (Behave as if defendant is guilty.)
If there is not enough evidence, do not reject the null hypothesis . (Behave as if defendant is not guilty.)

If the observed outcome, e.g., a sample statistic, is surprising under the assumption that the null hypothesis is true, but more probable if the alternative is true, then this outcome is evidence against H 0 and in favor of H A .

An observed effect so large that it would rarely occur by chance is called statistically significant (i.e., not likely to happen by chance).

Using the p -value to make the decision

The p -value represents how likely we would be to observe such an extreme sample if the null hypothesis were true. The p -value is a probability computed assuming the null hypothesis is true, that the test statistic would take a value as extreme or more extreme than that actually observed. Since it's a probability, it is a number between 0 and 1. The closer the number is to 0 means the event is “unlikely.” So if p -value is “small,” (typically, less than 0.05), we can then reject the null hypothesis.

Significance level and p -value

Significance level, α, is a decisive value for p -value. In this context, significant does not mean “important”, but it means “not likely to happened just by chance”.

α is the maximum probability of rejecting the null hypothesis when the null hypothesis is true. If α = 1 we always reject the null, if α = 0 we never reject the null hypothesis. In articles, journals, etc… you may read: “The results were significant ( p <0.05).” So if p =0.03, it's significant at the level of α = 0.05 but not at the level of α = 0.01. If we reject the H 0 at the level of α = 0.05 (which corresponds to 95% CI), we are saying that if H 0 is true, the observed phenomenon would happen no more than 5% of the time (that is 1 in 20). If we choose to compare the p -value to α = 0.01, we are insisting on a stronger evidence!

So, what kind of error could we make? No matter what decision we make, there is always a chance we made an error.

Errors in Criminal Trial:

Errors in Hypothesis Testing

Type I error (False positive): The null hypothesis is rejected when it is true.

α is the maximum probability of making a Type I error.

Type II error (False negative): The null hypothesis is not rejected when it is false.

β is the probability of making a Type II error

There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!

The power of a statistical test is its probability of rejecting the null hypothesis if the null hypothesis is false. That is, power is the ability to correctly reject H 0 and detect a significant effect. In other words, power is one minus the type II error risk.

$\text{Power }=1-\beta = P\left(\text{reject} H_0 | H_0 \text{is false } \right)$

Which error is worse?

Type I = you are innocent, yet accused of cheating on the test. Type II = you cheated on the test, but you are found innocent.

This depends on the context of the problem too. But in most cases scientists are trying to be “conservative”; it's worse to make a spurious discovery than to fail to make a good one. Our goal it to increase the power of the test that is to minimize the length of the CI.

We need to keep in mind:

the effect of the sample size,
the correctness of the underlying assumptions about the population,
statistical vs. practical significance, etc…

(see the handout). To study the tradeoffs between the sample size, α, and Type II error we can use power and operating characteristic curves.

What type of error might we have made?

Type I error is claiming that average student height is not 65 inches, when it really is. Type II error is failing to claim that the average student height is not 65in when it is.

We rejected the null hypothesis, i.e., claimed that the height is not 65, thus making potentially a Type I error. But sometimes the p -value is too low because of the large sample size, and we may have statistical significance but not really practical significance! That's why most statisticians are much more comfortable with using CI than tests.

There is a need for a further generalization. What if we can't assume that σ is known? In this case we would use s (the sample standard deviation) to estimate σ.

If the sample is very large, we can treat σ as known by assuming that σ = s . According to the law of large numbers, this is not too bad a thing to do. But if the sample is small, the fact that we have to estimate both the standard deviation and the mean adds extra uncertainty to our inference. In practice this means that we need a larger multiplier for the standard error.

We need one-sample t -test.

One sample t -test

Assume data are independently sampled from a normal distribution with unknown mean μ and variance σ 2 . Make an initial assumption, μ 0 .
t-statistic: $\frac{\bar{X}-\mu_0}{s / \sqrt{n}}$ where s is a sample st.dev.
t-statistic follows t -distribution with df = n - 1
Alpha = 0.05, we conclude ….

Testing for the population proportion

Let's go back to our CNN poll. Assume we have a SRS of 1,017 adults.

We are interested in testing the following hypothesis: H 0 : p = 0.50 vs. p > 0.50

What is the test statistic?

If alpha = 0.05, what do we conclude?

We will see more details in the next lesson on proportions, then distributions, and possible tests.

Tutorial Playlist

Statistics tutorial, everything you need to know about the probability density function in statistics, the best guide to understand central limit theorem, an in-depth guide to measures of central tendency : mean, median and mode, the ultimate guide to understand conditional probability.

A Comprehensive Look at Percentile in Statistics

The Best Guide to Understand Bayes Theorem

Everything you need to know about the normal distribution, an in-depth explanation of cumulative distribution function, a complete guide to chi-square test, a complete guide on hypothesis testing in statistics, understanding the fundamentals of arithmetic and geometric progression, the definitive guide to understand spearman’s rank correlation, a comprehensive guide to understand mean squared error, all you need to know about the empirical rule in statistics, the complete guide to skewness and kurtosis, a holistic look at bernoulli distribution.

All You Need to Know About Bias in Statistics

A Complete Guide to Get a Grasp of Time Series Analysis

The Key Differences Between Z-Test Vs. T-Test

The Complete Guide to Understand Pearson's Correlation

A complete guide on the types of statistical studies, everything you need to know about poisson distribution, your best guide to understand correlation vs. regression, the most comprehensive guide for beginners on what is correlation, what is hypothesis testing in statistics types and examples.

Lesson 10 of 24 By Avijeet Biswal

A Complete Guide on Hypothesis Testing in Statistics

In today’s data-driven world , decisions are based on data all the time. Hypothesis plays a crucial role in that process, whether it may be making business decisions, in the health sector, academia, or in quality improvement. Without hypothesis & hypothesis tests, you risk drawing the wrong conclusions and making bad decisions. In this tutorial, you will look at Hypothesis Testing in Statistics.

What Is Hypothesis Testing in Statistics?

Hypothesis Testing is a type of statistical analysis in which you put your assumptions about a population parameter to the test. It is used to estimate the relationship between 2 statistical variables.

Let's discuss few examples of statistical hypothesis from real-life -

A teacher assumes that 60% of his college's students come from lower-middle-class families.
A doctor believes that 3D (Diet, Dose, and Discipline) is 90% effective for diabetic patients.

Now that you know about hypothesis testing, look at the two types of hypothesis testing in statistics.

Hypothesis Testing Formula

Z = ( x̅ – μ0 ) / (σ /√n)

Here, x̅ is the sample mean,
μ0 is the population mean,
σ is the standard deviation,
n is the sample size.

How Hypothesis Testing Works?

An analyst performs hypothesis testing on a statistical sample to present evidence of the plausibility of the null hypothesis. Measurements and analyses are conducted on a random sample of the population to test a theory. Analysts use a random population sample to test two hypotheses: the null and alternative hypotheses.

The null hypothesis is typically an equality hypothesis between population parameters; for example, a null hypothesis may claim that the population means return equals zero. The alternate hypothesis is essentially the inverse of the null hypothesis (e.g., the population means the return is not equal to zero). As a result, they are mutually exclusive, and only one can be correct. One of the two possibilities, however, will always be correct.

Your Dream Career is Just Around The Corner!

Null Hypothesis and Alternate Hypothesis

The Null Hypothesis is the assumption that the event will not occur. A null hypothesis has no bearing on the study's outcome unless it is rejected.

H0 is the symbol for it, and it is pronounced H-naught.

The Alternate Hypothesis is the logical opposite of the null hypothesis. The acceptance of the alternative hypothesis follows the rejection of the null hypothesis. H1 is the symbol for it.

Let's understand this with an example.

A sanitizer manufacturer claims that its product kills 95 percent of germs on average.

To put this company's claim to the test, create a null and alternate hypothesis.

H0 (Null Hypothesis): Average = 95%.

Alternative Hypothesis (H1): The average is less than 95%.

Another straightforward example to understand this concept is determining whether or not a coin is fair and balanced. The null hypothesis states that the probability of a show of heads is equal to the likelihood of a show of tails. In contrast, the alternate theory states that the probability of a show of heads and tails would be very different.

Become a Data Scientist with Hands-on Training!

Hypothesis Testing Calculation With Examples

Let's consider a hypothesis test for the average height of women in the United States. Suppose our null hypothesis is that the average height is 5'4". We gather a sample of 100 women and determine that their average height is 5'5". The standard deviation of population is 2.

To calculate the z-score, we would use the following formula:

z = ( x̅ – μ0 ) / (σ /√n)

z = (5'5" - 5'4") / (2" / √100)

z = 0.5 / (0.045)

We will reject the null hypothesis as the z-score of 11.11 is very large and conclude that there is evidence to suggest that the average height of women in the US is greater than 5'4".

Steps of Hypothesis Testing

Step 1: specify your null and alternate hypotheses.

It is critical to rephrase your original research hypothesis (the prediction that you wish to study) as a null (Ho) and alternative (Ha) hypothesis so that you can test it quantitatively. Your first hypothesis, which predicts a link between variables, is generally your alternate hypothesis. The null hypothesis predicts no link between the variables of interest.

Step 2: Gather Data

For a statistical test to be legitimate, sampling and data collection must be done in a way that is meant to test your hypothesis. You cannot draw statistical conclusions about the population you are interested in if your data is not representative.

Step 3: Conduct a Statistical Test

Other statistical tests are available, but they all compare within-group variance (how to spread out the data inside a category) against between-group variance (how different the categories are from one another). If the between-group variation is big enough that there is little or no overlap between groups, your statistical test will display a low p-value to represent this. This suggests that the disparities between these groups are unlikely to have occurred by accident. Alternatively, if there is a large within-group variance and a low between-group variance, your statistical test will show a high p-value. Any difference you find across groups is most likely attributable to chance. The variety of variables and the level of measurement of your obtained data will influence your statistical test selection.

Step 4: Determine Rejection Of Your Null Hypothesis

Your statistical test results must determine whether your null hypothesis should be rejected or not. In most circumstances, you will base your judgment on the p-value provided by the statistical test. In most circumstances, your preset level of significance for rejecting the null hypothesis will be 0.05 - that is, when there is less than a 5% likelihood that these data would be seen if the null hypothesis were true. In other circumstances, researchers use a lower level of significance, such as 0.01 (1%). This reduces the possibility of wrongly rejecting the null hypothesis.

Step 5: Present Your Results

The findings of hypothesis testing will be discussed in the results and discussion portions of your research paper, dissertation, or thesis. You should include a concise overview of the data and a summary of the findings of your statistical test in the results section. You can talk about whether your results confirmed your initial hypothesis or not in the conversation. Rejecting or failing to reject the null hypothesis is a formal term used in hypothesis testing. This is likely a must for your statistics assignments.

Types of Hypothesis Testing

To determine whether a discovery or relationship is statistically significant, hypothesis testing uses a z-test. It usually checks to see if two means are the same (the null hypothesis). Only when the population standard deviation is known and the sample size is 30 data points or more, can a z-test be applied.

A statistical test called a t-test is employed to compare the means of two groups. To determine whether two groups differ or if a procedure or treatment affects the population of interest, it is frequently used in hypothesis testing.

Chi-Square

You utilize a Chi-square test for hypothesis testing concerning whether your data is as predicted. To determine if the expected and observed results are well-fitted, the Chi-square test analyzes the differences between categorical variables from a random sample. The test's fundamental premise is that the observed values in your data should be compared to the predicted values that would be present if the null hypothesis were true.

Hypothesis Testing and Confidence Intervals

Both confidence intervals and hypothesis tests are inferential techniques that depend on approximating the sample distribution. Data from a sample is used to estimate a population parameter using confidence intervals. Data from a sample is used in hypothesis testing to examine a given hypothesis. We must have a postulated parameter to conduct hypothesis testing.

Bootstrap distributions and randomization distributions are created using comparable simulation techniques. The observed sample statistic is the focal point of a bootstrap distribution, whereas the null hypothesis value is the focal point of a randomization distribution.

A variety of feasible population parameter estimates are included in confidence ranges. In this lesson, we created just two-tailed confidence intervals. There is a direct connection between these two-tail confidence intervals and these two-tail hypothesis tests. The results of a two-tailed hypothesis test and two-tailed confidence intervals typically provide the same results. In other words, a hypothesis test at the 0.05 level will virtually always fail to reject the null hypothesis if the 95% confidence interval contains the predicted value. A hypothesis test at the 0.05 level will nearly certainly reject the null hypothesis if the 95% confidence interval does not include the hypothesized parameter.

Simple and Composite Hypothesis Testing

Depending on the population distribution, you can classify the statistical hypothesis into two types.

Simple Hypothesis: A simple hypothesis specifies an exact value for the parameter.

Composite Hypothesis: A composite hypothesis specifies a range of values.

A company is claiming that their average sales for this quarter are 1000 units. This is an example of a simple hypothesis.

Suppose the company claims that the sales are in the range of 900 to 1000 units. Then this is a case of a composite hypothesis.

One-Tailed and Two-Tailed Hypothesis Testing

The One-Tailed test, also called a directional test, considers a critical region of data that would result in the null hypothesis being rejected if the test sample falls into it, inevitably meaning the acceptance of the alternate hypothesis.

In a one-tailed test, the critical distribution area is one-sided, meaning the test sample is either greater or lesser than a specific value.

In two tails, the test sample is checked to be greater or less than a range of values in a Two-Tailed test, implying that the critical distribution area is two-sided.

If the sample falls within this range, the alternate hypothesis will be accepted, and the null hypothesis will be rejected.

Become a Data Scientist With Real-World Experience

Right Tailed Hypothesis Testing

If the larger than (>) sign appears in your hypothesis statement, you are using a right-tailed test, also known as an upper test. Or, to put it another way, the disparity is to the right. For instance, you can contrast the battery life before and after a change in production. Your hypothesis statements can be the following if you want to know if the battery life is longer than the original (let's say 90 hours):

The null hypothesis is (H0 <= 90) or less change.
A possibility is that battery life has risen (H1) > 90.

The crucial point in this situation is that the alternate hypothesis (H1), not the null hypothesis, decides whether you get a right-tailed test.

Left Tailed Hypothesis Testing

Alternative hypotheses that assert the true value of a parameter is lower than the null hypothesis are tested with a left-tailed test; they are indicated by the asterisk "<".

Suppose H0: mean = 50 and H1: mean not equal to 50

According to the H1, the mean can be greater than or less than 50. This is an example of a Two-tailed test.

In a similar manner, if H0: mean >=50, then H1: mean <50

Here the mean is less than 50. It is called a One-tailed test.

Type 1 and Type 2 Error

A hypothesis test can result in two types of errors.

Type 1 Error: A Type-I error occurs when sample results reject the null hypothesis despite being true.

Type 2 Error: A Type-II error occurs when the null hypothesis is not rejected when it is false, unlike a Type-I error.

Suppose a teacher evaluates the examination paper to decide whether a student passes or fails.

H0: Student has passed

H1: Student has failed

Type I error will be the teacher failing the student [rejects H0] although the student scored the passing marks [H0 was true].

Type II error will be the case where the teacher passes the student [do not reject H0] although the student did not score the passing marks [H1 is true].

Level of Significance

The alpha value is a criterion for determining whether a test statistic is statistically significant. In a statistical test, Alpha represents an acceptable probability of a Type I error. Because alpha is a probability, it can be anywhere between 0 and 1. In practice, the most commonly used alpha values are 0.01, 0.05, and 0.1, which represent a 1%, 5%, and 10% chance of a Type I error, respectively (i.e. rejecting the null hypothesis when it is in fact correct).

Future-Proof Your AI/ML Career: Top Dos and Don'ts

A p-value is a metric that expresses the likelihood that an observed difference could have occurred by chance. As the p-value decreases the statistical significance of the observed difference increases. If the p-value is too low, you reject the null hypothesis.

Here you have taken an example in which you are trying to test whether the new advertising campaign has increased the product's sales. The p-value is the likelihood that the null hypothesis, which states that there is no change in the sales due to the new advertising campaign, is true. If the p-value is .30, then there is a 30% chance that there is no increase or decrease in the product's sales. If the p-value is 0.03, then there is a 3% probability that there is no increase or decrease in the sales value due to the new advertising campaign. As you can see, the lower the p-value, the chances of the alternate hypothesis being true increases, which means that the new advertising campaign causes an increase or decrease in sales.

Why is Hypothesis Testing Important in Research Methodology?

Hypothesis testing is crucial in research methodology for several reasons:

Provides evidence-based conclusions: It allows researchers to make objective conclusions based on empirical data, providing evidence to support or refute their research hypotheses.
Supports decision-making: It helps make informed decisions, such as accepting or rejecting a new treatment, implementing policy changes, or adopting new practices.
Adds rigor and validity: It adds scientific rigor to research using statistical methods to analyze data, ensuring that conclusions are based on sound statistical evidence.
Contributes to the advancement of knowledge: By testing hypotheses, researchers contribute to the growth of knowledge in their respective fields by confirming existing theories or discovering new patterns and relationships.

Limitations of Hypothesis Testing

Hypothesis testing has some limitations that researchers should be aware of:

It cannot prove or establish the truth: Hypothesis testing provides evidence to support or reject a hypothesis, but it cannot confirm the absolute truth of the research question.
Results are sample-specific: Hypothesis testing is based on analyzing a sample from a population, and the conclusions drawn are specific to that particular sample.
Possible errors: During hypothesis testing, there is a chance of committing type I error (rejecting a true null hypothesis) or type II error (failing to reject a false null hypothesis).
Assumptions and requirements: Different tests have specific assumptions and requirements that must be met to accurately interpret results.

After reading this tutorial, you would have a much better understanding of hypothesis testing, one of the most important concepts in the field of Data Science . The majority of hypotheses are based on speculation about observed behavior, natural phenomena, or established theories.

If you are interested in statistics of data science and skills needed for such a career, you ought to explore Simplilearn’s Post Graduate Program in Data Science.

If you have any questions regarding this ‘Hypothesis Testing In Statistics’ tutorial, do share them in the comment section. Our subject matter expert will respond to your queries. Happy learning!

1. What is hypothesis testing in statistics with example?

Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence. An example: testing if a new drug improves patient recovery (Ha) compared to the standard treatment (H0) based on collected patient data.

2. What is hypothesis testing and its types?

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. It involves formulating two hypotheses: the null hypothesis (H0), which represents the default assumption, and the alternative hypothesis (Ha), which contradicts H0. The goal is to assess the evidence and determine whether there is enough statistical significance to reject the null hypothesis in favor of the alternative hypothesis.

Types of hypothesis testing:

One-sample test: Used to compare a sample to a known value or a hypothesized value.
Two-sample test: Compares two independent samples to assess if there is a significant difference between their means or distributions.
Paired-sample test: Compares two related samples, such as pre-test and post-test data, to evaluate changes within the same subjects over time or under different conditions.
Chi-square test: Used to analyze categorical data and determine if there is a significant association between variables.
ANOVA (Analysis of Variance): Compares means across multiple groups to check if there is a significant difference between them.

3. What are the steps of hypothesis testing?

The steps of hypothesis testing are as follows:

Formulate the hypotheses: State the null hypothesis (H0) and the alternative hypothesis (Ha) based on the research question.
Set the significance level: Determine the acceptable level of error (alpha) for making a decision.
Collect and analyze data: Gather and process the sample data.
Compute test statistic: Calculate the appropriate statistical test to assess the evidence.
Make a decision: Compare the test statistic with critical values or p-values and determine whether to reject H0 in favor of Ha or not.
Draw conclusions: Interpret the results and communicate the findings in the context of the research question.

4. What are the 2 types of hypothesis testing?

One-tailed (or one-sided) test: Tests for the significance of an effect in only one direction, either positive or negative.
Two-tailed (or two-sided) test: Tests for the significance of an effect in both directions, allowing for the possibility of a positive or negative effect.

The choice between one-tailed and two-tailed tests depends on the specific research question and the directionality of the expected effect.

5. What are the 3 major types of hypothesis?

The three major types of hypotheses are:

Null Hypothesis (H0): Represents the default assumption, stating that there is no significant effect or relationship in the data.
Alternative Hypothesis (Ha): Contradicts the null hypothesis and proposes a specific effect or relationship that researchers want to investigate.
Nondirectional Hypothesis: An alternative hypothesis that doesn't specify the direction of the effect, leaving it open for both positive and negative possibilities.

Find our Data Analyst Online Bootcamp in top cities:

About the author.

Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football.

Recommended Resources

Free eBook: Top Programming Languages For A Data Scientist

Normality Test in Minitab: Minitab with Statistics

Machine Learning Career Guide: A Playbook to Becoming a Machine Learning Engineer

PMP, PMI, PMBOK, CAPM, PgMP, PfMP, ACP, PBA, RMP, SP, and OPM3 are registered marks of the Project Management Institute, Inc.

school Campus Bookshelves
menu_book Bookshelves
perm_media Learning Objects
login Login
how_to_reg Request Instructor Account
hub Instructor Commons
Download Page (PDF)
Download Full Book (PDF)
Periodic Table
Physics Constants
Scientific Calculator
Reference & Cite
Tools expand_more
Readability

selected template will load here

This action is not available.

9.1: Introduction to Hypothesis Testing

Last updated
Save as PDF
Page ID 10211

Kyle Siegrist
University of Alabama in Huntsville via Random Services

Basic Theory

Preliminaries.

As usual, our starting point is a random experiment with an underlying sample space and a probability measure $\P$. In the basic statistical model, we have an observable random variable $\bs{X}$ taking values in a set $S$. In general, $\bs{X}$ can have quite a complicated structure. For example, if the experiment is to sample $n$ objects from a population and record various measurements of interest, then \[ \bs{X} = (X_1, X_2, \ldots, X_n) \] where $X_i$ is the vector of measurements for the $i$th object. The most important special case occurs when $(X_1, X_2, \ldots, X_n)$ are independent and identically distributed. In this case, we have a random sample of size $n$ from the common distribution.

The purpose of this section is to define and discuss the basic concepts of statistical hypothesis testing . Collectively, these concepts are sometimes referred to as the Neyman-Pearson framework, in honor of Jerzy Neyman and Egon Pearson, who first formalized them.

A statistical hypothesis is a statement about the distribution of $\bs{X}$. Equivalently, a statistical hypothesis specifies a set of possible distributions of $\bs{X}$: the set of distributions for which the statement is true. A hypothesis that specifies a single distribution for $\bs{X}$ is called simple ; a hypothesis that specifies more than one distribution for $\bs{X}$ is called composite .

In hypothesis testing , the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis . The null hypothesis is usually denoted $H_0$ while the alternative hypothesis is usually denoted $H_1$.

An hypothesis test is a statistical decision ; the conclusion will either be to reject the null hypothesis in favor of the alternative, or to fail to reject the null hypothesis. The decision that we make must, of course, be based on the observed value $\bs{x}$ of the data vector $\bs{X}$. Thus, we will find an appropriate subset $R$ of the sample space $S$ and reject $H_0$ if and only if $\bs{x} \in R$. The set $R$ is known as the rejection region or the critical region . Note the asymmetry between the null and alternative hypotheses. This asymmetry is due to the fact that we assume the null hypothesis, in a sense, and then see if there is sufficient evidence in $\bs{x}$ to overturn this assumption in favor of the alternative.

An hypothesis test is a statistical analogy to proof by contradiction, in a sense. Suppose for a moment that $H_1$ is a statement in a mathematical theory and that $H_0$ is its negation. One way that we can prove $H_1$ is to assume $H_0$ and work our way logically to a contradiction. In an hypothesis test, we don't prove anything of course, but there are similarities. We assume $H_0$ and then see if the data $\bs{x}$ are sufficiently at odds with that assumption that we feel justified in rejecting $H_0$ in favor of $H_1$.

Often, the critical region is defined in terms of a statistic $w(\bs{X})$, known as a test statistic , where $w$ is a function from $S$ into another set $T$. We find an appropriate rejection region $R_T \subseteq T$ and reject $H_0$ when the observed value $w(\bs{x}) \in R_T$. Thus, the rejection region in $S$ is then $R = w^{-1}(R_T) = \left\{\bs{x} \in S: w(\bs{x}) \in R_T\right\}$. As usual, the use of a statistic often allows significant data reduction when the dimension of the test statistic is much smaller than the dimension of the data vector.

The ultimate decision may be correct or may be in error. There are two types of errors, depending on which of the hypotheses is actually true.

Types of errors:

A type 1 error is rejecting the null hypothesis $H_0$ when $H_0$ is true.
A type 2 error is failing to reject the null hypothesis $H_0$ when the alternative hypothesis $H_1$ is true.

Similarly, there are two ways to make a correct decision: we could reject $H_0$ when $H_1$ is true or we could fail to reject $H_0$ when $H_0$ is true. The possibilities are summarized in the following table:

Of course, when we observe $\bs{X} = \bs{x}$ and make our decision, either we will have made the correct decision or we will have committed an error, and usually we will never know which of these events has occurred. Prior to gathering the data, however, we can consider the probabilities of the various errors.

If $H_0$ is true (that is, the distribution of $\bs{X}$ is specified by $H_0$), then $\P(\bs{X} \in R)$ is the probability of a type 1 error for this distribution. If $H_0$ is composite, then $H_0$ specifies a variety of different distributions for $\bs{X}$ and thus there is a set of type 1 error probabilities.

The maximum probability of a type 1 error, over the set of distributions specified by $ H_0 $, is the significance level of the test or the size of the critical region.

The significance level is often denoted by $\alpha$. Usually, the rejection region is constructed so that the significance level is a prescribed, small value (typically 0.1, 0.05, 0.01).

If $H_1$ is true (that is, the distribution of $\bs{X}$ is specified by $H_1$), then $\P(\bs{X} \notin R)$ is the probability of a type 2 error for this distribution. Again, if $H_1$ is composite then $H_1$ specifies a variety of different distributions for $\bs{X}$, and thus there will be a set of type 2 error probabilities. Generally, there is a tradeoff between the type 1 and type 2 error probabilities. If we reduce the probability of a type 1 error, by making the rejection region $R$ smaller, we necessarily increase the probability of a type 2 error because the complementary region $S \setminus R$ is larger.

The extreme cases can give us some insight. First consider the decision rule in which we never reject $H_0$, regardless of the evidence $\bs{x}$. This corresponds to the rejection region $R = \emptyset$. A type 1 error is impossible, so the significance level is 0. On the other hand, the probability of a type 2 error is 1 for any distribution defined by $H_1$. At the other extreme, consider the decision rule in which we always rejects $H_0$ regardless of the evidence $\bs{x}$. This corresponds to the rejection region $R = S$. A type 2 error is impossible, but now the probability of a type 1 error is 1 for any distribution defined by $H_0$. In between these two worthless tests are meaningful tests that take the evidence $\bs{x}$ into account.

If $H_1$ is true, so that the distribution of $\bs{X}$ is specified by $H_1$, then $\P(\bs{X} \in R)$, the probability of rejecting $H_0$ is the power of the test for that distribution.

Thus the power of the test for a distribution specified by $ H_1 $ is the probability of making the correct decision.

Suppose that we have two tests, corresponding to rejection regions $R_1$ and $R_2$, respectively, each having significance level $\alpha$. The test with region $R_1$ is uniformly more powerful than the test with region $R_2$ if \[ \P(\bs{X} \in R_1) \ge \P(\bs{X} \in R_2) \text{ for every distribution of } \bs{X} \text{ specified by } H_1 \]

Naturally, in this case, we would prefer the first test. Often, however, two tests will not be uniformly ordered; one test will be more powerful for some distributions specified by $H_1$ while the other test will be more powerful for other distributions specified by $H_1$.

If a test has significance level $\alpha$ and is uniformly more powerful than any other test with significance level $\alpha$, then the test is said to be a uniformly most powerful test at level $\alpha$.

Clearly a uniformly most powerful test is the best we can do.

$P$-value

In most cases, we have a general procedure that allows us to construct a test (that is, a rejection region $R_\alpha$) for any given significance level $\alpha \in (0, 1)$. Typically, $R_\alpha$ decreases (in the subset sense) as $\alpha$ decreases.

The $P$-value of the observed value $\bs{x}$ of $\bs{X}$, denoted $P(\bs{x})$, is defined to be the smallest $\alpha$ for which $\bs{x} \in R_\alpha$; that is, the smallest significance level for which $H_0$ is rejected, given $\bs{X} = \bs{x}$.

Knowing $P(\bs{x})$ allows us to test $H_0$ at any significance level for the given data $\bs{x}$: If $P(\bs{x}) \le \alpha$ then we would reject $H_0$ at significance level $\alpha$; if $P(\bs{x}) \gt \alpha$ then we fail to reject $H_0$ at significance level $\alpha$. Note that $P(\bs{X})$ is a statistic . Informally, $P(\bs{x})$ can often be thought of as the probability of an outcome as or more extreme than the observed value $\bs{x}$, where extreme is interpreted relative to the null hypothesis $H_0$.

Analogy with Justice Systems

There is a helpful analogy between statistical hypothesis testing and the criminal justice system in the US and various other countries. Consider a person charged with a crime. The presumed null hypothesis is that the person is innocent of the crime; the conjectured alternative hypothesis is that the person is guilty of the crime. The test of the hypotheses is a trial with evidence presented by both sides playing the role of the data. After considering the evidence, the jury delivers the decision as either not guilty or guilty . Note that innocent is not a possible verdict of the jury, because it is not the point of the trial to prove the person innocent. Rather, the point of the trial is to see whether there is sufficient evidence to overturn the null hypothesis that the person is innocent in favor of the alternative hypothesis of that the person is guilty. A type 1 error is convicting a person who is innocent; a type 2 error is acquitting a person who is guilty. Generally, a type 1 error is considered the more serious of the two possible errors, so in an attempt to hold the chance of a type 1 error to a very low level, the standard for conviction in serious criminal cases is beyond a reasonable doubt .

Tests of an Unknown Parameter

Hypothesis testing is a very general concept, but an important special class occurs when the distribution of the data variable $\bs{X}$ depends on a parameter $\theta$ taking values in a parameter space $\Theta$. The parameter may be vector-valued, so that $\bs{\theta} = (\theta_1, \theta_2, \ldots, \theta_n)$ and $\Theta \subseteq \R^k$ for some $k \in \N_+$. The hypotheses generally take the form \[ H_0: \theta \in \Theta_0 \text{ versus } H_1: \theta \notin \Theta_0 \] where $\Theta_0$ is a prescribed subset of the parameter space $\Theta$. In this setting, the probabilities of making an error or a correct decision depend on the true value of $\theta$. If $R$ is the rejection region, then the power function $ Q $ is given by \[ Q(\theta) = \P_\theta(\bs{X} \in R), \quad \theta \in \Theta \] The power function gives a lot of information about the test.

The power function satisfies the following properties:

$Q(\theta)$ is the probability of a type 1 error when $\theta \in \Theta_0$.
$\max\left\{Q(\theta): \theta \in \Theta_0\right\}$ is the significance level of the test.
$1 - Q(\theta)$ is the probability of a type 2 error when $\theta \notin \Theta_0$.
$Q(\theta)$ is the power of the test when $\theta \notin \Theta_0$.

If we have two tests, we can compare them by means of their power functions.

Suppose that we have two tests, corresponding to rejection regions $R_1$ and $R_2$, respectively, each having significance level $\alpha$. The test with rejection region $R_1$ is uniformly more powerful than the test with rejection region $R_2$ if $ Q_1(\theta) \ge Q_2(\theta)$ for all $ \theta \notin \Theta_0 $.

Most hypothesis tests of an unknown real parameter $\theta$ fall into three special cases:

Suppose that $ \theta $ is a real parameter and $ \theta_0 \in \Theta $ a specified value. The tests below are respectively the two-sided test , the left-tailed test , and the right-tailed test .

$H_0: \theta = \theta_0$ versus $H_1: \theta \ne \theta_0$
$H_0: \theta \ge \theta_0$ versus $H_1: \theta \lt \theta_0$
$H_0: \theta \le \theta_0$ versus $H_1: \theta \gt \theta_0$

Thus the tests are named after the conjectured alternative. Of course, there may be other unknown parameters besides $\theta$ (known as nuisance parameters ).

Equivalence Between Hypothesis Test and Confidence Sets

There is an equivalence between hypothesis tests and confidence sets for a parameter $\theta$.

Suppose that $C(\bs{x})$ is a $1 - \alpha$ level confidence set for $\theta$. The following test has significance level $\alpha$ for the hypothesis $ H_0: \theta = \theta_0 $ versus $ H_1: \theta \ne \theta_0 $: Reject $H_0$ if and only if $\theta_0 \notin C(\bs{x})$

By definition, $\P[\theta \in C(\bs{X})] = 1 - \alpha$. Hence if $H_0$ is true so that $\theta = \theta_0$, then the probability of a type 1 error is $P[\theta \notin C(\bs{X})] = \alpha$.

Equivalently, we fail to reject $H_0$ at significance level $\alpha$ if and only if $\theta_0$ is in the corresponding $1 - \alpha$ level confidence set. In particular, this equivalence applies to interval estimates of a real parameter $\theta$ and the common tests for $\theta$ given above .

In each case below, the confidence interval has confidence level $1 - \alpha$ and the test has significance level $\alpha$.

Suppose that $\left[L(\bs{X}, U(\bs{X})\right]$ is a two-sided confidence interval for $\theta$. Reject $H_0: \theta = \theta_0$ versus $H_1: \theta \ne \theta_0$ if and only if $\theta_0 \lt L(\bs{X})$ or $\theta_0 \gt U(\bs{X})$.
Suppose that $L(\bs{X})$ is a confidence lower bound for $\theta$. Reject $H_0: \theta \le \theta_0$ versus $H_1: \theta \gt \theta_0$ if and only if $\theta_0 \lt L(\bs{X})$.
Suppose that $U(\bs{X})$ is a confidence upper bound for $\theta$. Reject $H_0: \theta \ge \theta_0$ versus $H_1: \theta \lt \theta_0$ if and only if $\theta_0 \gt U(\bs{X})$.

Pivot Variables and Test Statistics

Recall that confidence sets of an unknown parameter $\theta$ are often constructed through a pivot variable , that is, a random variable $W(\bs{X}, \theta)$ that depends on the data vector $\bs{X}$ and the parameter $\theta$, but whose distribution does not depend on $\theta$ and is known. In this case, a natural test statistic for the basic tests given above is $W(\bs{X}, \theta_0)$.

Statistics Tutorial

Descriptive statistics, inferential statistics, stat reference, statistics - hypothesis testing.

Hypothesis testing is a formal way of checking if a hypothesis about a population is true or not.

Hypothesis Testing

A hypothesis is a claim about a population parameter .

A hypothesis test is a formal procedure to check if a hypothesis is true or not.

Examples of claims that can be checked:

The average height of people in Denmark is more than 170 cm.

The share of left handed people in Australia is not 10%.

The average income of dentists is less the average income of lawyers.

The Null and Alternative Hypothesis

Hypothesis testing is based on making two different claims about a population parameter.

The null hypothesis ($H_{0} $) and the alternative hypothesis ($H_{1}$) are the claims.

The two claims needs to be mutually exclusive , meaning only one of them can be true.

The alternative hypothesis is typically what we are trying to prove.

For example, we want to check the following claim:

"The average height of people in Denmark is more than 170 cm."

In this case, the parameter is the average height of people in Denmark ($\mu$).

The null and alternative hypothesis would be:

Null hypothesis : The average height of people in Denmark is 170 cm.

Alternative hypothesis : The average height of people in Denmark is more than 170 cm.

The claims are often expressed with symbols like this:

$H_{0}$: $\mu = 170 \: cm $

$H_{1}$: $\mu > 170 \: cm $

If the data supports the alternative hypothesis, we reject the null hypothesis and accept the alternative hypothesis.

If the data does not support the alternative hypothesis, we keep the null hypothesis.

Note: The alternative hypothesis is also referred to as ($H_{A} $).

The Significance Level

The significance level ($\alpha$) is the uncertainty we accept when rejecting the null hypothesis in the hypothesis test.

The significance level is a percentage probability of accidentally making the wrong conclusion.

Typical significance levels are:

$\alpha = 0.1$ (10%)
$\alpha = 0.05$ (5%)
$\alpha = 0.01$ (1%)

A lower significance level means that the evidence in the data needs to be stronger to reject the null hypothesis.

There is no "correct" significance level - it only states the uncertainty of the conclusion.

Note: A 5% significance level means that when we reject a null hypothesis:

We expect to reject a true null hypothesis 5 out of 100 times.

The Test Statistic

The test statistic is used to decide the outcome of the hypothesis test.

The test statistic is a standardized value calculated from the sample.

Standardization means converting a statistic to a well known probability distribution .

The type of probability distribution depends on the type of test.

Common examples are:

Standard Normal Distribution (Z): used for Testing Population Proportions
Student's T-Distribution (T): used for Testing Population Means

Note: You will learn how to calculate the test statistic for each type of test in the following chapters.

The Critical Value and P-Value Approach

There are two main approaches used for hypothesis tests:

The critical value approach compares the test statistic with the critical value of the significance level.
The p-value approach compares the p-value of the test statistic and with the significance level.

The Critical Value Approach

The critical value approach checks if the test statistic is in the rejection region .

The rejection region is an area of probability in the tails of the distribution.

The size of the rejection region is decided by the significance level ($\alpha$).

The value that separates the rejection region from the rest is called the critical value .

Here is a graphical illustration:

If the test statistic is inside this rejection region, the null hypothesis is rejected .

For example, if the test statistic is 2.3 and the critical value is 2 for a significance level ($\alpha = 0.05$):

We reject the null hypothesis ($H_{0} $) at 0.05 significance level ($\alpha$)

The P-Value Approach

The p-value approach checks if the p-value of the test statistic is smaller than the significance level ($\alpha$).

The p-value of the test statistic is the area of probability in the tails of the distribution from the value of the test statistic.

If the p-value is smaller than the significance level, the null hypothesis is rejected .

The p-value directly tells us the lowest significance level where we can reject the null hypothesis.

For example, if the p-value is 0.03:

We reject the null hypothesis ($H_{0} $) at a 0.05 significance level ($\alpha$)

We keep the null hypothesis ($H_{0}$) at a 0.01 significance level ($\alpha$)

Note: The two approaches are only different in how they present the conclusion.

Steps for a Hypothesis Test

The following steps are used for a hypothesis test:

Check the conditions
Define the claims
Decide the significance level
Calculate the test statistic

One condition is that the sample is randomly selected from the population.

The other conditions depends on what type of parameter you are testing the hypothesis for.

Common parameters to test hypotheses are:

Proportions (for qualitative data)
Mean values (for numerical data)

You will learn the steps for both types in the following pages.

COLOR PICKER

Contact Sales

If you want to use W3Schools services as an educational institution, team or enterprise, send us an e-mail: [email protected]

Report Error

If you want to report an error, or if you want to make a suggestion, send us an e-mail: [email protected]

Hypothesis to Be Tested: Definition and 4 Steps for Testing with Example

What Is Hypothesis Testing?

Hypothesis testing, sometimes called significance testing, is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis.

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Such data may come from a larger population, or from a data-generating process. The word "population" will be used for both of these cases in the following descriptions.

Key Takeaways

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data.
The test provides evidence concerning the plausibility of the hypothesis, given the data.
Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed.
The four steps of hypothesis testing include stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.

How Hypothesis Testing Works

In hypothesis testing, an analyst tests a statistical sample, with the goal of providing evidence on the plausibility of the null hypothesis.

Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed. All analysts use a random population sample to test two different hypotheses: the null hypothesis and the alternative hypothesis.

The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a null hypothesis may state that the population mean return is equal to zero. The alternative hypothesis is effectively the opposite of a null hypothesis (e.g., the population mean return is not equal to zero). Thus, they are mutually exclusive , and only one can be true. However, one of the two hypotheses will always be true.

The null hypothesis is a statement about a population parameter, such as the population mean, that is assumed to be true.

4 Steps of Hypothesis Testing

All hypotheses are tested using a four-step process:

The first step is for the analyst to state the hypotheses.
The second step is to formulate an analysis plan, which outlines how the data will be evaluated.
The third step is to carry out the plan and analyze the sample data.
The final step is to analyze the results and either reject the null hypothesis, or state that the null hypothesis is plausible, given the data.

Real-World Example of Hypothesis Testing

If, for example, a person wants to test that a penny has exactly a 50% chance of landing on heads, the null hypothesis would be that 50% is correct, and the alternative hypothesis would be that 50% is not correct.

Mathematically, the null hypothesis would be represented as Ho: P = 0.5. The alternative hypothesis would be denoted as "Ha" and be identical to the null hypothesis, except with the equal sign struck-through, meaning that it does not equal 50%.

A random sample of 100 coin flips is taken, and the null hypothesis is then tested. If it is found that the 100 coin flips were distributed as 40 heads and 60 tails, the analyst would assume that a penny does not have a 50% chance of landing on heads and would reject the null hypothesis and accept the alternative hypothesis.

If, on the other hand, there were 48 heads and 52 tails, then it is plausible that the coin could be fair and still produce such a result. In cases such as this where the null hypothesis is "accepted," the analyst states that the difference between the expected results (50 heads and 50 tails) and the observed results (48 heads and 52 tails) is "explainable by chance alone."

Some staticians attribute the first hypothesis tests to satirical writer John Arbuthnot in 1710, who studied male and female births in England after observing that in nearly every year, male births exceeded female births by a slight proportion. Arbuthnot calculated that the probability of this happening by chance was small, and therefore it was due to “divine providence.”

What is Hypothesis Testing?

Hypothesis testing refers to a process used by analysts to assess the plausibility of a hypothesis by using sample data. In hypothesis testing, statisticians formulate two hypotheses: the null hypothesis and the alternative hypothesis. A null hypothesis determines there is no difference between two groups or conditions, while the alternative hypothesis determines that there is a difference. Researchers evaluate the statistical significance of the test based on the probability that the null hypothesis is true.

What are the Four Key Steps Involved in Hypothesis Testing?

Hypothesis testing begins with an analyst stating two hypotheses, with only one that can be right. The analyst then formulates an analysis plan, which outlines how the data will be evaluated. Next, they move to the testing phase and analyze the sample data. Finally, the analyst analyzes the results and either rejects the null hypothesis or states that the null hypothesis is plausible, given the data.

What are the Benefits of Hypothesis Testing?

Hypothesis testing helps assess the accuracy of new ideas or theories by testing them against data. This allows researchers to determine whether the evidence supports their hypothesis, helping to avoid false claims and conclusions. Hypothesis testing also provides a framework for decision-making based on data rather than personal opinions or biases. By relying on statistical analysis, hypothesis testing helps to reduce the effects of chance and confounding variables, providing a robust framework for making informed conclusions.

What are the Limitations of Hypothesis Testing?

Hypothesis testing relies exclusively on data and doesn’t provide a comprehensive understanding of the subject being studied. Additionally, the accuracy of the results depends on the quality of the available data and the statistical methods used. Inaccurate data or inappropriate hypothesis formulation may lead to incorrect conclusions or failed tests. Hypothesis testing can also lead to errors, such as analysts either accepting or rejecting a null hypothesis when they shouldn’t have. These errors may result in false conclusions or missed opportunities to identify significant patterns or relationships in the data.

The Bottom Line

Hypothesis testing refers to a statistical process that helps researchers and/or analysts determine the reliability of a study. By using a well-formulated hypothesis and set of statistical tests, individuals or businesses can make inferences about the population that they are studying and draw conclusions based on the data presented. There are different types of hypothesis testing, each with their own set of rules and procedures. However, all hypothesis testing methods have the same four step process, which includes stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result. Hypothesis testing plays a vital part of the scientific process, helping to test assumptions and make better data-based decisions.

Sage. " Introduction to Hypothesis Testing. " Page 4.

Elder Research. " Who Invented the Null Hypothesis? "

Formplus. " Hypothesis Testing: Definition, Uses, Limitations and Examples. "

Terms of Service
Editorial Policy
Privacy Policy
Your Privacy Choices

Machine Learning Tutorial
Data Analysis Tutorial
Python - Data visualization tutorial
Machine Learning Projects
Machine Learning Interview Questions
Machine Learning Mathematics
Deep Learning Tutorial
Deep Learning Project
Deep Learning Interview Questions
Computer Vision Tutorial
Computer Vision Projects
NLP Project
NLP Interview Questions
Statistics with Python
100 Days of Machine Learning
Data Analysis with Python

Introduction to Data Analysis

What is Data Analysis?
Data Analytics and its type
How to Install Numpy on Windows?
How to Install Pandas in Python?
How to Install Matplotlib on python?
How to Install Python Tensorflow in Windows?

Data Analysis Libraries

Pandas Tutorial
NumPy Tutorial - Python Library
Data Analysis with SciPy
Introduction to TensorFlow

Data Visulization Libraries

Matplotlib Tutorial
Python Seaborn Tutorial
Plotly tutorial
Introduction to Bokeh in Python

Exploratory Data Analysis (EDA)

Univariate, Bivariate and Multivariate data and its analysis
Measures of Central Tendency in Statistics
Measures of spread - Range, Variance, and Standard Deviation
Interquartile Range and Quartile Deviation using NumPy and SciPy
Anova Formula
Skewness of Statistical Data
How to Calculate Skewness and Kurtosis in Python?
Difference Between Skewness and Kurtosis
Histogram | Meaning, Example, Types and Steps to Draw
Interpretations of Histogram
Quantile Quantile plots
What is Univariate, Bivariate & Multivariate Analysis in Data Visualisation?
Using pandas crosstab to create a bar plot
Exploring Correlation in Python
Mathematics | Covariance and Correlation
Factor Analysis | Data Analysis
Data Mining - Cluster Analysis
MANOVA Test in R Programming
Python - Central Limit Theorem
Probability Distribution Function
Probability Density Estimation & Maximum Likelihood Estimation
Exponential Distribution in R Programming - dexp(), pexp(), qexp(), and rexp() Functions
Mathematics | Probability Distributions Set 4 (Binomial Distribution)
Poisson Distribution - Definition, Formula, Table and Examples
P-Value: Comprehensive Guide to Understand, Apply, and Interpret
Z-Score in Statistics
How to Calculate Point Estimates in R?
Confidence Interval
Chi-square test in Machine Learning

Understanding Hypothesis Testing

Data preprocessing.

ML | Data Preprocessing in Python
ML | Overview of Data Cleaning
ML | Handling Missing Values
Detect and Remove the Outliers using Python

Data Transformation

Data Normalization Machine Learning
Sampling distribution Using Python

Time Series Data Analysis

Data Mining - Time-Series, Symbolic and Biological Sequences Data
Basic DateTime Operations in Python
Time Series Analysis & Visualization in Python
How to deal with missing values in a Timeseries in Python?
How to calculate MOVING AVERAGE in a Pandas DataFrame?
What is a trend in time series?
How to Perform an Augmented Dickey-Fuller Test in R
AutoCorrelation

Case Studies and Projects

Top 8 Free Dataset Sources to Use for Data Science Projects
Step by Step Predictive Analysis - Machine Learning
6 Tips for Creating Effective Data Visualizations

Hypothesis testing involves formulating assumptions about population parameters based on sample statistics and rigorously evaluating these assumptions against empirical evidence. This article sheds light on the significance of hypothesis testing and the critical steps involved in the process.

What is Hypothesis Testing?

Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data.

Example: You say an average height in the class is 30 or a boy is taller than a girl. All of these is an assumption that we are assuming, and we need some statistical way to prove these. We need some mathematical conclusion whatever we are assuming is true.

Defining Hypotheses

$\mu$

Key Terms of Hypothesis Testing

$\alpha$

P-value: The P value , or calculated probability, is the probability of finding the observed/extreme results when the null hypothesis(H0) of a study-given problem is true. If your P-value is less than the chosen significance level then you reject the null hypothesis i.e. accept that your sample claims to support the alternative hypothesis.
Test Statistic: The test statistic is a numerical value calculated from sample data during a hypothesis test, used to determine whether to reject the null hypothesis. It is compared to a critical value or p-value to make decisions about the statistical significance of the observed results.
Critical value : The critical value in statistics is a threshold or cutoff point used to determine whether to reject the null hypothesis in a hypothesis test.
Degrees of freedom: Degrees of freedom are associated with the variability or freedom one has in estimating a parameter. The degrees of freedom are related to the sample size and determine the shape.

Why do we use Hypothesis Testing?

Hypothesis testing is an important procedure in statistics. Hypothesis testing evaluates two mutually exclusive population statements to determine which statement is most supported by sample data. When we say that the findings are statistically significant, thanks to hypothesis testing.

One-Tailed and Two-Tailed Test

One tailed test focuses on one direction, either greater than or less than a specified value. We use a one-tailed test when there is a clear directional expectation based on prior knowledge or theory. The critical region is located on only one side of the distribution curve. If the sample falls into this critical region, the null hypothesis is rejected in favor of the alternative hypothesis.

One-Tailed Test

There are two types of one-tailed test:

$\mu \geq 50$

Two-Tailed Test

A two-tailed test considers both directions, greater than and less than a specified value.We use a two-tailed test when there is no specific directional expectation, and want to detect any significant difference.

$\mu =$

What are Type 1 and Type 2 errors in Hypothesis Testing?

In hypothesis testing, Type I and Type II errors are two possible errors that researchers can make when drawing conclusions about a population based on a sample of data. These errors are associated with the decisions made regarding the null hypothesis and the alternative hypothesis.

$\alpha$

How does Hypothesis Testing work?

Step 1: define null and alternative hypothesis.

We first identify the problem about which we want to make an assumption keeping in mind that our assumption should be contradictory to one another, assuming Normally distributed data.

Step 2 – Choose significance level

$\alpha$

Step 3 – Collect and Analyze data.

Gather relevant data through observation or experimentation. Analyze the data using appropriate statistical methods to obtain a test statistic.

Step 4-Calculate Test Statistic

The data for the tests are evaluated in this step we look for various scores based on the characteristics of data. The choice of the test statistic depends on the type of hypothesis test being conducted.

There are various hypothesis tests, each appropriate for various goal to calculate our test. This could be a Z-test , Chi-square , T-test , and so on.

Z-test : If population means and standard deviations are known. Z-statistic is commonly used.
t-test : If population standard deviations are unknown. and sample size is small than t-test statistic is more appropriate.
Chi-square test : Chi-square test is used for categorical data or for testing independence in contingency tables
F-test : F-test is often used in analysis of variance (ANOVA) to compare variances or test the equality of means across multiple groups.

We have a smaller dataset, So, T-test is more appropriate to test our hypothesis.

T-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.

Step 5 – Comparing Test Statistic:

In this stage, we decide where we should accept the null hypothesis or reject the null hypothesis. There are two ways to decide where we should accept or reject the null hypothesis.

Method A: Using Crtical values

Comparing the test statistic and tabulated critical value we have,

If Test Statistic>Critical Value: Reject the null hypothesis.
If Test Statistic≤Critical Value: Fail to reject the null hypothesis.

Note: Critical values are predetermined threshold values that are used to make a decision in hypothesis testing. To determine critical values for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Method B: Using P-values

We can also come to an conclusion using the p-value,

$p\leq\alpha$

Note : The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed in the sample, assuming the null hypothesis is true. To determine p-value for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Step 7- Interpret the Results

At last, we can conclude our experiment using method A or B.

Calculating test statistic

To validate our hypothesis about a population parameter we use statistical functions . We use the z-score, p-value, and level of significance(alpha) to make evidence for our hypothesis for normally distributed data .

1. Z-statistics:

When population means and standard deviations are known.

$z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$

μ represents the population mean,
σ is the standard deviation
and n is the size of the sample.

2. T-Statistics

T test is used when n<30,

t-statistic calculation is given by:

$t=\frac{x̄-μ}{s/\sqrt{n}}$

t = t-score,
x̄ = sample mean
μ = population mean,
s = standard deviation of the sample,
n = sample size

3. Chi-Square Test

Chi-Square Test for Independence categorical Data (Non-normally distributed) using:

$\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}$

i,j are the rows and columns index respectively.

$E_{ij}$

Real life Hypothesis Testing example

Let’s examine hypothesis testing using two real life situations,

Case A: D oes a New Drug Affect Blood Pressure?

Imagine a pharmaceutical company has developed a new drug that they believe can effectively lower blood pressure in patients with hypertension. Before bringing the drug to market, they need to conduct a study to assess its impact on blood pressure.

Before Treatment: 120, 122, 118, 130, 125, 128, 115, 121, 123, 119
After Treatment: 115, 120, 112, 128, 122, 125, 110, 117, 119, 114

Step 1 : Define the Hypothesis

Null Hypothesis : (H 0 )The new drug has no effect on blood pressure.
Alternate Hypothesis : (H 1 )The new drug has an effect on blood pressure.

Step 2: Define the Significance level

Let’s consider the Significance level at 0.05, indicating rejection of the null hypothesis.

If the evidence suggests less than a 5% chance of observing the results due to random variation.

Step 3 : Compute the test statistic

Using paired T-test analyze the data to obtain a test statistic and a p-value.

The test statistic (e.g., T-statistic) is calculated based on the differences between blood pressure measurements before and after treatment.

t = m/(s/√n)

m = mean of the difference i.e X after, X before
s = standard deviation of the difference (d) i.e d i = X after, i − X before,
n = sample size,

then, m= -3.9, s= 1.8 and n= 10

we, calculate the , T-statistic = -9 based on the formula for paired t test

Step 4: Find the p-value

The calculated t-statistic is -9 and degrees of freedom df = 9, you can find the p-value using statistical software or a t-distribution table.

thus, p-value = 8.538051223166285e-06

Step 5: Result

If the p-value is less than or equal to 0.05, the researchers reject the null hypothesis.
If the p-value is greater than 0.05, they fail to reject the null hypothesis.

Conclusion: Since the p-value (8.538051223166285e-06) is less than the significance level (0.05), the researchers reject the null hypothesis. There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.

Python Implementation of Hypothesis Testing

Let’s create hypothesis testing with python, where we are testing whether a new drug affects blood pressure. For this example, we will use a paired T-test. We’ll use the scipy.stats library for the T-test.

Scipy is a mathematical library in Python that is mostly used for mathematical equations and computations.

We will implement our first real life problem via python,

In the above example, given the T-statistic of approximately -9 and an extremely small p-value, the results indicate a strong case to reject the null hypothesis at a significance level of 0.05.

The results suggest that the new drug, treatment, or intervention has a significant effect on lowering blood pressure.
The negative T-statistic indicates that the mean blood pressure after treatment is significantly lower than the assumed population mean before treatment.

Case B : Cholesterol level in a population

Data: A sample of 25 individuals is taken, and their cholesterol levels are measured.

Cholesterol Levels (mg/dL): 205, 198, 210, 190, 215, 205, 200, 192, 198, 205, 198, 202, 208, 200, 205, 198, 205, 210, 192, 205, 198, 205, 210, 192, 205.

Populations Mean = 200

Population Standard Deviation (σ): 5 mg/dL(given for this problem)

Step 1: Define the Hypothesis

Null Hypothesis (H 0 ): The average cholesterol level in a population is 200 mg/dL.
Alternate Hypothesis (H 1 ): The average cholesterol level in a population is different from 200 mg/dL.

As the direction of deviation is not given , we assume a two-tailed test, and based on a normal distribution table, the critical values for a significance level of 0.05 (two-tailed) can be calculated through the z-table and are approximately -1.96 and 1.96.

$(203.8 - 200) / (5 \div \sqrt{25})$

Step 4: Result

Since the absolute value of the test statistic (2.04) is greater than the critical value (1.96), we reject the null hypothesis. And conclude that, there is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL

Limitations of Hypothesis Testing

Although a useful technique, hypothesis testing does not offer a comprehensive grasp of the topic being studied. Without fully reflecting the intricacy or whole context of the phenomena, it concentrates on certain hypotheses and statistical significance.
The accuracy of hypothesis testing results is contingent on the quality of available data and the appropriateness of statistical methods used. Inaccurate data or poorly formulated hypotheses can lead to incorrect conclusions.
Relying solely on hypothesis testing may cause analysts to overlook significant patterns or relationships in the data that are not captured by the specific hypotheses being tested. This limitation underscores the importance of complimenting hypothesis testing with other analytical approaches.

Hypothesis testing stands as a cornerstone in statistical analysis, enabling data scientists to navigate uncertainties and draw credible inferences from sample data. By systematically defining null and alternative hypotheses, choosing significance levels, and leveraging statistical tests, researchers can assess the validity of their assumptions. The article also elucidates the critical distinction between Type I and Type II errors, providing a comprehensive understanding of the nuanced decision-making process inherent in hypothesis testing. The real-life example of testing a new drug’s effect on blood pressure using a paired T-test showcases the practical application of these principles, underscoring the importance of statistical rigor in data-driven decision-making.

Frequently Asked Questions (FAQs)

1. what are the 3 types of hypothesis test.

There are three types of hypothesis tests: right-tailed, left-tailed, and two-tailed. Right-tailed tests assess if a parameter is greater, left-tailed if lesser. Two-tailed tests check for non-directional differences, greater or lesser.

2.What are the 4 components of hypothesis testing?

Null Hypothesis ( ): No effect or difference exists. Alternative Hypothesis ( ): An effect or difference exists. Significance Level ( ): Risk of rejecting null hypothesis when it’s true (Type I error). Test Statistic: Numerical value representing observed evidence against null hypothesis.

3.What is hypothesis testing in ML?

Statistical method to evaluate the performance and validity of machine learning models. Tests specific hypotheses about model behavior, like whether features influence predictions or if a model generalizes well to unseen data.

4.What is the difference between Pytest and hypothesis in Python?

Pytest purposes general testing framework for Python code while Hypothesis is a Property-based testing framework for Python, focusing on generating test cases based on specified properties of the code.

Please Login to comment...

Improve your Coding Skills with Practice

What kind of Experience do you want to share?

Create an account

Create a free IEA account to download our reports or subcribe to a paid service.

Executive summary

Electric car sales
Electric car availability and affordability
Electric two- and three-wheelers
Electric light commercial vehicles
Electric truck and bus sales
Electric heavy-duty vehicle model availability
Charging for electric light-duty vehicles
Charging for electric heavy-duty vehicles
Battery supply and demand
Battery prices
Electric vehicle company strategy and market competition
Electric vehicle and battery start-ups
Vehicle outlook by mode
Vehicle outlook by region
The industry outlook
Light-duty vehicle charging
Heavy-duty vehicle charging
Battery demand
Electricity demand
Oil displacement
Well-to-wheel greenhouse gas emissions
Lifecycle impacts of electric cars

Cite report

IEA (2024), Global EV Outlook 2024 , IEA, Paris https://www.iea.org/reports/global-ev-outlook-2024, Licence: CC BY 4.0

Share this report

Share on Twitter Twitter
Share on Facebook Facebook
Share on LinkedIn LinkedIn
Share on Email Email
Share on Print Print

Report options

Growth in electric car sales remains robust as major markets progress and emerging economies ramp up.

Electric car sales keep rising and could reach around 17 million in 2024, accounting for more than one in five cars sold worldwide. Electric cars continue to make progress towards becoming a mass-market product in a larger number of countries. Tight margins, volatile battery metal prices, high inflation, and the phase-out of purchase incentives in some countries have sparked concerns about the industry’s pace of growth, but global sales data remain strong. In the first quarter of 2024, electric car sales grew by around 25% compared with the first quarter of 2023, similar to the year-on-year growth seen in the same period in 2022. In 2024, the market share of electric cars could reach up to 45% in China, 25% in Europe and over 11% in the United States, underpinned by competition among manufacturers, falling battery and car prices, and ongoing policy support.

Quarterly electric car sales by region, 2021-2024

Growth expectations for 2024 build on a record year: in 2023, global sales of electric cars neared 14 million, reaching 18% of all cars sold. This is up from 14% in 2022. Electric car sales in 2023 were 3.5 million higher than in 2022, a 35% year-on-year increase. This indicates robust growth even as many major markets enter a new phase, with uptake shifting from early adopters to the mass market. Over 250 000 electric cars were sold every week last year, more than the number sold in a year just a decade ago. Chinese carmakers produced more than half of all electric cars sold worldwide in 2023, despite accounting for just 10% of global sales of cars with internal combustion engines.

The pace at which electric car sales pick up in emerging and developing economies outside China will determine their global success. The vast majority of electric car sales in 2023 were in China (60%), Europe (25%) and the United States (10%). By comparison, these regions accounted for around 65% of total car sales worldwide, showing that sales of electric models remain more geographically concentrated than those of conventional ones. While electric car sales in emerging economies have been lagging those in the three big markets, growth picked up in 2023 in countries such as Viet Nam (around 15% of all cars sold) and Thailand (10%). In emerging economies with large car markets, shares are still relatively low, but several factors point to further growth. Policy measures such as purchase subsidies and incentives for electric vehicle (EV) and battery manufacturing are playing a key role. In India (where electric cars have a 2% market share), the Production Linked Incentives (PLI) Scheme is supporting domestic manufacturing. In Brazil (3% share), Indonesia, Malaysia (2% share each), and Thailand, cheaper models, mainly from Chinese brands, are underpinning uptake. In Mexico, EV supply chains are rapidly developing, stimulated by access to subsidies from the US Inflation Reduction Act (IRA).

Policy support is boosting industry investment, building confidence that rapid electrification will continue

Every other car sold globally in 2035 is set to be electric based on today’s energy, climate and industrial policy settings , as reflected in the IEA’s Stated Policies Scenario. This has significant impacts on the car fleet. As soon as 2030, almost one in three cars on the roads in China is electric in this scenario, and almost one in five in both the United States and European Union. The rapid uptake of EVs of all types – cars, vans, trucks, buses and two/three-wheelers – avoids 6 million barrels per day (mb/d) of oil demand in the Stated Policies Scenario in 2030, and over 10 mb/d in 2035. This is equivalent to the amount of oil used for road transport in the United States today. Recent policy developments continue to reinforce expectations for swift electrification, such as new emissions standards adopted in Canada, the European Union and the United States over the past year. Industrial incentives – such as those in the US IRA, the EU Net Zero Industry Act, China’s 14th Five-Year Plan, and India’s PLI scheme – also encourage adding value and creating jobs across EV supply chains in those economies. If all the national energy and climate targets made by governments are met in full and on time, as in the Announced Pledges Scenario, two-thirds of all vehicles sold in 2035 could be electric, avoiding around 12 mb/d of oil.

Expectations of strong growth are bolstering investment in the EV supply chain. Recent reporting shows that from 2022 to 2023, investment announcements in EV and battery manufacturing totalled almost USD 500 billion, of which around 40% has been committed. Over 20 major car manufacturers, representing more than 90% of global car sales in 2023, have set electrification targets. Taking the targets of all the largest automakers together, more than 40 million electric cars could be sold in 2030, which would meet the level of deployment projected under today’s policy settings.

Enough battery manufacturing capacity has reached a final investment decision to deliver on announced pledges from automakers and governments globally. Thanks to high levels of investment in the past 5 years, global EV battery manufacturing capacity far exceeded demand in 2023, at around 2.2 terawatt-hours and 750 gigawatt-hours, respectively. Demand is likely to grow quickly: up seven times by 2035 compared with 2023 in the Stated Policies Scenario, nine times in the Announced Pledges Scenario, and 12 times in the Net Zero Emissions by 2050 Scenario, which lays out a pathway to reach net zero energy sector emissions by mid-century. Manufacturing capacity appears capable of keeping pace with demand: committed and existing battery manufacturing capacity alone are practically aligned with the needs in a net zero pathway in 2030. Such prospects are opening significant opportunities across the supply chain for battery and mining companies, including in emerging markets outside China, although surplus capacity has been hurting margins and may lead to further market consolidation.

The pace of the transition to electric vehicles hinges on their affordability

Electric cars are getting cheaper as competition intensifies, particularly in China, but they remain more expensive than cars with internal combustion engines in other markets. A rapid transition to EVs will require bringing to market more affordable models. In China, we estimate that more than 60% of electric cars sold in 2023 were already cheaper than their average combustion engine equivalent. However, electric cars remain 10% to 50% more expensive than combustion engine equivalents in Europe and the United States, depending on the country and car segment. In 2023, two-thirds of available electric models globally were large cars, pick-up trucks or sports utility vehicles, pushing up average prices. When exactly price parity is reached is subject to a range of market variables, but current trends suggest that it could be reached by 2030 in major EV markets outside China for most models.

Price gap between the sales-weighted average price of conventional and electric cars in selected countries, before subsidy, by size, in 2018 and 2022

The pricing strategies of car manufacturers will be crucial for improving affordability, as will the pace of EV battery price decline. Turmoil in battery metal markets in 2022 led to the first price increase for lithium-ion packs, which became 7% more expensive than in 2021. In 2023, however, the prices of the key metals used to make batteries dropped, leading to a near-14% fall in pack prices year-on-year. China still supplies the cheapest batteries, but prices across regions are converging as batteries become a globalised commodity. Lithium-iron-phosphate batteries – which are significantly cheaper than those based on lithium, nickel, manganese and cobalt oxide – accounted for over 40% of global EV sales by capacity in 2023, more than double their share in 2020. Looking ahead, technological innovation will remain important for scaling up novel designs and chemistries such as sodium-ion batteries, which could cost as much as 20% less than lithium-based batteries without requiring any lithium.

In developing economies outside China, more affordable electric car models are arriving, and the future of electric two- and three-wheelers already looks bright. In 2023, 55% to 95% of the electric car sales across major emerging and developing economies were large models that are unaffordable for the average consumer, hindering mass-market uptake. However, smaller and much more affordable models launched in 2022 and 2023 have quickly become bestsellers, especially those by Chinese carmakers expanding overseas. Affordable electric two- and three-wheelers are also already available, helping deliver immediate benefits such as improved air quality and emissions reductions. Around 1.3 million electric two-wheelers were sold in India and Southeast Asia in 2023, accounting for 5% and 3% of total sales, respectively. One in five three-wheelers sold globally in 2023 was electric, and nearly 60% of those sold in India, boosted by the Faster Adoption and Manufacturing of Electric Vehicles (FAME II) subsidy scheme.

As electric vehicle markets mature, second-hand electric cars will become more widely available . In 2023, the market size for used electric cars was around 800 000 in China, 400 000 in the United States, and over 450 000 across France, Germany, Italy, Spain, the Netherlands and the United Kingdom. The prices of used electric cars are falling quickly and becoming competitive with combustion engine equivalents. Looking ahead, international trade of used electric cars is also expected to increase, including to emerging and developing economies outside of China.

The battery recycling industry is getting ready for the 2030s. Recycling and reuse are needed for supply chain sustainability and security. Many technology developers are seeking to position themselves in EV end-of-life markets, but planned locations do not always align with where EV retirement may occur. Global battery recycling capacity reached 300 gigawatt-hours in 2023. If all announced projects materialise, it could exceed 1 500 gigawatt-hours in 2030, of which 70% would be in China. Globally, announced recycling capacity is more than three times the supply of batteries that could potentially be recycled in 2030, as EVs reach their end of life in the Announced Pledges Scenario. However, EV battery retirement is expected to grow rapidly from the second half of the 2030s.

The roll-out of public charging needs to keep pace with EV sales

The global number of installed public charging points was up 40% in 2023 relative to 2022, and growth for fast chargers outpaced that of slower ones . In major EV markets, the deployment of charging points is continuing apace thanks to targeted policies. Broad, affordable access to public charging infrastructure will be needed for a mass-market switch to electric transport and to enable longer journeys – even if most charging continues to take place privately in residential and workplace settings. To reach EV deployment levels in the Announced Policies Scenario, public charging needs to increase sixfold by 2035.

As more electric heavy-duty vehicles such as trucks and large buses hit the road, dedicated and flexible charging is needed. In 2023, electric buses accounted for 3% of total bus sales. Electric truck sales jumped 35% compared with 2022, accounting for about 3% of truck sales in China and 1.5% in Europe. Under today’s policy settings, the stock of electric buses increases sevenfold by 2035 and that of electric trucks around thirtyfold, supported by tougher emissions standards in the United States and European Union. This level of deployment could require a twentyfold jump in charging capacity by 2035 – not only in depots, but also along main transit routes to enable long-distance trucking. Increasing heavy-duty charging has important implications for expanding and operating electrical grids, with opportunities for greater flexibility and renewables integration. Policy support, careful planning and co-ordination will be essential to ensure a secure, affordable and low-emissions supply of electricity with limited strain on local grids.

Subscription successful

Thank you for subscribing. You can unsubscribe at any time by clicking the link at the bottom of any IEA newsletter.

National Statistical

News and insight from the office for national statistics, understanding the new income statistics for local areas: different measures for better insights.

Ainslie Woods
April 30, 2024

The ONS has been busy producing faster, more detailed statistics using new data sources. Providing clearer insights at the local level is an important aspect of this work. Ainslie Woods explains how our local income statistics compare and why they offer a complementary picture of local areas.

At the ONS we have continued to deliver our ambitious project to transform our sub-national statistics and produce more timely, granular and harmonised figures. In March we published gross disposable household income (GDHI) statistics broken down to very local areas for the first time, another major success for the project.

Today we have published an analysis article comparing our admin-based local income statistics to our official survey-based statistics .

But why are there three different local income statistics, and how do they relate to each other?

To help users understand the differences between them, and the wider income landscape, we have published updated coherence and comparison guidance , which is available as a related download alongside our existing Income and earning statistics guide .

The key differences between our local income statistics include the frameworks used to produce them, whether we’re measuring household, individual or household sector income and their geographic coverage.

The frameworks for income statistics

Income statistics can be produced according to different frameworks. A framework is essentially a set of internationally agreed standards used to produce statistics. The frameworks for income statistics are either a “micro-framework” or a “macro-framework”. Statistics using the micro-framework are collected for individual people or households and aggregated to the area of interest. These statistics are often very rich as they can contain detailed information about participants. Both the survey-based and admin-based local income statistics are produced according to this framework.

By contrast, the macro-framework is used to compile measures of economic activity. Statistics using the macro-framework are produced according to the internationally agreed System of National Accounts . These are compiled by the ONS at the UK level using a multitude of different sources, including household and business surveys, as well as administrative records. This includes our GDHI figures.

Individual and household income, or income of the households sector?

Micro-framework income statistics can either be published for individuals or households (and sometimes occupied addresses). Those living in residential institutions, such as prisons and care homes, are not included. The survey-based measure uses household income, whereas the admin-based measure uses individual and occupied address income.

On the other hand, GDHI figures (which are produced according to the macro-framework) measure the total income of the households sector. This is a concept within the System of National Accounts, which measures the total income of all people living in the UK, including those in residential institutions.

Geographic coverage

Statistics can be produced at many different levels of geography, covering different countries of the UK. There are specific small area geographies that are often used for reporting official statistics. Our admin-based local income statistics and GDHI statistics are available at a ‘statistical building block’ level (called a Lower layer Super Output Area), and many other higher aggregates.

However, our survey-based local income statistics are only available for a slightly larger small area geography (called a Middle layer Super Output Area). More information can be found on our Statistical geographies webpage.

There are several other differences between the local income statistics which are described further in the Coherence and comparison guidance for income statistics .

Conclusion

There are rich sets of income statistics for local areas, which we are continuing to develop and improve to meet user needs. Explaining the coherence and comparability of these statistics is fundamental to ensuring that they continue to serve the public good.

If you would like to get in touch, or have any suggestions for how we can further improve the coherence, or explain the nuances and differences, of income statistics, please email us at [email protected]

Ainslie Woods is the income and earnings coherence lead at the Office for National Statistics

Share this post

FBI data shows America is seeing a 'considerable' drop in crime. Trump says the opposite.

Jeff Asher is a New Orleans-based crime data analyst who has worked at the CIA and Department of Defense. He leans towards caution when describing trends in his line of work.

Amid the heated crime rhetoric that is a staple of politics and is continuing this year – former President Donald Trump and his conservative allies in Congress and the media are using dire terms to describe crime trends in America – Asher has been carefully sifting through the data.

The story he tells has been slow to emerge but stands in stark contrast to Trump's narrative.

As early data showed murders declining nationwide last year, Asher was careful about overstating things. But as the big decline continued, he wrote in December that he had “seen enough” and was ready to declare that the U.S. was experiencing a major drop in killings.

“Murder plummeted in the United States in 2023, likely at one of the fastest rates of decline ever recorded,” Asher wrote online.

Prep for the polls: See who is running for president and compare where they stand on key issues in our Voter Guide

The decrease in murders is "potentially historically large," Asher told USA TODAY, and it's not just killings that are declining. Preliminary 2023 FBI data “paint the picture" of a big decrease in overall crime, he wrote.

That’s not the picture Trump and his supporters are painting on the campaign trail , with voters likely to hear plenty more in the coming months that attempts to cast President Joe Biden as weak on crime. A House Judiciary Committee field hearing scheduled for Friday in Philadelphia is expected to focus on the topic, picking up on a theme the GOP-led panel covered during similar sessions last year in New York and Chicago.

Trump’s crime rhetoric has been escalating as he faces his own criminal jeopardy, with the former president arguing that prosecutors are ignoring the real crime problem in America to pursue a political “witch hunt” against him.

He complained last summer about the "filth and the decay" in Washington, D.C. as he headed back to the nation's capital for his arraignment on federal criminal charges tied to the Jan. 6, 2021, riot at the U.S. Capitol. Campaigning in Georgia last month, where he's fighting additional state-based criminal charges , Trump declared that “crime is rampant and out of control like never, ever before.”

Yet even as the data contradicts Trump’s description of a nation in the grip of terrible crime wave, many Americans are inclined to agree with him, polls show , and crime could be a key issue this election cycle.

Trump is making public safety concerns, particularly crimes committed by undocumented immigrants , a centerpiece of his 2024 campaign. His message is targeted at swing voters who might have qualms about Trump and GOP policies on issues such as abortion but could find a tough-on-crime pitch appealing.

"The suburban housewives actually like Donald Trump. You know why? Because I'm the one who's gonna keep them safe,” Trump said recently , referencing a voting block that could swing the election.

The Trump campaign referred questions about Trump's rhetoric conflicting with FBI data to the Republican National Committee, which pointed to articles raising questions about the accuracy of the FBI data and conflicting information in federal reports.

RNC spokeswoman Anna Kelly said USA TODAY was "trying to gaslight Americans into believing that their lived experiences are wrong" and noted "families are rightfully concerned" about crime.

"Biden’s weakness has made Americans less safe, and his policies have failed," Kelly added.

Aggressive crime rhetoric has been a staple of GOP politics going back decades, but Trump’s comments clash with the reality laid out in FBI and other reports of a nation mending after a troubled period.

Pandemic crime wave

Republicans also ran on tackling crime during the 2022 midterm election cycle, and they had data to back up their claims that it was a growing problem.

The United States experienced a spike in violent crimes that coincided with the pandemic and social unrest surrounding police killings of George Floyd and other unarmed African American individuals.

The FBI reported that violent crimes increased an estimated 5.6% in 2020 and remained at that elevated level in 2021, dipping by just 1%.

The 29% estimated increase in murders in 2020 was particularly shocking, and murders jumped another 4.3% in 2021 estimates.

While the increase was alarming, crime was still well below levels seen a few decades ago, said Jeffrey Butts, a professor at the John Jay College of Criminal Justice and director of the school’s Research and Evaluation Center.

“When COVID hit we saw this spike, so from 2020 to 2022 it was bad but… it still came nowhere near where we were in the 1990s,” Butts said, noting crime soon began to drop again as expected.

Researchers believe the crime increase was a blip caused by pandemic disruptions, Butts said.

'Horrible things are happening'

Regardless, Republican leaders accused Biden and Democrats of triggering an “onslaught of violent crime that is ravaging our communities,” as GOP Reps. Steve Scalise and Scott Fitzgerald wrote in a Fox News column . They blamed "defund the police" rhetoric – which Biden rejected – and criminal justice reform efforts on the left, such as ending cash bail.

“House Republicans are ready to stand up to the criminals who think that this country is theirs for the taking, and the leftists in Washington who are enabling this outbreak of violent crime,” the lawmakers wrote.

Trump has continued that type of rhetoric, even as crime as ebbed.

The FBI’s national crime estimates for 2022 found that violent crime decreased 1.7% and there were 6.7% fewer murders. Complete FBI crime data for 2023 won’t be released until the fall, but quarterly reports show violent crime continuing to drop.

University of New Haven criminal justice Professor Maria Tcherni-Buzzeo said the downward trend in crime means "we are kind of returning to where we were before the pandemic."

Asher cited preliminary 2023 FBI data in predicting that the final numbers for the year could show a "considerable" drop. The third quarter FBI data showed murders dropping an estimated 15.6% compared with the same period in 2022 and violent crime dropping 8.2% overall.

The fourth quarter 2023 report released in March show an estimated 4% drop in property crime, 6% drop in violent crime and 13% decrease in murders from 2022. The fourth quarter report covers the entire 2023 calendar year, and can be considered a preliminary year-end report.

Questions about crime data

The FBI relies on voluntary reporting from police agencies to develop national crime statistics. Some police departments don't report their data, so the agency estimates crime levels in those regions to come up with a national number.

The FBI transitioned to a new system in 2021, causing problems for some police departments and a drop in agencies reporting their data. The agency had to lean more on estimates, and that has led to questions about the accuracy and completeness of the information.

The percentage of police departments reporting their data has been increasing since then, but the information is still incomplete – 79% of agencies reported in the fourth quarter of 2023 – and the agency uses methods to adjust for missing data and publish estimates.

While FBI data showed violent crime decreasing in 2022, another widely-cited crime barometer showed a different picture. The 2022 National Victimization Survey conducted by the Bureau of Justice Statistics, which captures both reported and unreported crimes, found a steep increase in the violent victimization rate.

That survey measures a different timespan, though, starting with crimes that occurred in July of 2021 when pandemic disruptions were more acute and continuing through November of 2022. The 2023 survey won't be released until the fall, and there aren't interim reports like the FBI data.

Because of the different timeframes and other factors, Asher believes the victimization survey and FBI reports shouldn't be compared. He and other crime statistics experts say they are confident in the overall trend laid out in the FBI data of crime decreasing. The data have a margin of error because they rely on estimates, but clearly point in the direction of reduced crime.

“The trends are clear, there’s no questioning the trend in good faith in my opinion," Asher said.

Some communities continue to experience elevated crime levels, or increases in certain crimes. Murders are up 9% in Los Angeles, 8% in St. Louis and 23% in Denver this year compared to the start of 2023, according to data compiled by Asher's firm, AH Datalytics. Nationwide, 2023 third and fourth quarter FBI crime reports show motor vehicles thefts increasing.

Overall, though, crime is going down and is at or near pre-pandemic levels, according to the FBI data, experts interviewed by USA TODAY and other leading groups that study the issue.

"Crime rates are largely returning to pre-COVID levels as the nation distances itself from the height of the pandemic, but there are notable exceptions," the non-partisan Council on Criminal Justice wrote in a year-end 2023 analysis of crime trends in 38 cities.

That trend is continuing into 2024.

Big drop in murders

There are 532 fewer murders so far this year in the 218 cities tracked by AH Datalytics, when compared with the same period last year, a 20% decline.

“I would have no problem walking around any big city in the United States right now,” said University of Miami criminologist Alex Piquero, who previously ran the federal Bureau of Justice Statistics, which publishes crime data. “The issue that we have is people are diving into social media and they’re not taking the time to digest really what’s happening.”

Piquero said people can "cherry pick" a few cities or crime types to argue crime still is a growing problem, but the national picture shows a steady decrease.

"There’s no doubt in my mind that 2022 was better than 2021, 2023 is going to be better than 2022 and 2024 will be better than 2023, I think every single data point we’re seeing is showing that," Piquero said.

Trump often focuses on crime in a few cities.

Murders are down 18% in New York City and 24% in Washington D.C. so far in 2024, according to police data, yet Trump continues to portray both cities as crime infested.

At his Georgia rally, Trump said businesses are going to leave New York “over crime” and described Washington D.C. as “a nightmare of murder and crime.”

“People from Georgia go down to Washington now and they get shot,” Trump said. “Horrible things are happening.”

Washington D.C. had 274 murders in 2023, the most since 1997, and up from 166 in 2019. So even if the 24% decrease holds for the entire year, the city would still be above the pre-pandemic murder level.

Yet the city's crime problem appears to be ebbing this year, with violent crimes down 25% and property crime down 13% so far when compared with the same period in 2023, according to the D.C. police department.

Politics of crime

Trump’s crime rhetoric continued as his trial kicked off earlier this month in New York City on charges stemming from alleged hush money payments to an adult film star to hide an affair.

After the first day of jury selection, Trump emerged from the courtroom and attacked Manhattan District Attorney Alvin Bragg.

“You go right outside and people are being mugged and killed all day long and he’s sitting here all day with about 10 or 12 prosecutors over nothing ," the former president said.

Trump later visited a New York City bodega where a man was stabbed to death to further argue that the city has a crime problem. Earlier this month he held an event in Michigan with law enforcement to highlight crimes committed by undocumented immigrants, seizing on the case of Ruby Garcia , who was killed by her undocumented partner.

Garcia’s sister rebuked Trump for claiming he spoke with the slain woman’s family, which she said is not true.

The downward crime trend could hurt Trump’s efforts to deflect attention away from his own criminal cases by claiming there are bigger crime concerns. But polls show many Americans aren’t convinced crime is down.

A Gallup survey released in November found that 77% of Americans believe there is more crime than a year ago, despite FBI data showing the contrary.

The survey also found that more people say crime is an extremely or very serious problem now than in 2021, when murders actually increased.

That’s despite the fact that only 17% of people say crime is a big problem where they live.

That may be a sign that people’s opinions are driven more by political and media narratives around crime in faraway places than by the data.

Tcherni-Buzzeo said that “it may seem that… Trump makes people feel like crime is up,” but in her experience the mistaken belief that crime is increasing predates the former president’s political campaigns.

Going back more than a decade, whenever she asked students if crime is on the rise, the majority of the class would say yes, even though crime generally has been trending down since the 1990s.

Such beliefs may be “fueled by the fact this is what they see on the news,” she said.

The reality with violent crime is that “the trend is positive” recently and that should be the baseline for policy debates, Asher said.

“That does not endorse the level (of crime), it doesn’t mean there’s not work to do,” he said, adding: “But we don’t argue about whether the Chiefs won the Super Bowl.”

Skip to main content
Accessibility help

Information

We use cookies to collect anonymous data to help us improve your site browsing experience.

Click 'Accept all cookies' to agree to all cookies that collect anonymous data. To only allow the cookies that make the site work, click 'Use essential cookies only.' Visit 'Set cookie preferences' to control specific cookies.

Your cookie preferences have been saved. You can change your cookie settings at any time.

Children’s Social Work Statistics 2022-23 – Looked After Children

Looked After Children Statistics for Scotland for 2022 to 2023 that cover data on children who are looked after, young people in continuing care, and young people eligible for aftercare services.

This document is part of a collection

Children's social work statistics

Introduction

These statistics include data collected from 32 local authorities in Scotland and cover statistics on: looked after children, young people in continuing care, and young people eligible for aftercare.

These figures relate to the reporting year 01 August 2022 to 31 July 2023. All data used to produce the tables and charts are available in the supporting documents , alongside additional breakdowns . Looked After Children statistics form part of our annual Children’s Social Work data collections. Statistics relating to Secure Care and Child Protection are published separately from this year onwards.

For purposes of presentation, table columns and chart axes refer to these as whole years (e.g., 2023 reflects 01 August 2022-31 July 2023 collection). This report includes some revisions to figures for the preceding reporting year (2021-22) in line with ongoing data validation improvements. As such, some 2021-22 figures may be different to those previously reported. Please refer to background notes on comparability over time.

Email: [email protected]

There is a problem

Thanks for your feedback

Your feedback helps us to improve this website. Do not give any personal information because we cannot reply to you directly.

IMAGES

Statistical Hypothesis Testing: Step by Step
PPT
Best Example of How to Write a Hypothesis 2024
Guide to Hypothesis Testing for Data Scientists
Choosing the Right Statistical Test
Hypothesis Testing- Meaning, Types & Steps

VIDEO

Concept of Hypothesis
What Is A Hypothesis?
Null Hypothesis and Alternative Hypothesis |Data Science and big data lecture-6 In Hindi|By Aakash
Introduction to Statistics: Hypothesis Testing
Testing of Hypothesis for Categorical Data
Hypothsis Testing in Statistics Part 2 Steps to Solving a Problem

COMMENTS

Hypothesis Testing
There are 5 main steps in hypothesis testing: State your research hypothesis as a null hypothesis and alternate hypothesis (H o) and (H a or H 1 ). Collect data in a way designed to test the hypothesis. Perform an appropriate statistical test. Decide whether to reject or fail to reject your null hypothesis. Present the findings in your results ...
7.1: Basics of Hypothesis Testing
Test Statistic: z = ¯ x − μo σ / √n since it is calculated as part of the testing of the hypothesis. Definition 7.1.4. p - value: probability that the test statistic will take on more extreme values than the observed test statistic, given that the null hypothesis is true.
Introduction to Hypothesis Testing
To test whether a statistical hypothesis about a population parameter is true, we obtain a random sample from the population and perform a hypothesis test on the sample data. There are two types of statistical hypotheses: The null hypothesis, denoted as H 0, is the hypothesis that the sample data occurs purely from chance. The alternative ...
6a.2
A hypothesis, in statistics, is a statement about a population parameter, where this statement typically is represented by some specific numerical value. In testing a hypothesis, we use a method where we gather data in an effort to gather evidence about the hypothesis.
Hypothesis Testing
The Four Steps in Hypothesis Testing. STEP 1: State the appropriate null and alternative hypotheses, Ho and Ha. STEP 2: Obtain a random sample, collect relevant data, and check whether the data meet the conditions under which the test can be used. If the conditions are met, summarize the data using a test statistic.
Statistical Hypothesis Testing Overview
Hypothesis testing is a crucial procedure to perform when you want to make inferences about a population using a random sample. These inferences include estimating population properties such as the mean, differences between means, proportions, and the relationships between variables. This post provides an overview of statistical hypothesis testing.
4.4: Hypothesis Testing
How do we test a claim or a hypothesis using statistical data? This webpage introduces the concept and procedure of hypothesis testing, a fundamental tool for inference in statistics. You will learn how to formulate null and alternative hypotheses, how to calculate test statistics and p-values, and how to interpret the results of hypothesis testing. This webpage is part of the Statistics ...
S.3 Hypothesis Testing
In statistics, the data are the evidence. The jury then makes a decision based on the available evidence: If the jury finds sufficient evidence — beyond a reasonable doubt — to make the assumption of innocence refutable, the jury rejects the null hypothesis and deems the defendant guilty.
Hypothesis Testing
Hypothesis testing is an indispensable tool in data science, allowing us to make data-driven decisions with confidence. By understanding its principles, conducting tests properly, and considering real-world applications, you can harness the power of hypothesis testing to unlock valuable insights from your data.
Understanding Hypothesis Testing
Image by Author. So, a one-tailed statistical test is one whose distribution has only one tail — either the left (left-tailed test) or the right (right-tailed test).A two-tailed statistical test is one whose distribution has two tails — both left and right.. The purpose of a tail in statistical tests is to see whether the test statistic obtained falls within the tail or outside it.
Hypothesis Testing
(In statistics, the data are the evidence.) Next, you make your initial assumption. Defendant is innocent until proven guilty. In statistics, we always assume the null hypothesis is true. Then, make a decision based on the available evidence. If there is sufficient evidence ("beyond a reasonable doubt"), reject the null hypothesis. (Behave ...
Hypothesis testing for data scientists
Hypothesis testing. In hypothesis testing, two mutually exclusive statements about a parameter or population (hypotheses) are evaluated to decide which statement is best supported by sample data. Parameters and statistics. In statistics, a parameter is a description of a population, while a statistic describes a small portion of a population ...
Hypothesis Testing
Hypothesis testing in statistics is a way for you to test the results of a survey or experiment to see if you have meaningful results. You're basically testing whether your results are valid by figuring out the odds that your results have happened by chance. ... For this set of data: z= (112.5 - 100) / (15/√30) = 4.56 . Step 6: If Step 6 ...
What is Hypothesis Testing in Statistics? Types and Examples
Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence.
Statistics
Statistics - Hypothesis Testing, Sampling, Analysis: Hypothesis testing is a form of statistical inference that uses data from a sample to draw conclusions about a population parameter or a population probability distribution. First, a tentative assumption is made about the parameter or distribution. This assumption is called the null hypothesis and is denoted by H0.
Statistical Inference and Hypothesis Testing in Data Science Applications
This course is part of the Data Science Foundations: Statistical Inference Specialization. When you enroll in this course, you'll also be enrolled in this Specialization. Learn new concepts from industry experts. Gain a foundational understanding of a subject or tool. Develop job-relevant skills with hands-on projects.
Choosing the Right Statistical Test
Statistical tests assume a null hypothesis of no relationship or no difference between groups. Then they determine whether the observed data fall outside of the range of values predicted by the null hypothesis. ... You can perform statistical tests on data that have been collected in a statistically valid manner - either through an experiment ...
9.1: Introduction to Hypothesis Testing
In hypothesis testing, the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis.The null hypothesis is usually denoted $H_0$ while the alternative hypothesis is usually denoted $H_1$. An hypothesis test is a statistical decision; the conclusion will either be to reject the null hypothesis in favor ...
Statistics
Hypothesis testing is based on making two different claims about a population parameter. The null hypothesis ( H 0) and the alternative hypothesis ( H 1) are the claims. The two claims needs to be mutually exclusive, meaning only one of them can be true. The alternative hypothesis is typically what we are trying to prove.
Hypothesis to Be Tested: Definition and 4 Steps for ...
Hypothesis testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used ...
Understanding Hypothesis Testing
Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data.
Executive summary
Electric cars continue to make progress towards becoming a mass-market product in a larger number of countries. Tight margins, volatile battery metal prices, high inflation, and the phase-out of purchase incentives in some countries have sparked concerns about the industry's pace of growth, but global sales data remain strong.
Understanding the new income statistics for local areas: Different
The ONS has been busy producing faster, more detailed statistics using new data sources. Providing clearer insights at the local level is an important aspect of this work. Ainslie Woods explains how our local income statistics compare and why they offer a complementary picture of local areas.
Children's Social Work Statistics 2022-23
For general enquiries about Scottish Government statistics please contact: Office of the Chief Statistician. E-mail: [email protected]. Join our mailing list. If you would like to receive notification about statistical publications, or find out about consultations on our statistics please join the ScotStat mailing list. Future ...
FBI crime data shows decrease, despite Trump's contrary claims
FBI data shows violent crime has been dropping in recent years after a spike in 2020, but Trump says crime is still 'out of control.' 📷 Key players Meteor shower up next 📷 Leaders at the ...
PDF Births: Provisional Data for 2023
SOURCE: National Center for Health Statistics, National Vital Statistics System, natality data file. 0 20 40 60 80 100 18-19 15-17 15-19 1991 1995 2000 2005 2010 2015 2020 2023 Rate per 1,000 females Figure 2. Birth rate for teenagers, by age of mother: United States, ﬁnal 1991-2022 and provisional 2023.
Children's Social Work Statistics 2022-23
We use cookies to collect anonymous data to help us improve your site browsing experience. Click 'Accept all cookies' to agree to all cookies that collect anonymous data. To only allow the cookies that make the site work, click 'Use essential cookies only.' Visit 'Set cookie preferences' to control specific cookies.
Children's Social Work Statistics 2022-23
Looked After Children statistics form part of our annual Children's Social Work data collections. Statistics relating to Secure Care and Child Protection are published separately from this year onwards. For purposes of presentation, table columns and chart axes refer to these as whole years (e.g., 2023 reflects 01 August 2022-31 July 2023 ...

Introduction to Hypothesis Testing

The Two Types of Statistical Hypotheses

Hypothesis Tests

The Two Types of Decision Errors

One-Tailed and Two-Tailed Tests

Types of Hypothesis Tests

Featured Posts

Leave a Reply Cancel reply

Hypothesis Testing – A Deep Dive into Hypothesis Testing, The Backbone of Statistical Inference

1. What is Hypothesis Testing?

2. Steps in Hypothesis Testing

2.1. Set up Hypotheses: Null and Alternative

2.2. Choose a Significance Level (α)

2.3. Calculate a test statistic and P-Value

2.4. Make a Decision

3. Example : Testing a new drug.

4. Example in python

5. Conclusion

More Articles

Machine Learning A-Z™: Hands-On Python & R In Data Science

User Preferences

Keyboard Shortcuts

Basic approach to hypothesis testing

Making the Decision

Example: Criminal Trial Analogy

Using the p -value to make the decision

Significance level and p -value

One sample t -test

Testing for the population proportion

Tutorial Playlist

The Best Guide to Understand Bayes Theorem

A Complete Guide to Get a Grasp of Time Series Analysis

The Complete Guide to Understand Pearson's Correlation

Table of Contents

What Is Hypothesis Testing in Statistics?

Hypothesis Testing Formula

How Hypothesis Testing Works?

Your Dream Career is Just Around The Corner!

Null Hypothesis and Alternate Hypothesis

Become a Data Scientist with Hands-on Training!

Hypothesis Testing Calculation With Examples

Steps of Hypothesis Testing

Step 2: Gather Data

Step 3: Conduct a Statistical Test

Step 4: Determine Rejection Of Your Null Hypothesis

Step 5: Present Your Results

Types of Hypothesis Testing

Chi-Square

Hypothesis Testing and Confidence Intervals

Simple and Composite Hypothesis Testing

One-Tailed and Two-Tailed Hypothesis Testing

Become a Data Scientist With Real-World Experience

Right Tailed Hypothesis Testing

Left Tailed Hypothesis Testing

Type 1 and Type 2 Error

Level of Significance

Future-Proof Your AI/ML Career: Top Dos and Don'ts

Why is Hypothesis Testing Important in Research Methodology?

Limitations of Hypothesis Testing

1. What is hypothesis testing in statistics with example?

2. What is hypothesis testing and its types?

3. What are the steps of hypothesis testing?

4. What are the 2 types of hypothesis testing?

5. What are the 3 major types of hypothesis?

Find our Data Analyst Online Bootcamp in top cities:

Recommended Resources

9.1: Introduction to Hypothesis Testing

Basic Theory

\(P\)-value

Analogy with Justice Systems

Tests of an Unknown Parameter

Equivalence Between Hypothesis Test and Confidence Sets

Pivot Variables and Test Statistics

Statistics Tutorial

Hypothesis Testing

The Null and Alternative Hypothesis

The Significance Level

The Test Statistic

The Critical Value and P-Value Approach

The Critical Value Approach