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8.10: Summary

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• Page ID 29494

• Danielle Navarro
• University of New South Wales

Null hypothesis testing is one of the most ubiquitous elements to statistical theory. The vast majority of scientific papers report the results of some hypothesis test or another. As a consequence it is almost impossible to get by in science without having at least a cursory understanding of what a p -value means, making this one of the most important chapters in the book. As usual, here's a quick recap of the key ideas that we’ve talked about:

• Research hypotheses and statistical hypotheses. Null and alternative hypotheses. (Section 8.1).
• Type 1 and Type 2 errors (Section 8.2)
• Test statistics and sampling distributions (Section 8.3)
• Hypothesis testing as a decision-making process (Section 8.4)
• p-values as “soft” decisions (Section 8.5)
• Writing up the results of a hypothesis test (Section 8.6)
• Effect size and power (Section 8.8)
• A few issues to consider regarding hypothesis testing (Section 8.9)

Later in the book, in Chapter 14, we’ll revisit the theory of null hypothesis tests from a Bayesian perspective, and introduce a number of new tools that you can use if you aren’t particularly fond of the orthodox approach. But for now, though, we’re done with the abstract statistical theory, and we can start discussing specific data analysis tools.

Cohen, J. 1988. Statistical Power Analysis for the Behavioral Sciences . 2nd ed. Lawrence Erlbaum.

Ellis, P. D. 2010. The Essential Guide to Effect Sizes: Statistical Power, Meta-Analysis, and the Interpretation of Research Results . Cambridge, UK: Cambridge University Press.

Lehmann, Erich L. 2011. Fisher, Neyman, and the Creation of Classical Statistics . Springer.

Gelman, A., and H. Stern. 2006. “The Difference Between ‘Significant’ and ‘Not Significant’ Is Not Itself Statistically Significant.” The American Statistician 60: 328–31.

• The quote comes from Wittgenstein’s (1922) text, Tractatus Logico-Philosphicus .
• A technical note. The description below differs subtly from the standard description given in a lot of introductory texts. The orthodox theory of null hypothesis testing emerged from the work of Sir Ronald Fisher and Jerzy Neyman in the early 20th century; but Fisher and Neyman actually had very different views about how it should work. The standard treatment of hypothesis testing that most texts use is a hybrid of the two approaches. The treatment here is a little more Neyman-style than the orthodox view, especially as regards the meaning of the p value.
• My apologies to anyone who actually believes in this stuff, but on my reading of the literature on ESP, it’s just not reasonable to think this is real. To be fair, though, some of the studies are rigorously designed; so it’s actually an interesting area for thinking about psychological research design. And of course it’s a free country, so you can spend your own time and effort proving me wrong if you like, but I wouldn’t think that’s a terribly practical use of your intellect.
• This analogy only works if you’re from an adversarial legal system like UK/US/Australia. As I understand these things, the French inquisitorial system is quite different.
• An aside regarding the language you use to talk about hypothesis testing. Firstly, one thing you really want to avoid is the word “prove”: a statistical test really doesn’t prove that a hypothesis is true or false. Proof implies certainty, and as the saying goes, statistics means never having to say you’re certain. On that point almost everyone would agree. However, beyond that there’s a fair amount of confusion. Some people argue that you’re only allowed to make statements like “rejected the null”, “failed to reject the null”, or possibly “retained the null”. According to this line of thinking, you can’t say things like “accept the alternative” or “accept the null”. Personally I think this is too strong: in my opinion, this conflates null hypothesis testing with Karl Popper’s falsificationist view of the scientific process. While there are similarities between falsificationism and null hypothesis testing, they aren’t equivalent. However, while I personally think it’s fine to talk about accepting a hypothesis (on the proviso that “acceptance” doesn’t actually mean that it’s necessarily true, especially in the case of the null hypothesis), many people will disagree. And more to the point, you should be aware that this particular weirdness exists, so that you’re not caught unawares by it when writing up your own results.
• Strictly speaking, the test I just constructed has α=.057, which is a bit too generous. However, if I’d chosen 39 and 61 to be the boundaries for the critical region, then the critical region only covers 3.5% of the distribution. I figured that it makes more sense to use 40 and 60 as my critical values, and be willing to tolerate a 5.7% type I error rate, since that’s as close as I can get to a value of α=.05.
• The internet seems fairly convinced that Ashley said this, though I can’t for the life of me find anyone willing to give a source for the claim.
• That’s p=.000000000000000000000000136 for folks that don’t like scientific notation!
• Note that the p here has nothing to do with a p value. The p argument in the binom.test() function corresponds to the probability of making a correct response, according to the null hypothesis. In other words, it’s the θ value.
• There’s an R package called compute.es that can be used for calculating a very broad range of effect size measures; but for the purposes of the current book we won’t need it: all of the effect size measures that I’ll talk about here have functions in the lsr package
• Although in practice a very small effect size is worrying, because even very minor methodological flaws might be responsible for the effect; and in practice no experiment is perfect, so there are always methodological issues to worry about.
• Notice that the true population parameter θ doesn’t necessarily correspond to an immutable fact of nature. In this context θ is just the true probability that people would correctly guess the colour of the card in the other room. As such the population parameter can be influenced by all sorts of things. Of course, this is all on the assumption that ESP actually exists!
• Although this book describes both Neyman’s and Fisher’s definition of the p value, most don’t. Most introductory textbooks will only give you the Fisher version.
• In this case, the Pearson chi-square test of independence (Chapter 12; chisq.test() in R) is what we use; see also the prop.test() function.

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6a.2 - steps for hypothesis tests, the logic of hypothesis testing section  .

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.

How do we decide whether to reject the null hypothesis?

• If the sample data are consistent with the null hypothesis, then we do not reject it.
• If the sample data are inconsistent with the null hypothesis, but consistent with the alternative, then we reject the null hypothesis and conclude that the alternative hypothesis is true.

Six Steps for Hypothesis Tests Section  

In hypothesis testing, there are certain steps one must follow. Below these are summarized into six such steps to conducting a test of a hypothesis.

• Set up the hypotheses and check conditions : Each hypothesis test includes two hypotheses about the population. One is the null hypothesis, notated as \(H_0 \), which is a statement of a particular parameter value. This hypothesis is assumed to be true until there is evidence to suggest otherwise. The second hypothesis is called the alternative, or research hypothesis, notated as \(H_a \). The alternative hypothesis is a statement of a range of alternative values in which the parameter may fall. One must also check that any conditions (assumptions) needed to run the test have been satisfied e.g. normality of data, independence, and number of success and failure outcomes.
• Decide on the significance level, \(\alpha \): This value is used as a probability cutoff for making decisions about the null hypothesis. This alpha value represents the probability we are willing to place on our test for making an incorrect decision in regards to rejecting the null hypothesis. The most common \(\alpha \) value is 0.05 or 5%. Other popular choices are 0.01 (1%) and 0.1 (10%).
• Calculate the test statistic: Gather sample data and calculate a test statistic where the sample statistic is compared to the parameter value. The test statistic is calculated under the assumption the null hypothesis is true and incorporates a measure of standard error and assumptions (conditions) related to the sampling distribution.
• Calculate probability value (p-value), or find the rejection region: A p-value is found by using the test statistic to calculate the probability of the sample data producing such a test statistic or one more extreme. The rejection region is found by using alpha to find a critical value; the rejection region is the area that is more extreme than the critical value. We discuss the p-value and rejection region in more detail in the next section.
• Make a decision about the null hypothesis: In this step, we decide to either reject the null hypothesis or decide to fail to reject the null hypothesis. Notice we do not make a decision where we will accept the null hypothesis.
• State an overall conclusion : Once we have found the p-value or rejection region, and made a statistical decision about the null hypothesis (i.e. we will reject the null or fail to reject the null), we then want to summarize our results into an overall conclusion for our test.

We will follow these six steps for the remainder of this Lesson. In the future Lessons, the steps will be followed but may not be explained explicitly.

Step 1 is a very important step to set up correctly. If your hypotheses are incorrect, your conclusion will be incorrect. In this next section, we practice with Step 1 for the one sample situations.

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What are Summary Statistics

Summary statistics definition, summary statistics examples, practice problems, summary statistics – explanation and examples.

Summary statistics are numbers or words that describe a data set or data sets simply.

This includes measures of centrality, dispersion, and correlation as well as descriptions of the overall shape of the data set.

Summary statistics are used in all branches of math and science that employ statistics. These include probability, economics, biology, psychology, and astronomy.

Before moving on with this section, make sure to review measures of central tendency and standard deviation .

This section covers:

• What are Summary Statistics?

How to Interpret Summary Statistics

Summary statistics are numbers or words that describe a data set as succinctly as possible.

These include measures of central tendency such as mean, median, and mode. They also include measures of dispersion such as range and standard deviation. Summary statistics for multivariate data sets may also include measures of correlation such as the correlation coefficient.

Descriptions of the overall data shape such as “normally distributed” or “skewed right” are also part of summary statistics.

Summary statistics give a small “snapshot” of a data set that is more approachable than large quantities of data and more easily generalized than random data points. Like the summary of a story, they analyze and describe even large data sets in just a few numbers and words.

It is best to interpret individual components of summary statistics in light of the other components.

In general, a larger range and larger standard deviation indicate a wider dispersion. A wider range with a smaller standard deviation indicates outliers.

Similarly, when it comes to measures of central tendency, a mean that is higher than the median indicates a skew to the right. Likewise, a mean that is less than the median indicates a skew to the left. If they are about the same, the data set is likely normally distributed.

Summary statistics are measures of central tendency, dispersion, and correlation combined with descriptions of shape that provide a simple overview of a data set or data sets.

These measures can include, mean, median, mode, standard deviation, range, and correlation coefficient.

One example of an important use for summary statistics is a census. In the United States, there are over $320$ million people. This means that a census includes a lot of data points. Since a census also usually includes information such as age, family size, address, occupation, etc., these are multivariate data points!

But, civil servants and politicians need to make decisions based on census results. The easiest way to do that is to provide decision makers with summary statistics of census results. These snapshots are easier to understand than a collection of $320$ million+ data points.

Common Examples

This section covers common examples of problems involving summary statistics and their step-by-step solutions.

A data set has a mean of $200$, a median of $50$, a mode of $40$, and a range of $1500$. What do the summary statistics say about this data set?

The summary statistics for this data set indicate a strong skew to the right. This means that there is one or more upper outliers.

How do they show this?

Outlier have a strong effect on the mean of a data set but very little effect on the median. This means that upper outliers will increase the average while the median stays in place. In fact, it is the main reason for a discrepancy in the median and mean of a data set.

Clearly, there is a large difference between $50$ and $200$, especially in light of the fact that the mode is $40$. This means that half of the data points are more than $50$ and half are less with $40$ being the most commonly occurring term. It certainly does not fit with that to say that a typical term is $200$.

Likewise, the wide range indicates large values are possible.

Additional summary statistics that would paint a fuller picture are the highest and lowest values along with the standard deviation.

Find the summary statistics for the following data set.

$(1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 9, 11, 13, 17, 25, 33)$

Common summary statistics include mean, median, mode, range, and standard deviation.

In this case, the mean is equal to:

$\frac{1(6)+2(3)+3(2)+4(2)+5+6(2)+7+8+9+11+13+17+25+33}{24} = \frac{166}{24} = \frac{83}{12}$.

This is about equal to $6.9167$.

The median in this case is equal to the average of the twelfth and thirteenth numbers. These are both four, however, so four is the median.

Since one appears more often than any other number, it is the mode.

These are the measures of central tendency. On the other hand, the common measures of dispersion are range and standard deviation.

The range is just equal to the largest number minus the smallest number. This is equal to $33-1 = 32$.

Standard deviation, however, is difficult to calculate. It is equal to:

$\sqrt{\frac{\sum_{i=1}^k (n_i – \mu)^2}{k}}$.

These calculations take a while. For larger data sets, it is often easier to use a standard deviation calculator.

Whether calculating by hand or with technology, however, the standard deviation is about $8.086.$

The total summary, then, is:

Mean: $6.9167$

Median: $4$

Range: $32$

Standard Deviation: $8.086$.

The summary statistics may also note that there are $24$ elements in the data set, with the largest value being $1$ and the smallest value being $33$.

Consider the following data set:

$(85, 86, 88, 88, 90, 91, 94, 94, 96, 97, 98, 98, 98, 99, 99, 99, 99, 100, 100, 100, 100, 100, 100, 101, 101, 101, 102, 102, 102, 103, 103, 104, 104, 105, 106, 106, 108, 109, 110, 110, 110, 113, 115)$.

What are the summary statistics for this data set? What do these statistics say about the data set?

This data set has 43 data points. The highest value is $115$, while the lowest value is $85$. This means that the range is $115-85=30$.

The median of this data set is going to be the twenty-second term, which is $100$.

Likewise, the mode of the data set is $100$ because it appears more than any other value.

The mean of this data set is equal to:

$\frac{4314}{43}$. This is about equal to $100.3$.

Plugging the standard deviation into a standard deviation calculator reveals that it is approximately $6.9$.

Therefore, the summary statistics on this data set are:

Mean: $100.3$

Median: $100$

Mode: $100$

Range: $30$

Standard Deviation: $6.9$

Number of Terms: $43$

Highest Value: $115$

Lowest Value: $85$.

Based on these statistics, the data is probably normally distributed because all of the measures of central tendency are almost exactly equal.

A shipping company weighs a sample of packages before they are sent out. They get the following results.

$(0.1, 0.1, 0.3, 0.5, 0.8, 0.9, 1.1, 1.2, 1.4, 1.5, 1.5, 1.5, 1.6, 1.7, 1.7, 1.8, 1.9, 2.1, 2.9, 3.3, 4.0, 5.3, 5.5, 6.8, 9.2, 21.8)$.

What are the summary statistics for the data? What do they say about the data in context?

The summary statistics for this data set are:

Number of Terms: $26$

Mode (most common value): $1.5$

Median (average of the thirteenth and fourteenth terms): $1.65$

Mean (sum of the terms divided by $26$): About $3.096$

Highest Value: $21.8$

Lowest Value: $0.1$

Range (difference of highest and lowest values): $21.7$

Standard Deviation (average variance from mean): $4.397$

In this data set, the median and mode are approximately the same, but the mean is a bit higher. It is not, however, a full standard deviation higher. This means that the data is slightly skewed to the right, but not too much. This is likely due to the presence of some outliers.

In context, this means that there are a few heavier packages that the company sends, but, for the most part, the packages weigh around $1.65$ pounds.

• A data set has standard deviation of $1$, a mean of $0$, a median of $4$ and a mode of $3.5$. What can be said about the data set?
• Another data set is approximately normally distributed. It has a median of $16$ and a standard deviation of $3$. In what range do the median and mode likely fall?
• Describe what the summary statistics would look like for a U-shaped data set.
• Find the summary statistics for the following data set: $(-5, -4, -4, -3, -3, -3, -2, -2, -2, -1, -1, -1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 5)$.
• A charity receives donations at an event. The donation amounts in dollars are: $(1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 5, 5, 5, 5, 5, 5, 5, 5, 10, 10, 10, 10, 11, 12, 15, 15, 20, 20, 20, 20, 40, 40, 45, 50, 50, 50, 100, 200)$. Find the summary statistics for the donations and interpret them in context.
• Since the difference between the median and the mean is greater than the standard deviation and the mode is close the mean, the data is likely skewed to the left.
• Since this data is normally distributed, the median and mode are likely within $3$ units in either direction of the mean. That is, they are likely in the range of $13$ to $19$.
• In such a data set, the mean and median would be about the same. The standard deviation would be large relative to the range. The mode would likely be very high or very low (or both).
• Number of terms: $28$. Mean is about $-0.3214$, median is 0, and mode is $0$ an $1$. The range between the highest and lowest values of $5$ and $-5$ is $10$, and the standard deviation is about $2.405$. The data is approximately normally distributed.
• There were $37$ donations averaging $21.08$ dollars. The most common donation was $5$ dollars, and the median donation was $10$. The range of donations was from $1$ to $200$, which means the range was $199$. In this case, the standard deviation was about $36.31$, which means that there was a lot of variance in the donation amount. The large difference between the mean and median donation indicates an outlier to the right, namely the $200$ dollar donation.

All mathematical illustrations/objects created with Geogreba.

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Lesson 10 of 24 By Avijeet Biswal

Table of Contents

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.

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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.

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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.

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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.

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Social Statistics for a Diverse Society

Student resources, chapter summary.

Statistical hypothesis testing is a deci­sion-making process that enables us to deter­mine whether a particular sample result falls within a range that can occur by an accept­able level of chance.The process of statistical hypothesis testing consists of five steps: (1) making assumptions, (2) stating the research and null hypotheses and selecting alpha, (3) selecting a sampling distribution and a test sta­tistic, (4) computing the test statistic, and (5) making a decision and interpreting the results.

Statistical hypothesis testing may involve a comparison between a sample mean and a population mean or a comparison between two sample means.If we know the population variance(s) when testing for dif­ferences between means, we can use the Z sta­tistic and the normal distribution.However, in practice, we are unlikely to have this information.

When testing for differences between means when the population variance(s) are unknown, we use the t statistic and the t distribution.

Tests involving differences between pro­portions follow the same procedure as tests for differences between means when population variances are known.The test statistic is Z , and the sampling distribution is approximated by the normal distribution.

Statistics Made Easy

How to Write Hypothesis Test Conclusions (With Examples)

A   hypothesis test is used to test whether or not some hypothesis about a population parameter is true.

To perform a hypothesis test in the real world, researchers obtain a random sample from the population and perform a hypothesis test on the sample data, using a null and alternative hypothesis:

• Null Hypothesis (H 0 ): The sample data occurs purely from chance.
• Alternative Hypothesis (H A ): The sample data is influenced by some non-random cause.

If the p-value of the hypothesis test is less than some significance level (e.g. α = .05), then we reject the null hypothesis .

Otherwise, if the p-value is not less than some significance level then we fail to reject the null hypothesis .

When writing the conclusion of a hypothesis test, we typically include:

• Whether we reject or fail to reject the null hypothesis.
• The significance level.
• A short explanation in the context of the hypothesis test.

For example, we would write:

We reject the null hypothesis at the 5% significance level.   There is sufficient evidence to support the claim that…

Or, we would write:

We fail to reject the null hypothesis at the 5% significance level.   There is not sufficient evidence to support the claim that…

The following examples show how to write a hypothesis test conclusion in both scenarios.

Example 1: Reject the Null Hypothesis Conclusion

Suppose a biologist believes that a certain fertilizer will cause plants to grow more during a one-month period than they normally do, which is currently 20 inches. To test this, she applies the fertilizer to each of the plants in her laboratory for one month.

She then performs a hypothesis test at a 5% significance level using the following hypotheses:

• H 0 : μ = 20 inches (the fertilizer will have no effect on the mean plant growth)
• H A : μ > 20 inches (the fertilizer will cause mean plant growth to increase)

Suppose the p-value of the test turns out to be 0.002.

Here is how she would report the results of the hypothesis test:

We reject the null hypothesis at the 5% significance level.   There is sufficient evidence to support the claim that this particular fertilizer causes plants to grow more during a one-month period than they normally do.

Example 2: Fail to Reject the Null Hypothesis Conclusion

Suppose the manager of a manufacturing plant wants to test whether or not some new method changes the number of defective widgets produced per month, which is currently 250. To test this, he measures the mean number of defective widgets produced before and after using the new method for one month.

He performs a hypothesis test at a 10% significance level using the following hypotheses:

• H 0 : μ after = μ before (the mean number of defective widgets is the same before and after using the new method)
• H A : μ after ≠ μ before (the mean number of defective widgets produced is different before and after using the new method)

Suppose the p-value of the test turns out to be 0.27.

Here is how he would report the results of the hypothesis test:

We fail to reject the null hypothesis at the 10% significance level.   There is not sufficient evidence to support the claim that the new method leads to a change in the number of defective widgets produced per month.

Additional Resources

The following tutorials provide additional information about hypothesis testing:

Introduction to Hypothesis Testing 4 Examples of Hypothesis Testing in Real Life How to Write a Null Hypothesis

Hey there. My name is Zach Bobbitt. I have a Master 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.

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Statistical functions ( scipy.stats ) #

This module contains a large number of probability distributions, summary and frequency statistics, correlation functions and statistical tests, masked statistics, kernel density estimation, quasi-Monte Carlo functionality, and more.

Statistics is a very large area, and there are topics that are out of scope for SciPy and are covered by other packages. Some of the most important ones are:

statsmodels : regression, linear models, time series analysis, extensions to topics also covered by scipy.stats .

Pandas : tabular data, time series functionality, interfaces to other statistical languages.

PyMC : Bayesian statistical modeling, probabilistic machine learning.

scikit-learn : classification, regression, model selection.

Seaborn : statistical data visualization.

rpy2 : Python to R bridge.

Probability distributions #

Each univariate distribution is an instance of a subclass of rv_continuous ( rv_discrete for discrete distributions):

Continuous distributions #

The fit method of the univariate continuous distributions uses maximum likelihood estimation to fit the distribution to a data set. The fit method can accept regular data or censored data . Censored data is represented with instances of the CensoredData class.

Multivariate distributions #

scipy.stats.multivariate_normal methods accept instances of the following class to represent the covariance.

Discrete distributions #

An overview of statistical functions is given below. Many of these functions have a similar version in scipy.stats.mstats which work for masked arrays.

Summary statistics #

Frequency statistics #, hypothesis tests and related functions #.

SciPy has many functions for performing hypothesis tests that return a test statistic and a p-value, and several of them return confidence intervals and/or other related information.

The headings below are based on common uses of the functions within, but due to the wide variety of statistical procedures, any attempt at coarse-grained categorization will be imperfect. Also, note that tests within the same heading are not interchangeable in general (e.g. many have different distributional assumptions).

One Sample Tests / Paired Sample Tests #

One sample tests are typically used to assess whether a single sample was drawn from a specified distribution or a distribution with specified properties (e.g. zero mean).

Paired sample tests are often used to assess whether two samples were drawn from the same distribution; they differ from the independent sample tests below in that each observation in one sample is treated as paired with a closely-related observation in the other sample (e.g. when environmental factors are controlled between observations within a pair but not among pairs). They can also be interpreted or used as one-sample tests (e.g. tests on the mean or median of differences between paired observations).

Association/Correlation Tests #

These tests are often used to assess whether there is a relationship (e.g. linear) between paired observations in multiple samples or among the coordinates of multivariate observations.

These association tests and are to work with samples in the form of contingency tables. Supporting functions are available in scipy.stats.contingency .

Independent Sample Tests #

Independent sample tests are typically used to assess whether multiple samples were independently drawn from the same distribution or different distributions with a shared property (e.g. equal means).

Some tests are specifically for comparing two samples.

Others are generalized to multiple samples.

Resampling and Monte Carlo Methods #

The following functions can reproduce the p-value and confidence interval results of most of the functions above, and often produce accurate results in a wider variety of conditions. They can also be used to perform hypothesis tests and generate confidence intervals for custom statistics. This flexibility comes at the cost of greater computational requirements and stochastic results.

Instances of the following object can be passed into some hypothesis test functions to perform a resampling or Monte Carlo version of the hypothesis test.

Multiple Hypothesis Testing and Meta-Analysis #

These functions are for assessing the results of individual tests as a whole. Functions for performing specific multiple hypothesis tests (e.g. post hoc tests) are listed above.

The following functions are related to the tests above but do not belong in the above categories.

Quasi-Monte Carlo #

• scipy.stats.qmc.QMCEngine
• scipy.stats.qmc.Sobol
• scipy.stats.qmc.Halton
• scipy.stats.qmc.LatinHypercube
• scipy.stats.qmc.PoissonDisk
• scipy.stats.qmc.MultinomialQMC
• scipy.stats.qmc.MultivariateNormalQMC
• scipy.stats.qmc.discrepancy
• scipy.stats.qmc.geometric_discrepancy
• scipy.stats.qmc.update_discrepancy
• scipy.stats.qmc.scale

Contingency Tables #

• chi2_contingency
• relative_risk
• association
• expected_freq

Masked statistics functions #

• hdquantiles
• hdquantiles_sd
• idealfourths
• plotting_positions
• find_repeats
• trimmed_mean
• trimmed_mean_ci
• trimmed_std
• trimmed_var
• scoreatpercentile
• pointbiserialr
• kendalltau_seasonal
• siegelslopes
• theilslopes
• sen_seasonal_slopes
• ttest_1samp
• ttest_onesamp
• mannwhitneyu
• kruskalwallis
• friedmanchisquare
• brunnermunzel
• kurtosistest
• obrientransform
• trimmed_stde
• argstoarray
• count_tied_groups
• compare_medians_ms
• median_cihs
• mquantiles_cimj

Other statistical functionality #

Transformations #, statistical distances #.

• scipy.stats.sampling.NumericalInverseHermite
• scipy.stats.sampling.NumericalInversePolynomial
• scipy.stats.sampling.TransformedDensityRejection
• scipy.stats.sampling.SimpleRatioUniforms
• scipy.stats.sampling.RatioUniforms
• scipy.stats.sampling.DiscreteAliasUrn
• scipy.stats.sampling.DiscreteGuideTable
• scipy.stats.sampling.UNURANError
• FastGeneratorInversion
• scipy.stats.sampling.FastGeneratorInversion.evaluate_error
• scipy.stats.sampling.FastGeneratorInversion.ppf
• scipy.stats.sampling.FastGeneratorInversion.qrvs
• scipy.stats.sampling.FastGeneratorInversion.rvs
• scipy.stats.sampling.FastGeneratorInversion.support

Random variate generation / CDF Inversion #

Fitting / survival analysis #, directional statistical functions #, sensitivity analysis #, plot-tests #, univariate and multivariate kernel density estimation #, warnings / errors used in scipy.stats #, result classes used in scipy.stats #.

These classes are private, but they are included here because instances of them are returned by other statistical functions. User import and instantiation is not supported.

• scipy.stats._result_classes.RelativeRiskResult
• scipy.stats._result_classes.BinomTestResult
• scipy.stats._result_classes.TukeyHSDResult
• scipy.stats._result_classes.DunnettResult
• scipy.stats._result_classes.PearsonRResult
• scipy.stats._result_classes.FitResult
• scipy.stats._result_classes.OddsRatioResult
• scipy.stats._result_classes.TtestResult
• scipy.stats._result_classes.ECDFResult
• scipy.stats._result_classes.EmpiricalDistributionFunction

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• Energy and climate change: evidence and analysis

Green Homes Grant Local Authority Delivery (LAD) and Home Upgrade Grant (HUG) release, April 2024

Data to monitor the installation of energy efficiency measures in domestic properties via Green Homes Grant Local Authority Delivery (LAD) and Home Upgrade Grant (HUG) schemes in England.

Applies to England

Summary of the green homes grant local authority delivery (lad) and home upgrade grant (hug) statistics: april 2024, green homes grant local authority delivery (lad) and home upgrade grant (hug) release, april 2024 (excel).

MS Excel Spreadsheet , 4.02 MB

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Green Homes Grant Local Authority Delivery (LAD) and Home Upgrade Grant (HUG) release, April 2024 (ODS)

ODS , 1.89 MB

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This release includes measures installed under the  Green Homes Grant Local Authority Delivery (GHG LAD)  and  Home Upgrade Grant (HUG)  schemes.

The statistical release includes analysis on:

• measures installed
• homes upgraded
• installation rates
• installations by region, local authority and parliamentary constituency
• carbon, bill and energy savings
• changes in Energy Performance Certificate

As part of the scheme monitoring, the analysis is shown by geographical region. The scheme covers England only. Data provided in the monthly release is 2 months in arrears.

These statistics are provisional and are subject to future revisions.

The next monthly publication on measures installed to the end of March 2024 is due for release on 23 May 2024.

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