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Hypothesis Testing | A Step-by-Step Guide with Easy Examples

Published on November 8, 2019 by Rebecca Bevans . Revised on June 22, 2023.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics . It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories.

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 and discussion section.

Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps.

Table of contents

Step 1: state your null and alternate hypothesis, step 2: collect data, step 3: perform a statistical test, step 4: decide whether to reject or fail to reject your null hypothesis, step 5: present your findings, other interesting articles, frequently asked questions about hypothesis testing.

After developing your initial research hypothesis (the prediction that you want to investigate), it is important to restate it as a null (H o ) and alternate (H a ) hypothesis so that you can test it mathematically.

The alternate hypothesis is usually your initial hypothesis that predicts a relationship between variables. The null hypothesis is a prediction of no relationship between the variables you are interested in.

• H 0 : Men are, on average, not taller than women. H a : Men are, on average, taller than women.

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For a statistical test to be valid , it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in.

There are a variety of statistical tests available, but they are all based on the comparison of within-group variance (how spread out the data is within a category) versus between-group variance (how different the categories are from one another).

If the between-group variance is large enough that there is little or no overlap between groups, then your statistical test will reflect that by showing a low p -value . This means it is unlikely that the differences between these groups came about by chance.

Alternatively, if there is high within-group variance and low between-group variance, then your statistical test will reflect that with a high p -value. This means it is likely that any difference you measure between groups is due to chance.

Your choice of statistical test will be based on the type of variables and the level of measurement of your collected data .

• an estimate of the difference in average height between the two groups.
• a p -value showing how likely you are to see this difference if the null hypothesis of no difference is true.

Based on the outcome of your statistical test, you will have to decide whether to reject or fail to reject your null hypothesis.

In most cases you will use the p -value generated by your statistical test to guide your decision. And in most cases, your predetermined level of significance for rejecting the null hypothesis will be 0.05 – that is, when there is a less than 5% chance that you would see these results if the null hypothesis were true.

In some cases, researchers choose a more conservative level of significance, such as 0.01 (1%). This minimizes the risk of incorrectly rejecting the null hypothesis ( Type I error ).

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The results of hypothesis testing will be presented in the results and discussion sections of your research paper , dissertation or thesis .

In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p -value). In the discussion , you can discuss whether your initial hypothesis was supported by your results or not.

In the formal language of hypothesis testing, we talk about rejecting or failing to reject the null hypothesis. You will probably be asked to do this in your statistics assignments.

However, when presenting research results in academic papers we rarely talk this way. Instead, we go back to our alternate hypothesis (in this case, the hypothesis that men are on average taller than women) and state whether the result of our test did or did not support the alternate hypothesis.

If your null hypothesis was rejected, this result is interpreted as “supported the alternate hypothesis.”

These are superficial differences; you can see that they mean the same thing.

You might notice that we don’t say that we reject or fail to reject the alternate hypothesis . This is because hypothesis testing is not designed to prove or disprove anything. It is only designed to test whether a pattern we measure could have arisen spuriously, or by chance.

If we reject the null hypothesis based on our research (i.e., we find that it is unlikely that the pattern arose by chance), then we can say our test lends support to our hypothesis . But if the pattern does not pass our decision rule, meaning that it could have arisen by chance, then we say the test is inconsistent with our hypothesis .

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Normal distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

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• Indian J Crit Care Med
• v.23(Suppl 3); 2019 Sep

An Introduction to Statistics: Understanding Hypothesis Testing and Statistical Errors

Priya ranganathan.

1 Department of Anesthesiology, Critical Care and Pain, Tata Memorial Hospital, Mumbai, Maharashtra, India

2 Department of Surgical Oncology, Tata Memorial Centre, Mumbai, Maharashtra, India

The second article in this series on biostatistics covers the concepts of sample, population, research hypotheses and statistical errors.

How to cite this article

Ranganathan P, Pramesh CS. An Introduction to Statistics: Understanding Hypothesis Testing and Statistical Errors. Indian J Crit Care Med 2019;23(Suppl 3):S230–S231.

Two papers quoted in this issue of the Indian Journal of Critical Care Medicine report. The results of studies aim to prove that a new intervention is better than (superior to) an existing treatment. In the ABLE study, the investigators wanted to show that transfusion of fresh red blood cells would be superior to standard-issue red cells in reducing 90-day mortality in ICU patients. 1 The PROPPR study was designed to prove that transfusion of a lower ratio of plasma and platelets to red cells would be superior to a higher ratio in decreasing 24-hour and 30-day mortality in critically ill patients. 2 These studies are known as superiority studies (as opposed to noninferiority or equivalence studies which will be discussed in a subsequent article).

SAMPLE VERSUS POPULATION

A sample represents a group of participants selected from the entire population. Since studies cannot be carried out on entire populations, researchers choose samples, which are representative of the population. This is similar to walking into a grocery store and examining a few grains of rice or wheat before purchasing an entire bag; we assume that the few grains that we select (the sample) are representative of the entire sack of grains (the population).

The results of the study are then extrapolated to generate inferences about the population. We do this using a process known as hypothesis testing. This means that the results of the study may not always be identical to the results we would expect to find in the population; i.e., there is the possibility that the study results may be erroneous.

HYPOTHESIS TESTING

A clinical trial begins with an assumption or belief, and then proceeds to either prove or disprove this assumption. In statistical terms, this belief or assumption is known as a hypothesis. Counterintuitively, what the researcher believes in (or is trying to prove) is called the “alternate” hypothesis, and the opposite is called the “null” hypothesis; every study has a null hypothesis and an alternate hypothesis. For superiority studies, the alternate hypothesis states that one treatment (usually the new or experimental treatment) is superior to the other; the null hypothesis states that there is no difference between the treatments (the treatments are equal). For example, in the ABLE study, we start by stating the null hypothesis—there is no difference in mortality between groups receiving fresh RBCs and standard-issue RBCs. We then state the alternate hypothesis—There is a difference between groups receiving fresh RBCs and standard-issue RBCs. It is important to note that we have stated that the groups are different, without specifying which group will be better than the other. This is known as a two-tailed hypothesis and it allows us to test for superiority on either side (using a two-sided test). This is because, when we start a study, we are not 100% certain that the new treatment can only be better than the standard treatment—it could be worse, and if it is so, the study should pick it up as well. One tailed hypothesis and one-sided statistical testing is done for non-inferiority studies, which will be discussed in a subsequent paper in this series.

STATISTICAL ERRORS

There are two possibilities to consider when interpreting the results of a superiority study. The first possibility is that there is truly no difference between the treatments but the study finds that they are different. This is called a Type-1 error or false-positive error or alpha error. This means falsely rejecting the null hypothesis.

The second possibility is that there is a difference between the treatments and the study does not pick up this difference. This is called a Type 2 error or false-negative error or beta error. This means falsely accepting the null hypothesis.

The power of the study is the ability to detect a difference between groups and is the converse of the beta error; i.e., power = 1-beta error. Alpha and beta errors are finalized when the protocol is written and form the basis for sample size calculation for the study. In an ideal world, we would not like any error in the results of our study; however, we would need to do the study in the entire population (infinite sample size) to be able to get a 0% alpha and beta error. These two errors enable us to do studies with realistic sample sizes, with the compromise that there is a small possibility that the results may not always reflect the truth. The basis for this will be discussed in a subsequent paper in this series dealing with sample size calculation.

Conventionally, type 1 or alpha error is set at 5%. This means, that at the end of the study, if there is a difference between groups, we want to be 95% certain that this is a true difference and allow only a 5% probability that this difference has occurred by chance (false positive). Type 2 or beta error is usually set between 10% and 20%; therefore, the power of the study is 90% or 80%. This means that if there is a difference between groups, we want to be 80% (or 90%) certain that the study will detect that difference. For example, in the ABLE study, sample size was calculated with a type 1 error of 5% (two-sided) and power of 90% (type 2 error of 10%) (1).

Table 1 gives a summary of the two types of statistical errors with an example

Statistical errors

In the next article in this series, we will look at the meaning and interpretation of ‘ p ’ value and confidence intervals for hypothesis testing.

Source of support: Nil

Conflict of interest: None

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14.3: Design Research Hypotheses and Experiment

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• Page ID 130572

• Maurice A. Geraghty
• De Anza College

After developing a general question and having some sense of the data that is available or that is collected, we then design and an experiment and a set of hypotheses . 

Hypotheses and Hypothesis

Testing For purposes of testing, we need to design hypotheses that are statements about population parameters.  Some examples of hypotheses:

At least 20% of juvenile offenders are caught and sentenced to prison.

• The mean monthly income for college graduates is over $5000.
• The mean standardized test score for schools in Cupertino is the same as the mean scores for schools in Los Altos.
• The lung cancer rates in California are lower than the rates in Texas.
• The standard deviation of the New York Stock Exchange today is greater than 10 percentage points per year.

These same hypotheses could be written in symbolic notation:

• \(p \geq 0.20\)
• \(\mu>5000\)
• \(\mu_{1}=\mu_{2}\)
• \(p_{1}<p_{2}\)
• \(\sigma>10\)

Hypothesis Testing is a procedure, based on sample evidence and probability theory, used to determine whether the hypothesis is a reasonable statement and should not be rejected, or is unreasonable and should be rejected. This hypothesis that is tested is called the Null Hypothesis   and is designated by the symbol Ho. If the Null Hypothesis is unreasonable and needs to be rejected, then the research supports an Alternative Hypothesis designated by the symbol Ha.

Definition: Null Hypothesis (\(H_o\))

A statement about the value of a population parameter that is assumed to be true for the purpose of testing.

Definition: Alternative Hypothesis (\(H_a\))

A statement about the value of a population parameter that is assumed to be true if the Null Hypothesis is rejected during testing.

From these definitions it is clear that the Alternative Hypothesis will necessarily contradict the Null Hypothesis; both cannot be true at the same time. Some other important points about hypotheses:

• Hypotheses must be statements about population parameters, never about sample statistics.
• In most hypotheses tests, equality (\(=, \leq, \geq\)) will be associated with the Null Hypothesis while non‐equality (\(\neq,<,>\)) will be associated with the Alternative Hypothesis.
• It is the Null Hypothesis that is always tested in attempt to “disprove” it and support the Alternative Hypothesis. This process is analogous in concept to a “proof by contradiction” in Mathematics or Logic, but supporting a hypothesis with a level of confidence is not the same as an absolute mathematical proof.

Examples of Null and Alternative Hypotheses:

• \(H_{o}: p \leq 0.20 \qquad H_{a}: p>0.20\)
• \(H_{o}: \mu \leq 5000 \qquad H_{a}: \mu>5000\)
• \(H_{o}: \mu_{1}=\mu_{2} \qquad H_{a}: \mu_{1} \neq \mu_{2}\)
• \(H_{o}: p_{1} \geq p_{2} \qquad H_{a}: p_{1}<p_{2}\)
• \(H_{o}: \sigma \leq 10 \qquad H_{a}: \sigma>10\)

Statistical Model and Test Statistic

To test a hypothesis we need to use a statistical model that describes the behavior for data and the type of population parameter being tested.  Because of the Central Limit Theorem, many statistical models are from the Normal Family, most importantly the \(Z, t, \chi^{2}\), and \(F\) distributions. Other models that are used when the Central Limit Theorem is not appropriate are called non‐parametric Models and will not be discussed here.

Each chosen model has requirements of the data called model assumptions that should be checked for appropriateness. For example, many models require that the sample mean have approximately a Normal Distribution, something that may not be true for some smaller or heavily skewed data sets.

Once the model is chosen, we can then determine a test statistic , a value derived from the data that is used to decide whether to reject or fail to reject the Null Hypothesis.

Examples of Statistical Models and Test Statistics

Errors in Decision Making

Whenever we make a decision or support a position, there is always a chance we make the wrong choice. The hypothesis testing process requires us to either to reject the Null Hypothesis and support the Alternative Hypothesis or fail to reject the Null Hypothesis. This creates the possibility of two types of error:

• Type I Error Rejecting the null hypothesis when it is actually true.
• Type II Error Failing to reject the null hypothesis when it is actually false.

In designing hypothesis tests, we need to carefully consider the probability of making either one of these errors.  

Example: Pharmaceutical research

Recall the two news stories discussed earlier. In the first story, a drug company marketed a suppository that was later found to be ineffective (and often dangerous) in treatment. Before marketing the drug, the company determined that the drug was effective in treatment, meaning that the company rejected a Null Hypothesis that the suppository had no effect on the disease. This is an example of Type I error.

In the second story, research was abandoned when the testing showed Interferon was ineffective in treating a lung disease. The company in this case failed to reject a Null Hypothesis that the drug was ineffective. What if the drug really was effective? Did the company make Type II error? Possibly, but since the drug was never marketed, we have no way of knowing the truth.

These stories highlight the problem of statistical research: errors can be analyzed using probability models, but there is often no way of identifying specific errors. For example, there are unknown innocent people in prison right now because a jury made Type I error in wrongfully convicting defendants. We must be open to the possibility of modification or rejection of currently accepted theories when new data is discovered.   

In designing an experiment, we set a maximum probability of making Type I error. This probability is called the level of significance or significance level of the test and is designated by the Greek letter \(\alpha\) , read as alpha.   The analysis of Type II error is more problematic since there are many possible values that would satisfy the Alternative Hypothesis. For a specific value of the Alternative Hypothesis, the design probability of making Type II error is called Beta (\(\beta\)) which will be analyzed in detail later in this section.

Critical Value and Rejection Region

Once the significance level of the test is chosen, it is then possible to find the region(s) of the probability distribution function of the test statistic that would allow the Null Hypothesis to be rejected. This is called the Rejection Region , and the boundary between the Rejection Region and the “Fail to Reject” is called the Critical Value .

There can be more than one critical value and rejection region. What matters is that the total area of the rejection region equals the significance level \(\alpha\).

One and Two tailed Tests

A test is one‐tailed when the Alternative Hypothesis, \(H_{a}\), states a direction, such as:

\(H_{o}\): The mean income of females is less than or equal to the mean income of males.   

\(H_{a}\): The mean income of females is greater than that of males.

Since equality is usually part of the Null Hypothesis, it is the Alternative Hypothesis which determines which tail to test.  

A test is two‐tailed when no direction is specified in the alternate hypothesis Ha , such as:

\(H_{o}\): The mean income of females is equal to the mean income of males.

\(H_{a}\): The mean income of females is not equal to the mean income of the males.  

In a two tailed‐test, the significance level is split into two parts since there are two rejection regions. In hypothesis testing, in which the statistical model is symmetrical ( eg: the Standard Normal \(Z\) or Student’s t distribution) these two regions would be equal. There is a relationship between a  confidence interval and a two‐tailed test: if the level of confidence for a confidence interval is equal to \(1-\alpha\), where \(\alpha\) is the significance level of the two‐tailed test, the critical values would be the same. 

Here are some examples for testing the mean \(\mu\) against a hypothesized value \(\mu_{0}\):

\(H_{a}: \mu>\mu_{0}\) means test the upper tail and is also called a right‐tailed test.

\(H_{a}: \mu<\mu_{0}\) means test the  lower tail and is also called a left‐tailed test.

\(H_{a}: \mu \neq \mu_{0}\) means test both tails.

Deciding when to conduct a one or two‐tailed test is often controversial and many authorities even go so far as to say that only two‐tailed tests should be conducted. Ultimately, the decision depends on the wording of the problem. If we want to show that a new diet reduces weight, we would conduct a lower tailed test, since we don’t care if the diet causes weight gain. If instead, we wanted to determine if mean crime rate in California was different from the mean crime rate in the United States, we would run a two‐tailed test, since different implies greater than or less than.

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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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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.

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Statistics and probability

Unit 1: analyzing categorical data, unit 2: displaying and comparing quantitative data, unit 3: summarizing quantitative data, unit 4: modeling data distributions, unit 5: exploring bivariate numerical data, unit 6: study design, unit 7: probability, unit 8: counting, permutations, and combinations, unit 9: random variables, unit 10: sampling distributions, unit 11: confidence intervals, unit 12: significance tests (hypothesis testing), unit 13: two-sample inference for the difference between groups, unit 14: inference for categorical data (chi-square tests), unit 15: advanced regression (inference and transforming), unit 16: analysis of variance (anova).

Teach yourself statistics

Hypothesis Test for a Mean

This lesson explains how to conduct a hypothesis test of a mean, when the following conditions are met:

• The sampling method is simple random sampling .
• The sampling distribution is normal or nearly normal.

Generally, the sampling distribution will be approximately normally distributed if any of the following conditions apply.

• The population distribution is normal.
• The population distribution is symmetric , unimodal , without outliers , and the sample size is 15 or less.
• The population distribution is moderately skewed , unimodal, without outliers, and the sample size is between 16 and 40.
• The sample size is greater than 40, without outliers.

This approach consists of four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results.

State the Hypotheses

Every hypothesis test requires the analyst to state a null hypothesis and an alternative hypothesis . The hypotheses are stated in such a way that they are mutually exclusive. That is, if one is true, the other must be false; and vice versa.

The table below shows three sets of hypotheses. Each makes a statement about how the population mean μ is related to a specified value M . (In the table, the symbol ≠ means " not equal to ".)

The first set of hypotheses (Set 1) is an example of a two-tailed test , since an extreme value on either side of the sampling distribution would cause a researcher to reject the null hypothesis. The other two sets of hypotheses (Sets 2 and 3) are one-tailed tests , since an extreme value on only one side of the sampling distribution would cause a researcher to reject the null hypothesis.

Formulate an Analysis Plan

The analysis plan describes how to use sample data to accept or reject the null hypothesis. It should specify the following elements.

• Significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.
• Test method. Use the one-sample t-test to determine whether the hypothesized mean differs significantly from the observed sample mean.

Analyze Sample Data

Using sample data, conduct a one-sample t-test. This involves finding the standard error, degrees of freedom, test statistic, and the P-value associated with the test statistic.

SE = s * sqrt{ ( 1/n ) * [ ( N - n ) / ( N - 1 ) ] }

SE = s / sqrt( n )

• Degrees of freedom. The degrees of freedom (DF) is equal to the sample size (n) minus one. Thus, DF = n - 1.

t = ( x - μ) / SE

• P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a t statistic, use the t Distribution Calculator to assess the probability associated with the t statistic, given the degrees of freedom computed above. (See sample problems at the end of this lesson for examples of how this is done.)

Sample Size Calculator

As you probably noticed, the process of hypothesis testing can be complex. When you need to test a hypothesis about a mean score, consider using the Sample Size Calculator. The calculator is fairly easy to use, and it is free. You can find the Sample Size Calculator in Stat Trek's main menu under the Stat Tools tab. Or you can tap the button below.

Interpret Results

If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level , and rejecting the null hypothesis when the P-value is less than the significance level.

Test Your Understanding

In this section, two sample problems illustrate how to conduct a hypothesis test of a mean score. The first problem involves a two-tailed test; the second problem, a one-tailed test.

Problem 1: Two-Tailed Test

An inventor has developed a new, energy-efficient lawn mower engine. He claims that the engine will run continuously for 5 hours (300 minutes) on a single gallon of regular gasoline. From his stock of 2000 engines, the inventor selects a simple random sample of 50 engines for testing. The engines run for an average of 295 minutes, with a standard deviation of 20 minutes. Test the null hypothesis that the mean run time is 300 minutes against the alternative hypothesis that the mean run time is not 300 minutes. Use a 0.05 level of significance. (Assume that run times for the population of engines are normally distributed.)

Solution: The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:

Null hypothesis: μ = 300

Alternative hypothesis: μ ≠ 300

• Formulate an analysis plan . For this analysis, the significance level is 0.05. The test method is a one-sample t-test .

SE = s / sqrt(n) = 20 / sqrt(50) = 20/7.07 = 2.83

DF = n - 1 = 50 - 1 = 49

t = ( x - μ) / SE = (295 - 300)/2.83 = -1.77

where s is the standard deviation of the sample, x is the sample mean, μ is the hypothesized population mean, and n is the sample size.

Since we have a two-tailed test , the P-value is the probability that the t statistic having 49 degrees of freedom is less than -1.77 or greater than 1.77. We use the t Distribution Calculator to find P(t < -1.77) is about 0.04.

• If you enter 1.77 as the sample mean in the t Distribution Calculator, you will find the that the P(t < 1.77) is about 0.04. Therefore, P(t >  1.77) is 1 minus 0.96 or 0.04. Thus, the P-value = 0.04 + 0.04 = 0.08.
• Interpret results . Since the P-value (0.08) is greater than the significance level (0.05), we cannot reject the null hypothesis.

Note: If you use this approach on an exam, you may also want to mention why this approach is appropriate. Specifically, the approach is appropriate because the sampling method was simple random sampling, the population was normally distributed, and the sample size was small relative to the population size (less than 5%).

Problem 2: One-Tailed Test

Bon Air Elementary School has 1000 students. The principal of the school thinks that the average IQ of students at Bon Air is at least 110. To prove her point, she administers an IQ test to 20 randomly selected students. Among the sampled students, the average IQ is 108 with a standard deviation of 10. Based on these results, should the principal accept or reject her original hypothesis? Assume a significance level of 0.01. (Assume that test scores in the population of engines are normally distributed.)

Null hypothesis: μ >= 110

Alternative hypothesis: μ < 110

• Formulate an analysis plan . For this analysis, the significance level is 0.01. The test method is a one-sample t-test .

SE = s / sqrt(n) = 10 / sqrt(20) = 10/4.472 = 2.236

DF = n - 1 = 20 - 1 = 19

t = ( x - μ) / SE = (108 - 110)/2.236 = -0.894

Here is the logic of the analysis: Given the alternative hypothesis (μ < 110), we want to know whether the observed sample mean is small enough to cause us to reject the null hypothesis.

The observed sample mean produced a t statistic test statistic of -0.894. We use the t Distribution Calculator to find P(t < -0.894) is about 0.19.

• This means we would expect to find a sample mean of 108 or smaller in 19 percent of our samples, if the true population IQ were 110. Thus the P-value in this analysis is 0.19.
• Interpret results . Since the P-value (0.19) is greater than the significance level (0.01), we cannot reject the null hypothesis.

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3.1: The Fundamentals of Hypothesis Testing

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• Page ID 2883

• Diane Kiernan
• SUNY College of Environmental Science and Forestry via OpenSUNY

The previous two chapters introduced methods for organizing and summarizing sample data, and using sample statistics to estimate population parameters. This chapter introduces the next major topic of inferential statistics: hypothesis testing.

A hypothesis is a statement or claim about a property of a population.

The Fundamentals of Hypothesis Testing

When conducting scientific research, typically there is some known information, perhaps from some past work or from a long accepted idea. We want to test whether this claim is believable. This is the basic idea behind a hypothesis test:

• State what we think is true.
• Quantify how confident we are about our claim.
• Use sample statistics to make inferences about population parameters.

For example, past research tells us that the average life span for a hummingbird is about four years. You have been studying the hummingbirds in the southeastern United States and find a sample mean lifespan of 4.8 years. Should you reject the known or accepted information in favor of your results? How confident are you in your estimate? At what point would you say that there is enough evidence to reject the known information and support your alternative claim? How far from the known mean of four years can the sample mean be before we reject the idea that the average lifespan of a hummingbird is four years?

Definition: hypothesis testing

Hypothesis testing is a procedure, based on sample evidence and probability, used to test claims regarding a characteristic of a population.

A hypothesis is a claim or statement about a characteristic of a population of interest to us. A hypothesis test is a way for us to use our sample statistics to test a specific claim.

Example \(\PageIndex{1}\):

The population mean weight is known to be 157 lb. We want to test the claim that the mean weight has increased.

Example \(\PageIndex{2}\):

Two years ago, the proportion of infected plants was 37%. We believe that a treatment has helped, and we want to test the claim that there has been a reduction in the proportion of infected plants.

Components of a Formal Hypothesis Test

The null hypothesis is a statement about the value of a population parameter, such as the population mean (µ) or the population proportion ( p ). It contains the condition of equality and is denoted as H 0 (H-naught).

H 0 : µ = 157 or H0 : p = 0.37

The alternative hypothesis is the claim to be tested, the opposite of the null hypothesis. It contains the value of the parameter that we consider plausible and is denoted as H 1 .

H 1 : µ > 157 or H1 : p ≠ 0.37

The test statistic is a value computed from the sample data that is used in making a decision about the rejection of the null hypothesis. The test statistic converts the sample mean ( x̄ ) or sample proportion ( p̂ ) to a Z- or t-score under the assumption that the null hypothesis is true. It is used to decide whether the difference between the sample statistic and the hypothesized claim is significant.

The p-value is the area under the curve to the left or right of the test statistic. It is compared to the level of significance (α).

The critical value is the value that defines the rejection zone (the test statistic values that would lead to rejection of the null hypothesis). It is defined by the level of significance.

The level of significance (α) is the probability that the test statistic will fall into the critical region when the null hypothesis is true. This level is set by the researcher.

The conclusion is the final decision of the hypothesis test. The conclusion must always be clearly stated, communicating the decision based on the components of the test. It is important to realize that we never prove or accept the null hypothesis. We are merely saying that the sample evidence is not strong enough to warrant the rejection of the null hypothesis. The conclusion is made up of two parts:

1) Reject or fail to reject the null hypothesis, and 2) there is or is not enough evidence to support the alternative claim.

Option 1) Reject the null hypothesis (H0). This means that you have enough statistical evidence to support the alternative claim (H 1 ).

Option 2) Fail to reject the null hypothesis (H0). This means that you do NOT have enough evidence to support the alternative claim (H 1 ).

Another way to think about hypothesis testing is to compare it to the US justice system. A defendant is innocent until proven guilty (Null hypothesis—innocent). The prosecuting attorney tries to prove that the defendant is guilty (Alternative hypothesis—guilty). There are two possible conclusions that the jury can reach. First, the defendant is guilty (Reject the null hypothesis). Second, the defendant is not guilty (Fail to reject the null hypothesis). This is NOT the same thing as saying the defendant is innocent! In the first case, the prosecutor had enough evidence to reject the null hypothesis (innocent) and support the alternative claim (guilty). In the second case, the prosecutor did NOT have enough evidence to reject the null hypothesis (innocent) and support the alternative claim of guilty.

The Null and Alternative Hypotheses

There are three different pairs of null and alternative hypotheses:

Table \(PageIndex{1}\): The rejection zone for a two-sided hypothesis test.

where c is some known value.

A Two-sided Test

This tests whether the population parameter is equal to, versus not equal to, some specific value.

Ho: μ = 12 vs. H 1 : μ ≠ 12

The critical region is divided equally into the two tails and the critical values are ± values that define the rejection zones.

Example \(\PageIndex{3}\):

A forester studying diameter growth of red pine believes that the mean diameter growth will be different if a fertilization treatment is applied to the stand.

• Ho: μ = 1.2 in./ year
• H 1 : μ ≠ 1.2 in./ year

This is a two-sided question, as the forester doesn’t state whether population mean diameter growth will increase or decrease.

A Right-sided Test

This tests whether the population parameter is equal to, versus greater than, some specific value.

Ho: μ = 12 vs. H 1 : μ > 12

The critical region is in the right tail and the critical value is a positive value that defines the rejection zone.

Example \(\PageIndex{4}\):

A biologist believes that there has been an increase in the mean number of lakes infected with milfoil, an invasive species, since the last study five years ago.

• Ho: μ = 15 lakes
• H1: μ >15 lakes

This is a right-sided question, as the biologist believes that there has been an increase in population mean number of infected lakes.

A Left-sided Test

This tests whether the population parameter is equal to, versus less than, some specific value.

Ho: μ = 12 vs. H 1 : μ < 12

The critical region is in the left tail and the critical value is a negative value that defines the rejection zone.

Example \(\PageIndex{5}\):

A scientist’s research indicates that there has been a change in the proportion of people who support certain environmental policies. He wants to test the claim that there has been a reduction in the proportion of people who support these policies.

• Ho: p = 0.57
• H 1 : p < 0.57

This is a left-sided question, as the scientist believes that there has been a reduction in the true population proportion.

Statistically Significant

When the observed results (the sample statistics) are unlikely (a low probability) under the assumption that the null hypothesis is true, we say that the result is statistically significant, and we reject the null hypothesis. This result depends on the level of significance, the sample statistic, sample size, and whether it is a one- or two-sided alternative hypothesis.

Types of Errors

When testing, we arrive at a conclusion of rejecting the null hypothesis or failing to reject the null hypothesis. Such conclusions are sometimes correct and sometimes incorrect (even when we have followed all the correct procedures). We use incomplete sample data to reach a conclusion and there is always the possibility of reaching the wrong conclusion. There are four possible conclusions to reach from hypothesis testing. Of the four possible outcomes, two are correct and two are NOT correct.

Table \(\PageIndex{2}\). Possible outcomes from a hypothesis test.

A Type I error is when we reject the null hypothesis when it is true. The symbol α (alpha) is used to represent Type I errors. This is the same alpha we use as the level of significance. By setting alpha as low as reasonably possible, we try to control the Type I error through the level of significance.

A Type II error is when we fail to reject the null hypothesis when it is false. The symbol β(beta) is used to represent Type II errors.

In general, Type I errors are considered more serious. One step in the hypothesis test procedure involves selecting the significance level ( α ), which is the probability of rejecting the null hypothesis when it is correct. So the researcher can select the level of significance that minimizes Type I errors. However, there is a mathematical relationship between α, β, and n (sample size).

• As α increases, β decreases
• As α decreases, β increases
• As sample size increases (n), both α and β decrease

The natural inclination is to select the smallest possible value for α, thinking to minimize the possibility of causing a Type I error. Unfortunately, this forces an increase in Type II errors. By making the rejection zone too small, you may fail to reject the null hypothesis, when, in fact, it is false. Typically, we select the best sample size and level of significance, automatically setting β.

Power of the Test

A Type II error (β) is the probability of failing to reject a false null hypothesis. It follows that 1-β is the probability of rejecting a false null hypothesis. This probability is identified as the power of the test, and is often used to gauge the test’s effectiveness in recognizing that a null hypothesis is false.

Definition: power of the test

The probability that at a fixed level α significance test will reject H0, when a particular alternative value of the parameter is true is called the power of the test.

Power is also directly linked to sample size. For example, suppose the null hypothesis is that the mean fish weight is 8.7 lb. Given sample data, a level of significance of 5%, and an alternative weight of 9.2 lb., we can compute the power of the test to reject μ = 8.7 lb. If we have a small sample size, the power will be low. However, increasing the sample size will increase the power of the test. Increasing the level of significance will also increase power. A 5% test of significance will have a greater chance of rejecting the null hypothesis than a 1% test because the strength of evidence required for the rejection is less. Decreasing the standard deviation has the same effect as increasing the sample size: there is more information about μ.

What is Hypothesis Testing in Statistics? Types and Examples

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Hypothesis testing in statistics involves testing an assumption about a population parameter using sample data. Learners can download Hypothesis Testing PDF to get instant access to all information!

What exactly is hypothesis testing, and how does it work in statistics? Can I find practical examples and understand the different types from this blog?

Hypothesis Testing : Ever wonder how researchers determine if a new medicine actually works or if a new marketing campaign effectively drives sales? They use hypothesis testing! It is at the core of how scientific studies, business experiments and surveys determine if their results are statistically significant or just due to chance.

Hypothesis testing allows us to make evidence-based decisions by quantifying uncertainty and providing a structured process to make data-driven conclusions rather than guessing. In this post, we will discuss hypothesis testing types, examples, and processes!

Table of Contents

Hypothesis Testing

Hypothesis testing is a statistical method used to evaluate the validity of a hypothesis using sample data. It involves assessing whether observed data provide enough evidence to reject a specific hypothesis about a population parameter. 

Hypothesis Testing in Data Science

Hypothesis testing in data science is a statistical method used to evaluate two mutually exclusive population statements based on sample data. The primary goal is to determine which statement is more supported by the observed data.

Hypothesis testing assists in supporting the certainty of findings in research and data science projects. This statistical inference aids in making decisions about population parameters using sample data. For those who are looking to deepen their knowledge in data science and expand their skillset, we highly recommend checking out Master Generative AI: Data Science Course by Physics Wallah .

Also Read: What is Encapsulation Explain in Details

What is the Hypothesis Testing Procedure in Data Science?

The hypothesis testing procedure in data science involves a structured approach to evaluating hypotheses using statistical methods. Here’s a step-by-step breakdown of the typical procedure:

1) State the Hypotheses:

• Null Hypothesis (H0): This is the default assumption or a statement of no effect or difference. It represents what you aim to test against.
• Alternative Hypothesis (Ha): This is the opposite of the null hypothesis and represents what you want to prove.

2) Choose a Significance Level (α):

• Decide on a threshold (commonly 0.05) beyond which you will reject the null hypothesis. This is your significance level.

3) Select the Appropriate Test:

• Depending on your data type (e.g., continuous, categorical) and the nature of your research question, choose the appropriate statistical test (e.g., t-test, chi-square test, ANOVA, etc.).

4) Collect Data:

• Gather data from your sample or population, ensuring that it’s representative and sufficiently large (or as per your experimental design).

5)Compute the Test Statistic:

• Using your data and the chosen statistical test, compute the test statistic that summarizes the evidence against the null hypothesis.

6) Determine the Critical Value or P-value:

• Based on your significance level and the test statistic’s distribution, determine the critical value from a statistical table or compute the p-value.

7) Make a Decision:

• If the p-value is less than α: Reject the null hypothesis.
• If the p-value is greater than or equal to α: Fail to reject the null hypothesis.

8) Draw Conclusions:

• Based on your decision, draw conclusions about your research question or hypothesis. Remember, failing to reject the null hypothesis doesn’t prove it true; it merely suggests that you don’t have sufficient evidence to reject it.

9) Report Findings:

• Document your findings, including the test statistic, p-value, conclusion, and any other relevant details. Ensure clarity so that others can understand and potentially replicate your analysis.

Also Read: Binary Search Algorithm

How Hypothesis Testing Works?

Hypothesis testing is a fundamental concept in statistics that aids analysts in making informed decisions based on sample data about a larger population. The process involves setting up two contrasting hypotheses, the null hypothesis and the alternative hypothesis, and then using statistical methods to determine which hypothesis provides a more plausible explanation for the observed data.

The Core Principles:

• The Null Hypothesis (H0): This serves as the default assumption or status quo. Typically, it posits that there is no effect or no difference, often represented by an equality statement regarding population parameters. For instance, it might state that a new drug’s effect is no different from a placebo.
• The Alternative Hypothesis (H1 or Ha): This is the counter assumption or what researchers aim to prove. It’s the opposite of the null hypothesis, indicating that there is an effect, a change, or a difference in the population parameters. Using the drug example, the alternative hypothesis would suggest that the new drug has a different effect than the placebo.

Testing the Hypotheses:

Once these hypotheses are established, analysts gather data from a sample and conduct statistical tests. The objective is to determine whether the observed results are statistically significant enough to reject the null hypothesis in favor of the alternative.

Examples to Clarify the Concept:

• Null Hypothesis (H0): The sanitizer’s average efficacy is 95%.
• By conducting tests, if evidence suggests that the sanitizer’s efficacy is significantly less than 95%, we reject the null hypothesis.
• Null Hypothesis (H0): The coin is fair, meaning the probability of heads and tails is equal.
• Through experimental trials, if results consistently show a skewed outcome, indicating a significantly different probability for heads and tails, the null hypothesis might be rejected.

What are the 3 types of Hypothesis Test?

Hypothesis testing is a cornerstone in statistical analysis, providing a framework to evaluate the validity of assumptions or claims made about a population based on sample data. Within this framework, several specific tests are utilized based on the nature of the data and the question at hand. Here’s a closer look at the three fundamental types of hypothesis tests:

The z-test is a statistical method primarily employed when comparing means from two datasets, particularly when the population standard deviation is known. Its main objective is to ascertain if the means are statistically equivalent. 

A crucial prerequisite for the z-test is that the sample size should be relatively large, typically 30 data points or more. This test aids researchers and analysts in determining the significance of a relationship or discovery, especially in scenarios where the data’s characteristics align with the assumptions of the z-test.

The t-test is a versatile statistical tool used extensively in research and various fields to compare means between two groups. It’s particularly valuable when the population standard deviation is unknown or when dealing with smaller sample sizes. 

By evaluating the means of two groups, the t-test helps ascertain if a particular treatment, intervention, or variable significantly impacts the population under study. Its flexibility and robustness make it a go-to method in scenarios ranging from medical research to business analytics.

3. Chi-Square Test:

The Chi-Square test stands distinct from the previous tests, primarily focusing on categorical data rather than means. This statistical test is instrumental when analyzing categorical variables to determine if observed data aligns with expected outcomes as posited by the null hypothesis. 

By assessing the differences between observed and expected frequencies within categorical data, the Chi-Square test offers insights into whether discrepancies are statistically significant. Whether used in social sciences to evaluate survey responses or in quality control to assess product defects, the Chi-Square test remains pivotal for hypothesis testing in diverse scenarios.

Also Read: Python vs Java: Which is Best for Machine learning algorithm

Hypothesis Testing in Statistics

Hypothesis testing is a fundamental concept in statistics used to make decisions or inferences about a population based on a sample of data. The process involves setting up two competing hypotheses, the null hypothesis H 0​ and the alternative hypothesis H 1​. 

Through various statistical tests, such as the t-test, z-test, or Chi-square test, analysts evaluate sample data to determine whether there’s enough evidence to reject the null hypothesis in favor of the alternative. The aim is to draw conclusions about population parameters or to test theories, claims, or hypotheses.

Hypothesis Testing in Research

In research, hypothesis testing serves as a structured approach to validate or refute theories or claims. Researchers formulate a clear hypothesis based on existing literature or preliminary observations. They then collect data through experiments, surveys, or observational studies. 

Using statistical methods, researchers analyze this data to determine if there’s sufficient evidence to reject the null hypothesis. By doing so, they can draw meaningful conclusions, make predictions, or recommend actions based on empirical evidence rather than mere speculation.

Hypothesis Testing in R

R, a powerful programming language and environment for statistical computing and graphics, offers a wide array of functions and packages specifically designed for hypothesis testing. Here’s how hypothesis testing is conducted in R:

• Data Collection : Before conducting any test, you need to gather your data and ensure it’s appropriately structured in R.
• Choose the Right Test : Depending on your research question and data type, select the appropriate hypothesis test. For instance, use the t.test() function for a t-test or chisq.test() for a Chi-square test.
• Set Hypotheses : Define your null and alternative hypotheses. Using R’s syntax, you can specify these hypotheses and run the corresponding test.
• Execute the Test : Utilize built-in functions in R to perform the hypothesis test on your data. For instance, if you want to compare two means, you can use the t.test() function, providing the necessary arguments like the data vectors and type of t-test (one-sample, two-sample, paired, etc.).
• Interpret Results : Once the test is executed, R will provide output, including test statistics, p-values, and confidence intervals. Based on these results and a predetermined significance level (often 0.05), you can decide whether to reject the null hypothesis.
• Visualization : R’s graphical capabilities allow users to visualize data distributions, confidence intervals, or test statistics, aiding in the interpretation and presentation of results.

Hypothesis testing is an integral part of statistics and research, offering a systematic approach to validate hypotheses. Leveraging R’s capabilities, researchers and analysts can efficiently conduct and interpret various hypothesis tests, ensuring robust and reliable conclusions from their data.

Do Data Scientists do Hypothesis Testing?

Yes, data scientists frequently engage in hypothesis testing as part of their analytical toolkit. Hypothesis testing is a foundational statistical technique used to make data-driven decisions, validate assumptions, and draw conclusions from data. Here’s how data scientists utilize hypothesis testing:

• Validating Assumptions : Before diving into complex analyses or building predictive models, data scientists often need to verify certain assumptions about the data. Hypothesis testing provides a structured approach to test these assumptions, ensuring that subsequent analyses or models are valid.
• Feature Selection : In machine learning and predictive modeling, data scientists use hypothesis tests to determine which features (or variables) are most relevant or significant in predicting a particular outcome. By testing hypotheses related to feature importance or correlation, they can streamline the modeling process and enhance prediction accuracy.
• A/B Testing : A/B testing is a common technique in marketing, product development, and user experience design. Data scientists employ hypothesis testing to compare two versions (A and B) of a product, feature, or marketing strategy to determine which performs better in terms of a specified metric (e.g., conversion rate, user engagement).
• Research and Exploration : In exploratory data analysis (EDA) or when investigating specific research questions, data scientists formulate hypotheses to test certain relationships or patterns within the data. By conducting hypothesis tests, they can validate these relationships, uncover insights, and drive data-driven decision-making.
• Model Evaluation : After building machine learning or statistical models, data scientists use hypothesis testing to evaluate the model’s performance, assess its predictive power, or compare different models. For instance, hypothesis tests like the t-test or F-test can help determine if a new model significantly outperforms an existing one based on certain metrics.
• Business Decision-making : Beyond technical analyses, data scientists employ hypothesis testing to support business decisions. Whether it’s evaluating the effectiveness of a marketing campaign, assessing customer preferences, or optimizing operational processes, hypothesis testing provides a rigorous framework to validate assumptions and guide strategic initiatives.

Hypothesis Testing Examples and Solutions

Let’s delve into some common examples of hypothesis testing and provide solutions or interpretations for each scenario.

Example: Testing the Mean

Scenario : A coffee shop owner believes that the average waiting time for customers during peak hours is 5 minutes. To test this, the owner takes a random sample of 30 customer waiting times and wants to determine if the average waiting time is indeed 5 minutes.

Hypotheses :

• H 0​ (Null Hypothesis): 5 μ =5 minutes (The average waiting time is 5 minutes)
• H 1​ (Alternative Hypothesis): 5 μ =5 minutes (The average waiting time is not 5 minutes)

Solution : Using a t-test (assuming population variance is unknown), calculate the t-statistic based on the sample mean, sample standard deviation, and sample size. Then, determine the p-value and compare it with a significance level (e.g., 0.05) to decide whether to reject the null hypothesis.

Example: A/B Testing in Marketing

Scenario : An e-commerce company wants to determine if changing the color of a “Buy Now” button from blue to green increases the conversion rate.

• H 0​: Changing the button color does not affect the conversion rate.
• H 1​: Changing the button color affects the conversion rate.

Solution : Split website visitors into two groups: one sees the blue button (control group), and the other sees the green button (test group). Track the conversion rates for both groups over a specified period. Then, use a chi-square test or z-test (for large sample sizes) to determine if there’s a statistically significant difference in conversion rates between the two groups.

Hypothesis Testing Formula

The formula for hypothesis testing typically depends on the type of test (e.g., z-test, t-test, chi-square test) and the nature of the data (e.g., mean, proportion, variance). Below are the basic formulas for some common hypothesis tests:

Z-Test for Population Mean :

Z=(σ/n​)(xˉ−μ0​)​

• ˉ x ˉ = Sample mean
• 0 μ 0​ = Population mean under the null hypothesis
• σ = Population standard deviation
• n = Sample size

T-Test for Population Mean :

t= (s/ n ​ ) ( x ˉ −μ 0 ​ ) ​ 

s = Sample standard deviation 

Chi-Square Test for Goodness of Fit :

χ2=∑Ei​(Oi​−Ei​)2​

• Oi ​ = Observed frequency
• Ei ​ = Expected frequency

Also Read: Full Form of OOPS

Hypothesis Testing Calculator

While you can perform hypothesis testing manually using the above formulas and statistical tables, many online tools and software packages simplify this process. Here’s how you might use a calculator or software:

• Z-Test and T-Test Calculators : These tools typically require you to input sample statistics (like sample mean, population mean, standard deviation, and sample size). Once you input these values, the calculator will provide you with the test statistic (Z or t) and a p-value.
• Chi-Square Calculator : For chi-square tests, you’d input observed and expected frequencies for different categories or groups. The calculator then computes the chi-square statistic and provides a p-value.
• Software Packages (e.g., R, Python with libraries like scipy, or statistical software like SPSS) : These platforms offer more comprehensive tools for hypothesis testing. You can run various tests, get detailed outputs, and even perform advanced analyses, including regression models, ANOVA, and more.

When using any calculator or software, always ensure you understand the underlying assumptions of the test, interpret the results correctly, and consider the broader context of your research or analysis.

Hypothesis Testing FAQs

What are the key components of a hypothesis test.

The key components include: Null Hypothesis (H0): A statement of no effect or no difference. Alternative Hypothesis (H1 or Ha): A statement that contradicts the null hypothesis. Test Statistic: A value computed from the sample data to test the null hypothesis. Significance Level (α): The threshold for rejecting the null hypothesis. P-value: The probability of observing the given data, assuming the null hypothesis is true.

What is the significance level in hypothesis testing?

The significance level (often denoted as α) is the probability threshold used to determine whether to reject the null hypothesis. Commonly used values for α include 0.05, 0.01, and 0.10, representing a 5%, 1%, or 10% chance of rejecting the null hypothesis when it's actually true.

How do I choose between a one-tailed and two-tailed test?

The choice between one-tailed and two-tailed tests depends on your research question and hypothesis. Use a one-tailed test when you're specifically interested in one direction of an effect (e.g., greater than or less than). Use a two-tailed test when you want to determine if there's a significant difference in either direction.

What is a p-value, and how is it interpreted?

The p-value is a probability value that helps determine the strength of evidence against the null hypothesis. A low p-value (typically ≤ 0.05) suggests that the observed data is inconsistent with the null hypothesis, leading to its rejection. Conversely, a high p-value suggests that the data is consistent with the null hypothesis, leading to no rejection.

Can hypothesis testing prove a hypothesis true?

No, hypothesis testing cannot prove a hypothesis true. Instead, it helps assess the likelihood of observing a given set of data under the assumption that the null hypothesis is true. Based on this assessment, you either reject or fail to reject the null hypothesis.

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4. Probability, Inferential Statistics, and Hypothesis Testing

4a. probability and inferential statistics, video lesson.

In this chapter, we will focus on connecting concepts of probability with the logic of inferential statistics. “The whole problem with the world is that fools and fanatics are always so certain of themselves, and wiser people so full of doubts.” — Bertrand Russel (1872-1970)

These notable quotes represent why probability is critical for a basic understanding of scientific reasoning.

“Medicine is a science of uncertainty and an art of probability.” — William Osler (1849–1919) In many ways, the process of postsecondary education is all about instilling a sense of doubt and wonder, and the ability to estimate probabilities . As a matter of fact, that essentially sums up the entire reason why you are in this course. So let us tackle probability .

We will be keeping our coverage of probability to a very simple level, because the introductory statistics we will cover rely on only simple probability . That said, I encourage you to read further on compound and conditional probabilities , because they will certainly make you smarter at real-life decision making. We will briefly touch on examples of how bad people can be at using probability in real life, and we will then address what probability has to do with inferential statistics. Finally, I will introduce you to the central limit theorem . This is probably one of the heftiest math concepts in the course, but worry not. Its implications are easy to learn, and the concepts behind it can be demonstrated empirically in the interactive exercises.

First, we need to define probability . In a situation where several different outcomes are possible, the probability of any specific outcome is a fraction or proportion of all possible outcomes. Another way of saying that is this. If you wish to answer the question, “What are the chances that outcome would have happened?”, you can calculate the probability as the ratio of possible successful outcomes to all possible outcomes.

Concept Practice: define probability

People often use the rolling of dice as examples of simple probability problems.

If you were to roll one typical die, which has a number on each side from 1 to 6, then the simple probability of rolling a 1 would be 1/6. There are six possible outcomes, but only 1 of them is the successful outcome, that of rolling a 1.

Concept Practice: calculate probability

Another common example used to introduce simple probability is cards. In a standard deck of casino cards, there are 52 cards. There are 4 aces in such a deck of cards (Aces are the “1” card, and there is 1 in each suit – hearts, spades, diamonds and clubs.)

If you were to ask the question “what is the probability that a card drawn at random from a deck of cards will be an ace?”, and you know all outcomes are equally likely, the probability would be the ratio of the number of times one could draw and ace divided by the number of all possible outcomes. In this example, then, the probability would be 4/52. This ratio can be converted into a decimal: 4 divided by 52 is 0.077, or 7.7%. (Remember, to turn a decimal to a percent, you need to move the decimal place twice to the right.)

Probability seems pretty straightforward, right? But people often misunderstand probability in real life. Take the idea of the lucky streak, for example. Let’s say someone is rolling dice and they get 4 6’s in a row. Lots of people might say that’s a lucky streak and they might go as far as to say they should continue, because their luck is so good at the moment! According to the rules of probability , though, the next die roll has a 1/6 chance of being a 6, just like all the others. True, the probability of a 4-in-a-row streak occurring is fairly slim: 1/6 x 1/6 x 1/6 x 1/6. But the fact is that this rare event does not predict future events (unless it is an unfair die!). Each time you roll a die, the probability of that event remains the same. That is what the human brain seems to have a really hard time accepting.

Concept Practice: lucky streak

When someone makes a prediction attached to a certain probability (e.g. there is only a 1% chance of an earthquake in the next week), and then that event occurs in spite of that low probability estimate (e.g. there is actually an earthquake the day after the prediction was made)… was that person wrong? No, not really, because they allowed for the possibility. Had they said there was a 0% chance, they would have been wrong.

Probabilities are often used to express likelihood of outcomes under conditions of uncertainty. Like Bertrand Russell said, wise people rarely speak in terms of certainties. Because people so often misunderstand probability , or find irrational actions so hard to resist despite some understanding of probability , decision making in the realm of sciences needs to be designed to combat our natural human tendencies. What we are discussing now in terms of how to think about and calculate probabilities will form a core component of our decision-making framework as we move forward in the course.

Now, let’s take a look at how probability is used in statistics.

Concept Practice: area under normal curve as probability

We saw that percentiles are expressions of area under a normal curve. Areas under the curve can be expressed as probability , too. For example, if we say the 50th percentile for IQ is 100, that can be expressed as: “If I chose a person at random, there is a 50% chance that they will have an IQ score below 100.”

If we find the 84th percentile for IQ is 115 there is another way to say that “If I chose a person at random, there is an 84% chance that they will have an IQ score below 115.”

Concept Practice: find percentiles

Any time you are dealing with area under the normal curve, I encourage you to express that percentage in terms of probabilities . That will help you think clearly about what that area under the curve means once we get into the exercise of making decisions based on that information.

Concept Practice: interpreting percentile as probability

Probabilities , of course, range from 0 to 1 as proportions or fractions, and from 0% to 100% when expressed in percentage terms. In inferential statistics, we often express in terms of probability the likelihood that we would observe a particular score under a given normal curve model.

Concept Practice: applying probability

Although I encourage you to think of probabilities as percentages, the convention in statistics is to report to the probability of a score as a proportion, or decimal. The symbol used for “probability of score” is p . In statistics, the interpretation of “ p ” is a delicate subject. Generations of researchers have been lazy in our understanding of what “ p ”: tells us, and we have tended to over-interpret this statistic. As we begin to work with “ p ”, I will ask you to memorize a mantra that will help you report its meaning accurately. For now, just keep in mind that most psychologists and psychology students still make mistakes in how they express and understand the meaning of “ p ” values. This will take time and effort to fix, but I am confident that your generation will learn to do better at a precise and careful understanding of what statistics like “ p ” tell us… and what they do not.

To give you a sense of what a statement of p < .05 might mean, let us think back to our rat weights example.

If I were to take a rat from our high-grain food group and place it on the distribution of untreated rat weights, and if it placed at Z = .9, we could look at the area under the curve from that point and above. That would tell us how likely it would be to observe such a heavy rat in the general population of nontreated rats — those that eat a normal diet.

Think of it this way. When we select a rat from our treatment group (those that ate the grain-heavy diet), and it is heavier than the average for a nontreated rat, there are two possible explanations for that observation. One is that the diet made him that way. As a scientist whose hypothesis is that a grain-heavy diet will make the rats weigh more, I’m actually motivated to interpret the observation that way. I want to believe this event is meaningful, because it is consistent with my hypothesis! But the other possibility is that, by random chance, we picked a rat that was heavy to begin with. There are plenty of rats in the distribution of nontreated rats that were at least that heavy. So there is always some probability that we just randomly selected a heavier rat. In this case, if my treated rat’s weight was less than one standard deviation above the mean, we saw in the chapter on normal curves that the probability of observing a rat weight that high or higher in the nontreated population was about 18%. That is not so unusual. It would not be terribly surprising if that outcome were simply the result of random chance rather than a result of the diet the rat had been eating.

If, on the other hand, the rat we measured was 2.5 standard deviations above the mean, the tail probability beyond that Z-score would be vanishingly small.

The probability of observing such a rat weight in the nontreated population is very low, so it is far less likely that observation can be accounted for just by random chance alone. As we accumulate more evidence, the probability they could have come at random from the nontreated population will weigh into our decision making about whether the grain-heavy diet indeed causes rats to become heavier. This is the way probabilities are used in the process of hypothesis testing , the logic of inferential statistics that we will look at soon.

Concept Practice: statistics as probability

Now that you have seen the relevance of probability to the decision making process that comprises inferential statistics, we have one more major learning objective: to become familiar with the central limit theorem .

However, before we get to the central limit theorem , we need to be clear on the distinction between two concepts:  sample and population . In the world of statistics, the population is defined as all possible individuals or scores about which we would ideally draw conclusions. When we refer to the characteristics, or parameters, that describe a population , we will use Greek letters. A sample is defined as the individuals or scores about which we are actually drawing conclusions. When we refer to the characteristics, or statistics, that describe a sample , we will use English letters.

It is important to understand the difference between a population and a sample , and how they relate to one another, in order to comprehend the central limit theorem and its usefulness for statistics. From a population we can draw multiple samples . The larger sample , the more closely our sample will represent the population .

Think of a Venn diagram.  There is a circle that is a population . Inside that large circle, you can draw an infinite number of smaller circles, each of which represents a sample .

The larger that inner circle, the more of the population it contains, and thus the more representative it is.

Let us take a concrete example. A population might be the depression screening scores for all current postsecondary students in Canada. A sample from that population might be depression screening scores for 500 randomly selected postsecondary students from several institutions across Canada. That seems a more reasonable proportion of the two million students in the population than a sample that contains only 5 students. The 500 student sample has a better shot at adequately representing the entire population than does the 5 student sample , right? You can see that intuitively… and once you learn the central limit theorem , you will see the mathematical demonstration of the importance of sample size for representing the population .

To conduct the inferential statistics we are using in this course, we will be using the normal curve model to estimate probabilities associated with particular scores. To do that, we need to assume that data are normally distributed. However, in real life, our data are almost never actually a perfect match for the normal curve.

So how can we reasonably make the normality assumption? Here’s the thing. The central limit theorem is a mathematical principle that assures us that the normality assumption is a reasonable one as long as we have a decent sample size.

According to the theorem, as long as we take a decent-sized sample , if we took many samples (10,000) of large enough size (30+) and took the mean each time, the distribution of those means w ill approach a normal distribution, even if the scores from each sample are not normally distributed. To see this for yourself, take a look at the histograms shown on the right. The top histogram came from taking from a population 10,000 samples of just one score each, and plotting them on a histogram. See how it has a flat, or rectangular shape? No way we could call that a shape approximating a normal  curve. Next is a histogram that came from taking the means of 10,000 samples , if each sample included 4 scores. Looks slightly better, but still not very convincing. With a sample size of 7, it looks a bit better. Once our sample size is 10, we at least have something pretty close. Mathematically speaking, as long as the sample size is no smaller than 30, then the assumption of normality holds. The other way we can reasonably make the normality assumption is if we know the population itself follows a normal curve. In that case, even if individual samples do not have a nice shaped histogram, that is okay, because the normality assumption is one apply to the population in question, not to the sample itself.

Now, you can play around with an online demonstration so you can really convince yourself that the central limit theorem works in practice. The goal here is to see what sample size is sufficient to generate a histogram that closely approximates a normal curve. And to trust that even if real-life data look wonky, the normal curve may still be a reasonable model for data analysis for purposes of inference.

Concept Practice: Central Limit Theorem

4b. hypothesis testing.

We are finally ready for your first introduction to a formal decision making procedure often used in statistics, known as hypothesis testing .

In this course, we started off with descriptive statistics, so that you would become familiar with ways to summarize the important characteristics of datasets. Then we explored the concepts standardizing scores, and relating those to probability as area under the normal curve model. With all those tools, we are now ready to make something!

Okay, not furniture, exactly, but decisions.

We are now into the portion of the course that deals with inferential statistics. Just to get you thinking in terms of making decisions on the basis of data, let us take a slightly silly example. Suppose I have discovered a pill that cures hangovers!

Well, it greatly lessened symptoms of hangover in 10 of the 15 people I tested it on. I am charging 50 dollars per pill. Will you buy it the next time you go out for a night of drinking? Or recommend it to a friend? … If you said yes, I wonder if you are thinking very critically? Should we think about the cost-benefit ratio here on the basis of what information you have? If you said no, I bet some of the doubts I bring up popped to your mind as well. If 10 out of 15 people saw lessened symptoms, that’s 2/3 of people – so some people saw no benefits. Also, what does “greatly lessened symptoms of hangover” mean? Which symptoms? How much is greatly? Was the reduction by two or more standard deviations from the mean? Or was it less than one standard deviation improvement? Given the cost of 50 dollars per pill, I have to say I would be skeptical about buying it without seeing some statistics!

On this list is a preview of the basic concepts to which you will be introduced as we go through the rest of this chapter.

Hypothesis Testing Basic Concepts

• Null Hypothesis
• Research Hypothesis (alternative hypothesis)
• Statistical significance
• Conventional levels of significance
• Cutoff sample score (critical value)
• Directional vs. non-directional hypotheses
• One-tailed and two-tailed tests
• Type I and Type II errors

You can see that there are lots of new concepts to master. In my experience, each concept makes the most sense in context, within its place in the hypothesis testing workflow. We will start with defining our null and research hypotheses , then discuss the levels of statistical significance and their conventional usage. Next, we will look at how to find the cutoff sample score that will form the critical value for our decision criterion. We will look at how that differs for directional vs. non-directional hypotheses , which will lend themselves to one- or two-tailed tests , respectively.

The hypothesis testing procedure, or workflow, can be broken down into five discrete steps.

Steps of Hypothesis Testing

• Restate question as a research hypothesis and a null hypothesis about populations.
• Determine characteristics of the comparison distribution.
• Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.
• Determine your sample’s score on the comparison distribution.
• Decide whether to reject the null hypothesis.

These steps are something we will be using pretty much the rest of the semester, so it is worth memorizing them now. My favourite approach to that is to create a mnemonic device. I recommend the following key words from which to form your mnemonic device: hypothesis, characteristics, cutoff, score, and decide. Not very memorable? Try association those with more memorable words that start with the same letter or sound. How about “ Happy Chickens Cure Sad Days .” Or you can put the words into a mnemonic device generator on the internet and get something truly bizarre. I just tried one and got “ Hairless Carsick Chewbacca Slapped Demons ”. Another good one: “ Hamlet Chose Cranky Sushi Drunkenly .” Anyway, you play around with it or brainstorm until you hit upon one that works for you. Who knew statistics could be this much fun!

The first step in hypothesis testing is always to formulate hypotheses. The first rule that will help you do so correctly, is that hypotheses are always about populations . We study samples in order to make conclusions about populations, so our predictions should be about the populations themselves. First, we define population 1 and population 2. Population 1 is always defined as people like the ones in our research study, the ones we are truly interested in. Population 2 is the comparison population , the status quo to which we are looking to compare our research population . Now, remember, when referring to populations , we always use Greek letters. So if we formulate our hypotheses in symbols, we need to use Greek letters.

It is a good idea to state our hypotheses both in symbols and in words. We need to make them specific and disprovable. If you follow my tips, you will have it down with just a little practice.

We need to state two hypotheses. First, we state the research hypothesis , which is sometimes referred to as the alternative hypothesis. The research hypothesis (often called the alternative hypothesis) is a statement of inequality, or that Something happened! This hypothesis makes the prediction that the population from which the research sample came is different from the comparison population . In other words, there is a really high probability that the sample comes from a different distribution than the comparison one.

The null hypothesis , on the other hand, is a statement of equality, or that nothing happened. This hypothesis makes the prediction that the population from which sample came is not different from the comparison population . We set up the null hypothesis as a so-called straw man, that we hope to tear down. Just remember, null means nothing – that nothing is different between the populations .

Step two of hypothesis testing is to determine the characteristics of the comparison distribution. This is where our descriptive statistics, the mean and standard deviation, come in. We need to ensure our normal curve model to which we are comparing our research sample is mapped out according to the particular characteristics of the population of comparison, which is population 2.

Next it is time to set our decision rule. Step 3 is to determine the cutoff sample score , which is derived from two pieces of information. The first is the conventional significance level that applies. By convention, the probability level that we are willing to accept as a risk that the score from our research sample might occur by random chance within the comparison distribution is set to one of three levels: 10%, 5%, or 1%. The most common choice of significance level is 5%. Typically the significance level will be provided to you in the problem for your statistics courses, but if it is not, just default to a significance level of .05. Sometimes researchers will choose a more conservative significance level , like 1%, if they are particularly risk averse. If the researcher chooses a 10% significance level , they are likely conducting a more exploratory study, perhaps a pilot study, and are not too worried about the probability that the score might be fairly common under the comparison distribution.

The second piece of information we need to know in order to find our cutoff sample score is which tail we are looking at. Is this a directional hypothesis , and thus one-tailed test ? Or a non-directional hypothesis , and thus a two-tailed test ? This depends on the research hypothesis from step 1. Look for directional keywords in the problem. If the researcher prediction involves words like “greater than” or “larger than”, this signals that we should be doing a one-tailed test and that our cutoff sample score should be in the top tail of the distribution. If the researcher prediction involves words like “lower than” or “smaller than”, this signals that we should be doing a one-tailed test and that our cutoff sample score should be in the bottom tail of the distribution. If the prediction is neutral in directionality, and uses a word like “different”, that signals a non-directional hypothesis . In that case, we would need to use a two-tailed test, and our cutoff scores would need to be indicated on both tails of the distribution. To do that, we take our area under the curve, which matches our significance level , and split it into both tails.

For example, if we have a two-tailed test with a .05 significance level , then we would split the 5% area under the curve into the two tails, so two and a half percent in each tail.

Concept Practice: deciding on one-tailed vs. two-tailed tests

We can find the Z-score that forms the border of the tail area we have identified based on significance level and directionality by looking it up in a table or an online calculator . I always recommend mapping this cutoff score onto a drawing of the comparison distribution as shown above. This should help you visualize the setup of the hypothesis test clearly and accurately.

Concept Practice: inference through hypothesis testing

The next step in the hypothesis testing procedure is to determine your sample’s score on the comparison distribution. To do this, we calculate a test statistic from the sample raw score, mark it on the comparison distribution, and determine whether it falls in the shaded tail or not. In reality, we would always have a sample with more than one score in it. However, for the sake of keeping our test statistic formula a familiar one, we will use a sample size of one. We will use our Z-score formula to translate the sample’s raw score into a Z-score – in other words, we will figure out how many standard deviations above or below the comparison distribution’s mean the sample score is.

Finally, it’s time to decide whether to reject the null hypothesis . This decision is based on whether our sample’s data point was more extreme than the cutoff score , in other words, “did it fall in the shaded tail?” If the sample score is more extreme than the cutoff score , then we must reject the null hypothesis. Our research hypothesis is supported! (Not proven… remember, there is still some probability that that score could have occurred randomly within the comparison distribution.) But it is sound to say that it appears quite likely that the population from which our sample came is different from the comparison population. Another way to express this decision is to say that the result was statistically significant , which is to say that there is less than a 5% chance of this result occurring randomly within the comparison distribution (here I just filled in the blank with the significance level).

What if the research sample score did not fall in the shaded tail? In the case that the sample score is less extreme than the cutoff score , then our research hypothesis is not supported. We do not reject the null hypothesis . It appears that the population from which our sample came is not different from the comparison population . Note that we do not typically express this result as “accept the null hypothesis” or “we have proved the null hypothesis”. From this test, we do not have evidence that the null hypothesis is correct, rather we simply did not have enough evidence to reject it. Another way to express this decision is to say that the result was not statistically significant , which is to say that there is more than a 5% chance of this result occurring randomly within the comparison distribution (here I just used the most common significance level ).

Concept Practice: interpreting conclusions of hypothesis tests

So we have described the hypothesis testing process from beginning to end. The whole process of null hypothesis testing can feel like pretty tortured logic at first. So let us zoom out, and look at the whole process another way. Essentially what we are seeking to do in such a hypothesis test is to compare two populations . We want to find out if the populations are distinct enough to confidently state that there is a difference between population 1 and population 2. In our example, we wanted to know if the population of people using a new medication, population 1, sleep longer than the population of people who are not using that new medication, population 2. We ended up finding that the research evidence to hand suggests population 1’s distribution is distinct enough from population 2 that we could reject the null hypothesis of similarity.

In other words, we were able to conclude that the difference between the centres of the two distributions was statistically significant .

If, on the other hand, the distributions were a bit less distinct, we would not have been able to make that claim of a significant difference.

We would not have rejected the null hypothesis if evidence indicated the populations were too similar.

Just how different do the two distributions need to be? That criterion is set by the cutoff score , which depends on the significance level , and whether it is a one-tailed or two-tailed hypothesis test .

Concept Practice: Putting hypothesis test elements together

That was a lot of new concepts to take on! As a reward, assuming you enjoy memes, there are a plethora of statistics memes , some of which you may find funny now that you have made it into inferential statistics territory. Welcome to the exclusive club of people who have this rather peculiar sense of humour. Enjoy!

Chapter Summary

In this chapter we examined probability and how it can be used to make inferences about data in the framework of hypothesis testing . We now have a sense of how two populations can be compared and the difference between their means evaluated for statistical significance .

Concept Practice

Return to text.

Return to 4a. Probability and Inferential Statistics

Try interactive Worksheet 4a or download Worksheet 4a

Return to 4b. Hypothesis Testing

Try interactive Worksheet 4b  or download Worksheet 4b

in a situation where several different outcomes are possible, the probability of any specific outcome is a fraction or proportion of all possible outcomes

mathematical theorem that proposes the following: as long as we take a decent-sized sample, if we took many samples (10,000) of large enough size (30+) and took the mean each time, the distribution of those means will approach a normal distribution, even if the scores from each sample are not normally distributed

all possible individuals or scores about which we would ideally draw conclusions

a formal decision making procedure often used in inferential statistics

the individuals or scores about which we are actually drawing conclusions

the probability level that we are willing to accept as a risk that the score from our research sample might occur by random chance within the comparison distribution. By convention, it is set to one of three levels: 10%, 5%, or 1%.

critical value that serves as a decision criterion in hypothesis testing

prediction that the population from which the research sample came is different from the comparison population

the prediction that the population from which sample came is not different from the comparison population

a research prediction that the research population mean will be “greater than” or "less than" the comparison population mean

a hypothesis test in which there is only one cutoff sample score on either the lower or the upper end of the comparison distribution

a research prediction that the research population mean will be “different from" the comparison population mean, but allows for the possibility that the research population mean may be either greater than or less than the comparison population mean

a hypothesis test in which there are two cutoff sample scores, one on either end of the comparison distribution

a decision in hypothesis testing that concludes statistical significance because the sample score is more extreme than the cutoff score

the conclusion from a hypothesis test that probability of the observed result occurring randomly within the comparison distribution is less than the significance level

a decision in hypothesis testing that is inconclusive because the sample score is less extreme than the cutoff score

Beginner Statistics for Psychology Copyright © 2021 by Nicole Vittoz is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License , except where otherwise noted.

Statistics and Analysis of Scientific Data pp 147–179 Cite as

Hypothesis Testing and Fundamental Statistics

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• First Online: 13 July 2022

Part of the book series: Graduate Texts in Physics ((GTP))

Data-based statistics are subject to the random fluctuations of the measurements, and their sampling distribution describes the probability of occurrence of the values of the statistic. The method of hypothesis testing and the associated p -value describe the process of using the distribution of a statistic to quantitatively test the agreement of data with a statistical hypothesis. This chapter also introduces a few fundamental statistics that play a central role in data analysis, such as the \(\chi ^2\) statistic, Fisher’s F -statistic, and Student ’s t -statistic.

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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

Key Terms of Hypothesis Testing

• 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:

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.

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.

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

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,

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.

• 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 = 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:

• i,j are the rows and columns index respectively.

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.

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.

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