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A hypothesis test is a statistical inference method used to test the significance of a proposed (hypothesized) relation between population statistics (parameters) and their corresponding sample estimators . In other words, hypothesis tests are used to determine if there is enough evidence in a sample to prove a hypothesis true for the entire population.

The test considers two hypotheses: the null hypothesis , which is a statement meant to be tested, usually something like "there is no effect" with the intention of proving this false, and the alternate hypothesis , which is the statement meant to stand after the test is performed. The two hypotheses must be mutually exclusive ; moreover, in most applications, the two are complementary (one being the negation of the other). The test works by comparing the \(p\)-value to the level of significance (a chosen target). If the \(p\)-value is less than or equal to the level of significance, then the null hypothesis is rejected.

When analyzing data, only samples of a certain size might be manageable as efficient computations. In some situations the error terms follow a continuous or infinite distribution, hence the use of samples to suggest accuracy of the chosen test statistics. The method of hypothesis testing gives an advantage over guessing what distribution or which parameters the data follows.

Definitions and Methodology

Hypothesis test and confidence intervals.

In statistical inference, properties (parameters) of a population are analyzed by sampling data sets. Given assumptions on the distribution, i.e. a statistical model of the data, certain hypotheses can be deduced from the known behavior of the model. These hypotheses must be tested against sampled data from the population.

The null hypothesis \((\)denoted \(H_0)\) is a statement that is assumed to be true. If the null hypothesis is rejected, then there is enough evidence (statistical significance) to accept the alternate hypothesis \((\)denoted \(H_1).\) Before doing any test for significance, both hypotheses must be clearly stated and non-conflictive, i.e. mutually exclusive, statements. Rejecting the null hypothesis, given that it is true, is called a type I error and it is denoted \(\alpha\), which is also its probability of occurrence. Failing to reject the null hypothesis, given that it is false, is called a type II error and it is denoted \(\beta\), which is also its probability of occurrence. Also, \(\alpha\) is known as the significance level , and \(1-\beta\) is known as the power of the test. \(H_0\) \(\textbf{is true}\)\(\hspace{15mm}\) \(H_0\) \(\textbf{is false}\) \(\textbf{Reject}\) \(H_0\)\(\hspace{10mm}\) Type I error Correct Decision \(\textbf{Reject}\) \(H_1\) Correct Decision Type II error The test statistic is the standardized value following the sampled data under the assumption that the null hypothesis is true, and a chosen particular test. These tests depend on the statistic to be studied and the assumed distribution it follows, e.g. the population mean following a normal distribution. The \(p\)-value is the probability of observing an extreme test statistic in the direction of the alternate hypothesis, given that the null hypothesis is true. The critical value is the value of the assumed distribution of the test statistic such that the probability of making a type I error is small.

Methodologies: Given an estimator \(\hat \theta\) of a population statistic \(\theta\), following a probability distribution \(P(T)\), computed from a sample \(\mathcal{S},\) and given a significance level \(\alpha\) and test statistic \(t^*,\) define \(H_0\) and \(H_1;\) compute the test statistic \(t^*.\) \(p\)-value Approach (most prevalent): Find the \(p\)-value using \(t^*\) (right-tailed). If the \(p\)-value is at most \(\alpha,\) reject \(H_0\). Otherwise, reject \(H_1\). Critical Value Approach: Find the critical value solving the equation \(P(T\geq t_\alpha)=\alpha\) (right-tailed). If \(t^*>t_\alpha\), reject \(H_0\). Otherwise, reject \(H_1\). Note: Failing to reject \(H_0\) only means inability to accept \(H_1\), and it does not mean to accept \(H_0\).

Assume a normally distributed population has recorded cholesterol levels with various statistics computed. From a sample of 100 subjects in the population, the sample mean was 214.12 mg/dL (milligrams per deciliter), with a sample standard deviation of 45.71 mg/dL. Perform a hypothesis test, with significance level 0.05, to test if there is enough evidence to conclude that the population mean is larger than 200 mg/dL. Hypothesis Test We will perform a hypothesis test using the \(p\)-value approach with significance level \(\alpha=0.05:\) Define \(H_0\): \(\mu=200\). Define \(H_1\): \(\mu>200\). Since our values are normally distributed, the test statistic is \(z^*=\frac{\bar X - \mu_0}{\frac{s}{\sqrt{n}}}=\frac{214.12 - 200}{\frac{45.71}{\sqrt{100}}}\approx 3.09\). Using a standard normal distribution, we find that our \(p\)-value is approximately \(0.001\). Since the \(p\)-value is at most \(\alpha=0.05,\) we reject \(H_0\). Therefore, we can conclude that the test shows sufficient evidence to support the claim that \(\mu\) is larger than \(200\) mg/dL.

If the sample size was smaller, the normal and \(t\)-distributions behave differently. Also, the question itself must be managed by a double-tail test instead.

Assume a population's cholesterol levels are recorded and various statistics are computed. From a sample of 25 subjects, the sample mean was 214.12 mg/dL (milligrams per deciliter), with a sample standard deviation of 45.71 mg/dL. Perform a hypothesis test, with significance level 0.05, to test if there is enough evidence to conclude that the population mean is not equal to 200 mg/dL. Hypothesis Test We will perform a hypothesis test using the \(p\)-value approach with significance level \(\alpha=0.05\) and the \(t\)-distribution with 24 degrees of freedom: Define \(H_0\): \(\mu=200\). Define \(H_1\): \(\mu\neq 200\). Using the \(t\)-distribution, the test statistic is \(t^*=\frac{\bar X - \mu_0}{\frac{s}{\sqrt{n}}}=\frac{214.12 - 200}{\frac{45.71}{\sqrt{25}}}\approx 1.54\). Using a \(t\)-distribution with 24 degrees of freedom, we find that our \(p\)-value is approximately \(2(0.068)=0.136\). We have multiplied by two since this is a two-tailed argument, i.e. the mean can be smaller than or larger than. Since the \(p\)-value is larger than \(\alpha=0.05,\) we fail to reject \(H_0\). Therefore, the test does not show sufficient evidence to support the claim that \(\mu\) is not equal to \(200\) mg/dL.

The complement of the rejection on a two-tailed hypothesis test (with significance level \(\alpha\)) for a population parameter \(\theta\) is equivalent to finding a confidence interval \((\)with confidence level \(1-\alpha)\) for the population parameter \(\theta\). If the assumption on the parameter \(\theta\) falls inside the confidence interval, then the test has failed to reject the null hypothesis \((\)with \(p\)-value greater than \(\alpha).\) Otherwise, if \(\theta\) does not fall in the confidence interval, then the null hypothesis is rejected in favor of the alternate \((\)with \(p\)-value at most \(\alpha).\)

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What is Hypothesis Testing?

A statistical hypothesis is an assumption about a population parameter . This assumption may or may not be true. Hypothesis testing refers to the formal procedures used by statisticians to accept or reject statistical hypotheses.

Statistical Hypotheses

The best way to determine whether a statistical hypothesis is true would be to examine the entire population. Since that is often impractical, researchers typically examine a random sample from the population. If sample data are not consistent with the statistical hypothesis, the hypothesis is rejected.

There are two types of statistical hypotheses.

• Null hypothesis . The null hypothesis, denoted by H o , is usually the hypothesis that sample observations result purely from chance.
• Alternative hypothesis . The alternative hypothesis, denoted by H 1 or H a , is the hypothesis that sample observations are influenced by some non-random cause.

For example, suppose we wanted to determine whether a coin was fair and balanced. A null hypothesis might be that half the flips would result in Heads and half, in Tails. The alternative hypothesis might be that the number of Heads and Tails would be very different. Symbolically, these hypotheses would be expressed as

H o : P = 0.5 H a : P ≠ 0.5

Suppose we flipped the coin 50 times, resulting in 40 Heads and 10 Tails. Given this result, we would be inclined to reject the null hypothesis. We would conclude, based on the evidence, that the coin was probably not fair and balanced.

Can We Accept the Null Hypothesis?

Some researchers say that a hypothesis test can have one of two outcomes: you accept the null hypothesis or you reject the null hypothesis. Many statisticians, however, take issue with the notion of "accepting the null hypothesis." Instead, they say: you reject the null hypothesis or you fail to reject the null hypothesis.

Why the distinction between "acceptance" and "failure to reject?" Acceptance implies that the null hypothesis is true. Failure to reject implies that the data are not sufficiently persuasive for us to prefer the alternative hypothesis over the null hypothesis.

Hypothesis Tests

Statisticians follow a formal process to determine whether to reject a null hypothesis, based on sample data. This process, called hypothesis testing , consists of four steps.

• State the hypotheses. This involves stating the null and alternative hypotheses. The hypotheses are stated in such a way that they are mutually exclusive. That is, if one is true, the other must be false.
• Formulate an analysis plan. The analysis plan describes how to use sample data to evaluate the null hypothesis. The evaluation often focuses around a single test statistic.
• Analyze sample data. Find the value of the test statistic (mean score, proportion, t statistic, z-score, etc.) described in the analysis plan.
• Interpret results. Apply the decision rule described in the analysis plan. If the value of the test statistic is unlikely, based on the null hypothesis, reject the null hypothesis.

Decision Errors

Two types of errors can result from a hypothesis test.

• Type I error . A Type I error occurs when the researcher rejects a null hypothesis when it is true. The probability of committing a Type I error is called the significance level . This probability is also called alpha , and is often denoted by α.
• Type II error . A Type II error occurs when the researcher fails to reject a null hypothesis that is false. The probability of committing a Type II error is called Beta , and is often denoted by β. The probability of not committing a Type II error is called the Power of the test.

Decision Rules

The analysis plan for a hypothesis test must include decision rules for rejecting the null hypothesis. In practice, statisticians describe these decision rules in two ways - with reference to a P-value or with reference to a region of acceptance.

• P-value. The strength of evidence in support of a null hypothesis is measured by the P-value . Suppose the test statistic is equal to S . The P-value is the probability of observing a test statistic as extreme as S , assuming the null hypothesis is true. If the P-value is less than the significance level, we reject the null hypothesis.

The set of values outside the region of acceptance is called the region of rejection . If the test statistic falls within the region of rejection, the null hypothesis is rejected. In such cases, we say that the hypothesis has been rejected at the α level of significance.

These approaches are equivalent. Some statistics texts use the P-value approach; others use the region of acceptance approach.

One-Tailed and Two-Tailed Tests

A test of a statistical hypothesis, where the region of rejection is on only one side of the sampling distribution , is called a one-tailed test . For example, suppose the null hypothesis states that the mean is less than or equal to 10. The alternative hypothesis would be that the mean is greater than 10. The region of rejection would consist of a range of numbers located on the right side of sampling distribution; that is, a set of numbers greater than 10.

A test of a statistical hypothesis, where the region of rejection is on both sides of the sampling distribution, is called a two-tailed test . For example, suppose the null hypothesis states that the mean is equal to 10. The alternative hypothesis would be that the mean is less than 10 or greater than 10. The region of rejection would consist of a range of numbers located on both sides of sampling distribution; that is, the region of rejection would consist partly of numbers that were less than 10 and partly of numbers that were greater than 10.

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• 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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What Is Hypothesis Testing?

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Hypothesis Testing: 4 Steps and Example

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

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

Key Takeaways

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

How Hypothesis Testing Works

In hypothesis testing, an  analyst  tests a statistical sample, intending to provide evidence on the plausibility of the null hypothesis. Statistical analysts measure and examine a random sample of the population being analyzed. All analysts use a random population sample to test two different hypotheses: the null hypothesis and the alternative hypothesis.

The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a null hypothesis may state that the population mean return is equal to zero. The alternative hypothesis is effectively the opposite of a null hypothesis. Thus, they are mutually exclusive , and only one can be true. However, one of the two hypotheses will always be true.

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

• State the hypotheses.
• Formulate an analysis plan, which outlines how the data will be evaluated.
• Carry out the plan and analyze the sample data.
• Analyze the results and either reject the null hypothesis, or state that the null hypothesis is plausible, given the data.

Example of Hypothesis Testing

If an individual wants to test that a penny has exactly a 50% chance of landing on heads, the null hypothesis would be that 50% is correct, and the alternative hypothesis would be that 50% is not correct. Mathematically, the null hypothesis is represented as Ho: P = 0.5. The alternative hypothesis is shown as "Ha" and is identical to the null hypothesis, except with the equal sign struck-through, meaning that it does not equal 50%.

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

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

When Did Hypothesis Testing Begin?

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

What are the Benefits of Hypothesis Testing?

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

What are the Limitations of Hypothesis Testing?

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

Hypothesis testing refers to a statistical process that helps researchers determine the reliability of a study. By using a well-formulated hypothesis and set of statistical tests, individuals or businesses can make inferences about the population that they are studying and draw conclusions based on the data presented. All hypothesis testing methods have the same four-step process, which includes stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.

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

Elder Research. " Who Invented the Null Hypothesis? "

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

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Hypothesis Testing

Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid.

A null hypothesis and an alternative hypothesis are set up before performing the hypothesis testing. This helps to arrive at a conclusion regarding the sample obtained from the population. In this article, we will learn more about hypothesis testing, its types, steps to perform the testing, and associated examples.

What is Hypothesis Testing in Statistics?

Hypothesis testing uses sample data from the population to draw useful conclusions regarding the population probability distribution . It tests an assumption made about the data using different types of hypothesis testing methodologies. The hypothesis testing results in either rejecting or not rejecting the null hypothesis.

Hypothesis Testing Definition

Hypothesis testing can be defined as a statistical tool that is used to identify if the results of an experiment are meaningful or not. It involves setting up a null hypothesis and an alternative hypothesis. These two hypotheses will always be mutually exclusive. This means that if the null hypothesis is true then the alternative hypothesis is false and vice versa. An example of hypothesis testing is setting up a test to check if a new medicine works on a disease in a more efficient manner.

Null Hypothesis

The null hypothesis is a concise mathematical statement that is used to indicate that there is no difference between two possibilities. In other words, there is no difference between certain characteristics of data. This hypothesis assumes that the outcomes of an experiment are based on chance alone. It is denoted as \(H_{0}\). Hypothesis testing is used to conclude if the null hypothesis can be rejected or not. Suppose an experiment is conducted to check if girls are shorter than boys at the age of 5. The null hypothesis will say that they are the same height.

Alternative Hypothesis

The alternative hypothesis is an alternative to the null hypothesis. It is used to show that the observations of an experiment are due to some real effect. It indicates that there is a statistical significance between two possible outcomes and can be denoted as \(H_{1}\) or \(H_{a}\). For the above-mentioned example, the alternative hypothesis would be that girls are shorter than boys at the age of 5.

Hypothesis Testing P Value

In hypothesis testing, the p value is used to indicate whether the results obtained after conducting a test are statistically significant or not. It also indicates the probability of making an error in rejecting or not rejecting the null hypothesis.This value is always a number between 0 and 1. The p value is compared to an alpha level, \(\alpha\) or significance level. The alpha level can be defined as the acceptable risk of incorrectly rejecting the null hypothesis. The alpha level is usually chosen between 1% to 5%.

Hypothesis Testing Critical region

All sets of values that lead to rejecting the null hypothesis lie in the critical region. Furthermore, the value that separates the critical region from the non-critical region is known as the critical value.

Hypothesis Testing Formula

Depending upon the type of data available and the size, different types of hypothesis testing are used to determine whether the null hypothesis can be rejected or not. The hypothesis testing formula for some important test statistics are given below:

• z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). \(\overline{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation and n is the size of the sample.
• t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\). s is the sample standard deviation.
• \(\chi ^{2} = \sum \frac{(O_{i}-E_{i})^{2}}{E_{i}}\). \(O_{i}\) is the observed value and \(E_{i}\) is the expected value.

We will learn more about these test statistics in the upcoming section.

Types of Hypothesis Testing

Selecting the correct test for performing hypothesis testing can be confusing. These tests are used to determine a test statistic on the basis of which the null hypothesis can either be rejected or not rejected. Some of the important tests used for hypothesis testing are given below.

Hypothesis Testing Z Test

A z test is a way of hypothesis testing that is used for a large sample size (n ≥ 30). It is used to determine whether there is a difference between the population mean and the sample mean when the population standard deviation is known. It can also be used to compare the mean of two samples. It is used to compute the z test statistic. The formulas are given as follows:

• One sample: z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).
• Two samples: z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing t Test

The t test is another method of hypothesis testing that is used for a small sample size (n < 30). It is also used to compare the sample mean and population mean. However, the population standard deviation is not known. Instead, the sample standard deviation is known. The mean of two samples can also be compared using the t test.

• One sample: t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\).
• Two samples: t = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing Chi Square

The Chi square test is a hypothesis testing method that is used to check whether the variables in a population are independent or not. It is used when the test statistic is chi-squared distributed.

One Tailed Hypothesis Testing

One tailed hypothesis testing is done when the rejection region is only in one direction. It can also be known as directional hypothesis testing because the effects can be tested in one direction only. This type of testing is further classified into the right tailed test and left tailed test.

Right Tailed Hypothesis Testing

The right tail test is also known as the upper tail test. This test is used to check whether the population parameter is greater than some value. The null and alternative hypotheses for this test are given as follows:

\(H_{0}\): The population parameter is ≤ some value

\(H_{1}\): The population parameter is > some value.

If the test statistic has a greater value than the critical value then the null hypothesis is rejected

Left Tailed Hypothesis Testing

The left tail test is also known as the lower tail test. It is used to check whether the population parameter is less than some value. The hypotheses for this hypothesis testing can be written as follows:

\(H_{0}\): The population parameter is ≥ some value

\(H_{1}\): The population parameter is < some value.

The null hypothesis is rejected if the test statistic has a value lesser than the critical value.

Two Tailed Hypothesis Testing

In this hypothesis testing method, the critical region lies on both sides of the sampling distribution. It is also known as a non - directional hypothesis testing method. The two-tailed test is used when it needs to be determined if the population parameter is assumed to be different than some value. The hypotheses can be set up as follows:

\(H_{0}\): the population parameter = some value

\(H_{1}\): the population parameter ≠ some value

The null hypothesis is rejected if the test statistic has a value that is not equal to the critical value.

Hypothesis Testing Steps

Hypothesis testing can be easily performed in five simple steps. The most important step is to correctly set up the hypotheses and identify the right method for hypothesis testing. The basic steps to perform hypothesis testing are as follows:

• Step 1: Set up the null hypothesis by correctly identifying whether it is the left-tailed, right-tailed, or two-tailed hypothesis testing.
• Step 2: Set up the alternative hypothesis.
• Step 3: Choose the correct significance level, \(\alpha\), and find the critical value.
• Step 4: Calculate the correct test statistic (z, t or \(\chi\)) and p-value.
• Step 5: Compare the test statistic with the critical value or compare the p-value with \(\alpha\) to arrive at a conclusion. In other words, decide if the null hypothesis is to be rejected or not.

Hypothesis Testing Example

The best way to solve a problem on hypothesis testing is by applying the 5 steps mentioned in the previous section. Suppose a researcher claims that the mean average weight of men is greater than 100kgs with a standard deviation of 15kgs. 30 men are chosen with an average weight of 112.5 Kgs. Using hypothesis testing, check if there is enough evidence to support the researcher's claim. The confidence interval is given as 95%.

Step 1: This is an example of a right-tailed test. Set up the null hypothesis as \(H_{0}\): \(\mu\) = 100.

Step 2: The alternative hypothesis is given by \(H_{1}\): \(\mu\) > 100.

Step 3: As this is a one-tailed test, \(\alpha\) = 100% - 95% = 5%. This can be used to determine the critical value.

1 - \(\alpha\) = 1 - 0.05 = 0.95

0.95 gives the required area under the curve. Now using a normal distribution table, the area 0.95 is at z = 1.645. A similar process can be followed for a t-test. The only additional requirement is to calculate the degrees of freedom given by n - 1.

Step 4: Calculate the z test statistic. This is because the sample size is 30. Furthermore, the sample and population means are known along with the standard deviation.

z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).

\(\mu\) = 100, \(\overline{x}\) = 112.5, n = 30, \(\sigma\) = 15

z = \(\frac{112.5-100}{\frac{15}{\sqrt{30}}}\) = 4.56

Step 5: Conclusion. As 4.56 > 1.645 thus, the null hypothesis can be rejected.

Hypothesis Testing and Confidence Intervals

Confidence intervals form an important part of hypothesis testing. This is because the alpha level can be determined from a given confidence interval. Suppose a confidence interval is given as 95%. Subtract the confidence interval from 100%. This gives 100 - 95 = 5% or 0.05. This is the alpha value of a one-tailed hypothesis testing. To obtain the alpha value for a two-tailed hypothesis testing, divide this value by 2. This gives 0.05 / 2 = 0.025.

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• Data Handling

Important Notes on Hypothesis Testing

• Hypothesis testing is a technique that is used to verify whether the results of an experiment are statistically significant.
• It involves the setting up of a null hypothesis and an alternate hypothesis.
• There are three types of tests that can be conducted under hypothesis testing - z test, t test, and chi square test.
• Hypothesis testing can be classified as right tail, left tail, and two tail tests.

Examples on Hypothesis Testing

• Example 1: The average weight of a dumbbell in a gym is 90lbs. However, a physical trainer believes that the average weight might be higher. A random sample of 5 dumbbells with an average weight of 110lbs and a standard deviation of 18lbs. Using hypothesis testing check if the physical trainer's claim can be supported for a 95% confidence level. Solution: As the sample size is lesser than 30, the t-test is used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) > 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 5, s = 18. \(\alpha\) = 0.05 Using the t-distribution table, the critical value is 2.132 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = 2.484 As 2.484 > 2.132, the null hypothesis is rejected. Answer: The average weight of the dumbbells may be greater than 90lbs
• Example 2: The average score on a test is 80 with a standard deviation of 10. With a new teaching curriculum introduced it is believed that this score will change. On random testing, the score of 38 students, the mean was found to be 88. With a 0.05 significance level, is there any evidence to support this claim? Solution: This is an example of two-tail hypothesis testing. The z test will be used. \(H_{0}\): \(\mu\) = 80, \(H_{1}\): \(\mu\) ≠ 80 \(\overline{x}\) = 88, \(\mu\) = 80, n = 36, \(\sigma\) = 10. \(\alpha\) = 0.05 / 2 = 0.025 The critical value using the normal distribution table is 1.96 z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) z = \(\frac{88-80}{\frac{10}{\sqrt{36}}}\) = 4.8 As 4.8 > 1.96, the null hypothesis is rejected. Answer: There is a difference in the scores after the new curriculum was introduced.
• Example 3: The average score of a class is 90. However, a teacher believes that the average score might be lower. The scores of 6 students were randomly measured. The mean was 82 with a standard deviation of 18. With a 0.05 significance level use hypothesis testing to check if this claim is true. Solution: The t test will be used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) < 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 6, s = 18 The critical value from the t table is -2.015 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = \(\frac{82-90}{\frac{18}{\sqrt{6}}}\) t = -1.088 As -1.088 > -2.015, we fail to reject the null hypothesis. Answer: There is not enough evidence to support the claim.

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FAQs on Hypothesis Testing

What is hypothesis testing.

Hypothesis testing in statistics is a tool that is used to make inferences about the population data. It is also used to check if the results of an experiment are valid.

What is the z Test in Hypothesis Testing?

The z test in hypothesis testing is used to find the z test statistic for normally distributed data . The z test is used when the standard deviation of the population is known and the sample size is greater than or equal to 30.

What is the t Test in Hypothesis Testing?

The t test in hypothesis testing is used when the data follows a student t distribution . It is used when the sample size is less than 30 and standard deviation of the population is not known.

What is the formula for z test in Hypothesis Testing?

The formula for a one sample z test in hypothesis testing is z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) and for two samples is z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

What is the p Value in Hypothesis Testing?

The p value helps to determine if the test results are statistically significant or not. In hypothesis testing, the null hypothesis can either be rejected or not rejected based on the comparison between the p value and the alpha level.

What is One Tail Hypothesis Testing?

When the rejection region is only on one side of the distribution curve then it is known as one tail hypothesis testing. The right tail test and the left tail test are two types of directional hypothesis testing.

What is the Alpha Level in Two Tail Hypothesis Testing?

To get the alpha level in a two tail hypothesis testing divide \(\alpha\) by 2. This is done as there are two rejection regions in the curve.

Statistics Tutorial

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

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

Hypothesis Testing

A hypothesis is a claim about a population parameter .

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

Examples of claims that can be checked:

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

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

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

The Null and Alternative Hypothesis

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

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

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

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

For example, we want to check the following claim:

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

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

The null and alternative hypothesis would be:

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

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

The claims are often expressed with symbols like this:

\(H_{0}\): \(\mu = 170 \: cm \)

\(H_{1}\): \(\mu > 170 \: cm \)

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

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

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

The Significance Level

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

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

Typical significance levels are:

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

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

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

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

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

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The Test Statistic

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

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

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

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

Common examples are:

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

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

The Critical Value and P-Value Approach

There are two main approaches used for hypothesis tests:

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

The Critical Value Approach

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

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

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

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

Here is a graphical illustration:

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

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

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

The P-Value Approach

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

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

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

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

For example, if the p-value is 0.03:

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

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

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

Steps for a Hypothesis Test

The following steps are used for a hypothesis test:

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

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

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

Common parameters to test hypotheses are:

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

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

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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 and 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.

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Importance of Hypothesis Testing in Data Analysis

Here is what makes hypothesis testing so important in data analysis and why it is key to making better decisions:

Avoiding Misleading Conclusions (Type I and Type II Errors)

One of the biggest benefits of hypothesis testing is that it helps you avoid jumping to the wrong conclusions. For instance, a Type I error could occur if a company launches a new product thinking it will be a hit, only to find out later that the data misled them. A Type II error might happen when a company overlooks a potentially successful product because their testing wasn’t thorough enough. By setting up the right significance level and carefully calculating the p-value, hypothesis testing minimizes the chances of these errors, leading to more accurate results.

Making Smarter Choices

Hypothesis testing is key to making smarter, evidence-based decisions. Let’s say a city planner wants to determine if building a new park will increase community engagement. By testing the hypothesis using data from similar projects, they can make an informed choice. Similarly, a teacher might use hypothesis testing to see if a new teaching method actually improves student performance. It’s about taking the guesswork out of decisions and relying on solid evidence instead.

Optimizing Business Tactics

In business, hypothesis testing is invaluable for testing new ideas and strategies before fully committing to them. For example, an e-commerce company might want to test whether offering free shipping increases sales. By using hypothesis testing, they can compare sales data from customers who received free shipping offers and those who didn’t. This allows them to base their business decisions on data, not hunches, reducing the risk of costly mistakes.

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 Alternative 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 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 in Hypothesis Testing

Hypothesis testing is a statistical method to determine if there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. Here’s a breakdown of the typical steps involved in hypothesis testing:

Formulate Hypotheses

• Null Hypothesis (H0): This hypothesis states that there is no effect or difference, and it is the hypothesis you attempt to reject with your test.
• Alternative Hypothesis (H1 or Ha): This hypothesis is what you might believe to be true or hope to prove true. It is usually considered the opposite of the null hypothesis.

Choose the Significance Level (α)

The significance level, often denoted by alpha (α), is the probability of rejecting the null hypothesis when it is true. Common choices for α are 0.05 (5%), 0.01 (1%), and 0.10 (10%).

Select the Appropriate Test

Choose a statistical test based on the type of data and the hypothesis. Common tests include t-tests, chi-square tests, ANOVA, and regression analysis. The selection depends on data type, distribution, sample size, and whether the hypothesis is one-tailed or two-tailed.

Collect Data

Gather the data that will be analyzed in the test. To infer conclusions accurately, this data should be representative of the population.

Calculate the Test Statistic

Based on the collected data and the chosen test, calculate a test statistic that reflects how much the observed data deviates from the null hypothesis.

Determine the p-value

The p-value is the probability of observing test results at least as extreme as the results observed, assuming the null hypothesis is correct. It helps determine the strength of the evidence against the null hypothesis.

Make a Decision

Compare the p-value to the chosen significance level:

• If the p-value ≤ α: Reject the null hypothesis, suggesting sufficient evidence in the data supports the alternative hypothesis.
• If the p-value > α: Do not reject the null hypothesis, suggesting insufficient evidence to support the alternative hypothesis.

Report the Results

Present the findings from the hypothesis test, including the test statistic, p-value, and the conclusion about the hypotheses.

Perform Post-hoc Analysis (if necessary)

Depending on the results and the study design, further analysis may be needed to explore the data more deeply or to address multiple comparisons if several hypotheses were tested simultaneously.

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.

3. 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.

ANOVA , or Analysis of Variance, is a statistical method used to compare the means of three or more groups. It’s particularly useful when you want to see if there are significant differences between multiple groups. For instance, in business, a company might use ANOVA to analyze whether three different stores are performing differently in terms of sales. It’s also widely used in fields like medical research and social sciences, where comparing group differences can provide valuable insights.

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.

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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].

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Practice Problems on Hypothesis Testing

Here are the practice problems on hypothesis testing that will help you understand how to apply these concepts in real-world scenarios:

A telecom service provider claims that customers spend an average of ₹400 per month, with a standard deviation of ₹25. However, a random sample of 50 customer bills shows a mean of ₹250 and a standard deviation of ₹15. Does this sample data support the service provider’s claim?

Solution: Let’s break this down:

• Null Hypothesis (H0): The average amount spent per month is ₹400.
• Alternate Hypothesis (H1): The average amount spent per month is not ₹400.
• Population Standard Deviation (σ): ₹25
• Sample Size (n): 50
• Sample Mean (x̄): ₹250

1. Calculate the z-value:

z=250-40025/50 −42.42

2. Compare with critical z-values: For a 5% significance level, critical z-values are -1.96 and +1.96. Since -42.42 is far outside this range, we reject the null hypothesis. The sample data suggests that the average amount spent is significantly different from ₹400.

Out of 850 customers, 400 made online grocery purchases. Can we conclude that more than 50% of customers are moving towards online grocery shopping?

Solution: Here’s how to approach it:

• Proportion of customers who shopped online (p): 400 / 850 = 0.47
• Null Hypothesis (H0): The proportion of online shoppers is 50% or more.
• Alternate Hypothesis (H1): The proportion of online shoppers is less than 50%.
• Sample Size (n): 850
• Significance Level (α): 5%

z=p-PP(1-P)/n

z=0.47-0.500.50.5/850  −1.74

2. Compare with the critical z-value: For a 5% significance level (one-tailed test), the critical z-value is -1.645. Since -1.74 is less than -1.645, we reject the null hypothesis. This means the data does not support the idea that most customers are moving towards online grocery shopping.

In a study of code quality, Team A has 250 errors in 1000 lines of code, and Team B has 300 errors in 800 lines of code. Can we say Team B performs worse than Team A?

Solution: Let’s analyze it:

• Proportion of errors for Team A (pA): 250 / 1000 = 0.25
• Proportion of errors for Team B (pB): 300 / 800 = 0.375
• Null Hypothesis (H0): Team B’s error rate is less than or equal to Team A’s.
• Alternate Hypothesis (H1): Team B’s error rate is greater than Team A’s.
• Sample Size for Team A (nA): 1000
• Sample Size for Team B (nB): 800

p=nApA+nBpBnA+nB

p=10000.25+8000.3751000+800 ≈ 0.305

z=​pA−pB​p(1-p)(1nA+1nB)

z=​0.25−0.375​0.305(1-0.305) (11000+1800) ≈ −5.72

2. Compare with the critical z-value: For a 5% significance level (one-tailed test), the critical z-value is +1.645. Since -5.72 is far less than +1.645, we reject the null hypothesis. The data indicates that Team B’s performance is significantly worse than Team A’s.

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Applications of Hypothesis Testing

Apart from the practical problems, let's look at the real-world applications of hypothesis testing across various fields:

Medicine and Healthcare

In medicine, hypothesis testing plays a pivotal role in assessing the success of new treatments. For example, researchers may want to find out if a new exercise regimen improves heart health. By comparing data from patients who followed the program to those who didn’t, they can determine if the exercise significantly improves health outcomes. Such rigorous testing allows medical professionals to rely on proven methods rather than assumptions.

Quality Control and Manufacturing

In manufacturing, ensuring product quality is vital, and hypothesis testing helps maintain those standards. Suppose a beverage company introduces a new bottling process and wants to verify if it reduces contamination. By analyzing samples from the new and old processes, hypothesis testing can reveal whether the new method reduces the risk of contamination. This allows manufacturers to implement improvements that enhance product safety and quality confidently.

Education and Learning

In education and learning, hypothesis testing is a tool to evaluate the impact of innovative teaching techniques. Imagine a situation where teachers introduce project-based learning to boost critical thinking skills. By comparing the performance of students who engaged in project-based learning with those in traditional settings, educators can test their hypothesis. The results can help educators make informed choices about adopting new teaching strategies.

Environmental Science

Hypothesis testing is essential in environmental science for evaluating the effectiveness of conservation measures. For example, scientists might explore whether a new water management strategy improves river health. By collecting and comparing data on water quality before and after the implementation of the strategy, they can determine whether the intervention leads to positive changes. Such findings are crucial for guiding environmental decisions that have long-term impacts.

Marketing and Advertising

In marketing, businesses use hypothesis testing to refine their approaches. For instance, a clothing brand might test if offering limited-time discounts increases customer loyalty. By running campaigns with and without the discount and analyzing the outcomes, they can assess if the strategy boosts customer retention. Data-driven insights from hypothesis testing enable companies to design marketing strategies that resonate with their audience and drive growth.

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.

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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.

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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 H0 and H1 in statistics?

In statistics, H0​ and H1​ represent the null and alternative hypotheses. The null hypothesis, H0​, is the default assumption that no effect or difference exists between groups or conditions. The alternative hypothesis, H1​, is the competing claim suggesting an effect or a difference. Statistical tests determine whether to reject the null hypothesis in favor of the alternative hypothesis based on the data.

3. What is a simple hypothesis with an example?

A simple hypothesis is a specific statement predicting a single relationship between two variables. It posits a direct and uncomplicated outcome. For example, a simple hypothesis might state, "Increased sunlight exposure increases the growth rate of sunflowers." Here, the hypothesis suggests a direct relationship between the amount of sunlight (independent variable) and the growth rate of sunflowers (dependent variable), with no additional variables considered.

4. 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.

5. What software tools can assist with hypothesis testing?

Several software tools offering distinct features can help with hypothesis testing. R and RStudio are popular for their advanced statistical capabilities. The Python ecosystem, including libraries like SciPy and Statsmodels, also supports hypothesis testing. SAS and SPSS are well-established tools for comprehensive statistical analysis. For basic testing, Excel offers simple built-in functions.

6. How do I interpret the results of a hypothesis test?

Interpreting hypothesis test results involves comparing the p-value to the significance level (alpha). If the p-value is less than or equal to alpha, you can reject the null hypothesis, indicating statistical significance. This suggests that the observed effect is unlikely to have occurred by chance, validating your analysis findings.

7. Why is sample size important in hypothesis testing?

Sample size is crucial in hypothesis testing as it affects the test’s power. A larger sample size increases the likelihood of detecting a true effect, reducing the risk of Type II errors. Conversely, a small sample may lack the statistical power needed to identify differences, potentially leading to inaccurate conclusions.

8. Can hypothesis testing be used for non-numerical data?

Yes, hypothesis testing can be applied to non-numerical data through non-parametric tests. These tests are ideal when data doesn't meet parametric assumptions or when dealing with categorical data. Non-parametric tests, like the Chi-square or Mann-Whitney U test, provide robust methods for analyzing non-numerical data and drawing meaningful conclusions.

9. How do I choose the proper hypothesis test?

Selecting the right hypothesis test depends on several factors: the objective of your analysis, the type of data (numerical or categorical), and the sample size. Consider whether you're comparing means, proportions, or associations, and whether your data follows a normal distribution. The correct choice ensures accurate results tailored to your research question.

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• Knowledge Base
• Choosing the Right Statistical Test | Types & Examples

Choosing the Right Statistical Test | Types & Examples

Published on January 28, 2020 by Rebecca Bevans . Revised on June 22, 2023.

Statistical tests are used in hypothesis testing . They can be used to:

• determine whether a predictor variable has a statistically significant relationship with an outcome variable.
• estimate the difference between two or more groups.

Statistical tests assume a null hypothesis of no relationship or no difference between groups. Then they determine whether the observed data fall outside of the range of values predicted by the null hypothesis.

If you already know what types of variables you’re dealing with, you can use the flowchart to choose the right statistical test for your data.

Statistical tests flowchart

Table of contents

What does a statistical test do, when to perform a statistical test, choosing a parametric test: regression, comparison, or correlation, choosing a nonparametric test, flowchart: choosing a statistical test, other interesting articles, frequently asked questions about statistical tests.

Statistical tests work by calculating a test statistic – a number that describes how much the relationship between variables in your test differs from the null hypothesis of no relationship.

It then calculates a p value (probability value). The p -value estimates how likely it is that you would see the difference described by the test statistic if the null hypothesis of no relationship were true.

If the value of the test statistic is more extreme than the statistic calculated from the null hypothesis, then you can infer a statistically significant relationship between the predictor and outcome variables.

If the value of the test statistic is less extreme than the one calculated from the null hypothesis, then you can infer no statistically significant relationship between the predictor and outcome variables.

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You can perform statistical tests on data that have been collected in a statistically valid manner – either through an experiment , or through observations made using probability sampling methods .

For a statistical test to be valid , your sample size needs to be large enough to approximate the true distribution of the population being studied.

To determine which statistical test to use, you need to know:

• whether your data meets certain assumptions.
• the types of variables that you’re dealing with.

Statistical assumptions

Statistical tests make some common assumptions about the data they are testing:

• Independence of observations (a.k.a. no autocorrelation): The observations/variables you include in your test are not related (for example, multiple measurements of a single test subject are not independent, while measurements of multiple different test subjects are independent).
• Homogeneity of variance : the variance within each group being compared is similar among all groups. If one group has much more variation than others, it will limit the test’s effectiveness.
• Normality of data : the data follows a normal distribution (a.k.a. a bell curve). This assumption applies only to quantitative data .

If your data do not meet the assumptions of normality or homogeneity of variance, you may be able to perform a nonparametric statistical test , which allows you to make comparisons without any assumptions about the data distribution.

If your data do not meet the assumption of independence of observations, you may be able to use a test that accounts for structure in your data (repeated-measures tests or tests that include blocking variables).

Types of variables

The types of variables you have usually determine what type of statistical test you can use.

Quantitative variables represent amounts of things (e.g. the number of trees in a forest). Types of quantitative variables include:

• Continuous (aka ratio variables): represent measures and can usually be divided into units smaller than one (e.g. 0.75 grams).
• Discrete (aka integer variables): represent counts and usually can’t be divided into units smaller than one (e.g. 1 tree).

Categorical variables represent groupings of things (e.g. the different tree species in a forest). Types of categorical variables include:

• Ordinal : represent data with an order (e.g. rankings).
• Nominal : represent group names (e.g. brands or species names).
• Binary : represent data with a yes/no or 1/0 outcome (e.g. win or lose).

Choose the test that fits the types of predictor and outcome variables you have collected (if you are doing an experiment , these are the independent and dependent variables ). Consult the tables below to see which test best matches your variables.

Parametric tests usually have stricter requirements than nonparametric tests, and are able to make stronger inferences from the data. They can only be conducted with data that adheres to the common assumptions of statistical tests.

The most common types of parametric test include regression tests, comparison tests, and correlation tests.

Regression tests

Regression tests look for cause-and-effect relationships . They can be used to estimate the effect of one or more continuous variables on another variable.

Comparison tests

Comparison tests look for differences among group means . They can be used to test the effect of a categorical variable on the mean value of some other characteristic.

T-tests are used when comparing the means of precisely two groups (e.g., the average heights of men and women). ANOVA and MANOVA tests are used when comparing the means of more than two groups (e.g., the average heights of children, teenagers, and adults).

Correlation tests

Correlation tests check whether variables are related without hypothesizing a cause-and-effect relationship.

These can be used to test whether two variables you want to use in (for example) a multiple regression test are autocorrelated.

Non-parametric tests don’t make as many assumptions about the data, and are useful when one or more of the common statistical assumptions are violated. However, the inferences they make aren’t as strong as with parametric tests.

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This flowchart helps you choose among parametric tests. For nonparametric alternatives, check the table above.

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
• Null hypothesis

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

Statistical tests commonly assume that:

• the data are normally distributed
• the groups that are being compared have similar variance
• the data are independent

If your data does not meet these assumptions you might still be able to use a nonparametric statistical test , which have fewer requirements but also make weaker inferences.

A test statistic is a number calculated by a  statistical test . It describes how far your observed data is from the  null hypothesis  of no relationship between  variables or no difference among sample groups.

The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis . Different test statistics are used in different statistical tests.

Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test . Significance is usually denoted by a p -value , or probability value.

Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. The most common threshold is p < 0.05, which means that the data is likely to occur less than 5% of the time under the null hypothesis .

When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.

Quantitative variables are any variables where the data represent amounts (e.g. height, weight, or age).

Categorical variables are any variables where the data represent groups. This includes rankings (e.g. finishing places in a race), classifications (e.g. brands of cereal), and binary outcomes (e.g. coin flips).

You need to know what type of variables you are working with to choose the right statistical test for your data and interpret your results .

Discrete and continuous variables are two types of quantitative variables :

• Discrete variables represent counts (e.g. the number of objects in a collection).
• Continuous variables represent measurable amounts (e.g. water volume or weight).

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Test Statistics: Definition, Formulas & Examples

03.17.2022 • 10 min read

Sarah Thomas

Subject Matter Expert

This article explains what a test statistic is, how to complete one with formulas, and how to find the value for t-tests.

In This Article

What is a Standardized Test Statistic?

The general formula for calculating test statistics, types of test statistics with formulas, difference between t-tests and z-tests and when to use each, how to interpret a test statistic, don't overpay for college statistics.

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A test statistic is a standardized score used in hypothesis testing. It tells you how likely the results obtained from your sample data are under the assumption that the null hypothesis is true. The more unlikely your results are under this assumption, the easier it becomes to reject the null hypothesis in favor of an alternative hypothesis. The more likely your results are, the harder it becomes to reject the null hypothesis.

There are different kinds of test statistics, but they all work the same way. A test statistic maps the value of a particular sample statistic (such as a sample mean or a sample proportion) to a value on a standardized distribution, such as the Standard Normal Distribution or the t-distribution. This allows you to determine how likely or unlikely it is to observe the particular value of the statistic you obtained.

Olanrewaju Michael Akande reviews normal distribution in the following lesson clip:

As a quick example, say you have a null hypothesis that the average wait time to get seated at your favorite restaurant—at a table for two without a reservation on a Friday night—is 45 minutes. You select a random sample of 100 parties that got seated under these conditions and ask them what their wait times were. You find that the average wait time for your sample is 55 minutes ( x ˉ \bar{x} x ˉ = 55 minutes). A test statistic will convert this sample statistic x ˉ \bar{x} x ˉ into a standardized number that helps you answer this question:

“Assuming that my null hypothesis is true—assuming that the average wait time at the restaurant actually is 45 minutes—what is the likelihood that I found an average wait time of 55 minutes for my randomly drawn sample?”

Remember, the lower the likelihood of observing your sample statistic, the more confident you can be rejecting the null hypothesis.

The type of test statistic you use in a hypothesis test depends on several factors including:

The type of statistic you are using in the test

The size of your sample

Assumptions you can make about the distribution of your data

Assumptions you can make about the distribution of the statistic used in the test

The formula for calculating test statistics takes the following general form:

Remember, a statistic is a measure calculated from a single sample or many samples. Examples include the sample mean x ˉ \bar{x} x ˉ , the difference between two sample means x 1 ˉ − x 2 ˉ \bar{x_{1}} - \bar{x_{2}} x 1 ​ ˉ ​ − x 2 ​ ˉ ​ , or a sample proportion p ^ \hat{p} p ^ ​ .

A parameter is a measure calculated from a single population or many populations. Examples include the population mean μ \mu μ , the difference between two population means μ 1 − μ 2 \mu_{1}-\mu_{2} μ 1 ​ − μ 2 ​ , or a population proportion p p p .

In the denominator of the equation, you have the standard deviation—or the approximated standard deviation—of the statistic used in the numerator. If you use the sample mean x ˉ \bar{x} x ˉ , in the numerator, you should use the standard deviation of x ˉ \bar{x} x ˉ or an approximation of it in the denominator.

The test statistics you are most likely to encounter in an introductory statistics class are:

The Z-test statistic for a single sample mean

The Z-test statistic for population proportions

The t-test statistic for a single sample mean

The t-test statistic for two sample means

Z-test for a Sample Mean

We use the Z-test statistic (or Z-statistic) for a sample mean in hypothesis tests involving a sample mean x ˉ \bar{x} x ˉ , calculated for a single sample.

You use this test statistic when:

Your sample size is greater than or equal to 30 (n ≥ \geq ≥ 30)

The sampling distribution of the sample mean is assumed to be normal

The standard deviation of the population parameter σ \sigma σ is known.

The formula for this type of Z-test statistic is:

Z Z Z is the symbol for the Z-test statistic

x ˉ \bar{x} x ˉ is the sample mean

μ 0 \mu_{0} μ 0 ​ is the hypothesized value of the population mean according to the null hypothesis

σ \sigma σ is the population standard deviation

n n n is the sample size

σ n \frac{\sigma}{\sqrt{n}} n ​ σ ​ is the standard error of x ˉ \bar{x} x ˉ . The standard error is just the standard deviation of the sampling distribution of the sample mean.

You may notice that a Z-test statistic is just a z-score for a particular value of a normally distributed statistic. There are many variations of the Z-test statistic. We can use these in hypothesis tests, where the sample statistic is being used in the test is approximately normally distributed. One such variation of the Z-test statistic is the Z-test for proportions.

Z-test for Proportions

We use the Z-test statistic for proportions in hypothesis tests where a sample proportion p ^ \hat{p} p ^ ​ is being tested against the hypothesized value of the population proportion, p 0 p_{0} p 0 ​ . We use the Z-test for proportions when your sample size is greater than or equal to 30 (n ≥ \geq ≥ 30), and the distribution of the sample statistic is assumed to be normal. The formula for the Z-test statistic for population proportions is:

Z is the symbol for the Z-test statistic for population proportions

p ^ \hat{p} p ^ ​ is the sample proportion

p 0 p_{0} p 0 ​ is the hypothesized value of the population proportion according to the null hypothesis

When your sample size is smaller than 30 (n<30)—or when you cannot assume that the distribution of your sample statistic is normally distributed—you’ll often use a t-test statistic rather than a Z-test.

T-test for a Single Sample Mean

We use the t-test statistic (or t-statistic) for a sample mean in hypothesis tests involving a sample mean calculated for a single sample drawn from a population. Unlike the Z-test for a single sample mean, you use the t-test when:

Your sample size is less than 30 (n<30)

The distribution of the sample statistic is not approximated by a normal distribution

The standard deviation of the population parameter σ \sigma σ is unknown

A t-test statistic maps your statistics to a t-distribution as opposed to the normal distribution with a Z-test. A t-distribution is like a standard normal distribution, but it has thicker tails and changes depending on your sample size n n n . When n n n is large, the t-distribution is closer to the normal distribution; and as the sample size gets larger and larger, a t-distribution will converge to the normal distribution. As n n n gets smaller, the t-distribution gets flatter with thicker tails.

The formula for the t-test statistic for a sample mean is:

t t t is the symbol for the t-test statistic

μ 0 \mu_0 μ 0 ​ is the value of the population mean according to the null hypothesis

s s s is the sample standard deviation

s n \frac{s}{\sqrt{n}} n ​ s ​ is an approximation of the standard error of x ˉ \bar{x} x ˉ . In a t-test, because you do not know the value of the population standard deviation, you need to approximate the standard error of x ˉ \bar{x} x ˉ using the sample standard deviation s s s .

T-test for Two Sample Means

We can also use t-test statistics in hypothesis tests where the values of two sample means ( x 1 x_{1} x 1 ​ and x 2 x_{2} x 2 ​ ) are being compared. You do this to test the null hypothesis that the two samples are drawn from the same underlying population. If the null hypothesis is true, then any difference between the sample means is due to random variations in the data. Rejecting the null hypothesis suggests that the samples were drawn from two distinct populations and that the difference in the sample means reflects actual differences in the characteristics of subjects in one population compared to the other.

Like the t-test for a single sample mean, you use the t-test for two sample means when:

Your sample sizes are less than 30 (n<30)

The distribution of the sample statistics are not approximated by a normal distribution

The formula for the t-test statistic for two sample means is‌:

x 1 ˉ \bar{x_1} x 1 ​ ˉ ​ is the sample mean of sample 1

x 2 ˉ \bar{x_2} x 2 ​ ˉ ​ is the sample mean of sample 2

μ 1 \mu_1 μ 1 ​ is the mean of the population from which sample 1 was drawn

μ 2 \mu_2 μ 2 ​ is the mean of the population from which sample 2 was drawn

s 1 2 s_1^2 s 1 2 ​ is the variance of sample 1

s 2 2 s_2^2 s 2 2 ​ is the variance of sample 2

n 1 n_{1} n 1 ​ is the sample size for sample 1

n 2 n_{2} n 2 ​ is the sample size for sample 2

T-tests are generally used in place of Z-tests when one or more of the following conditions hold: The sample size is less than 30 (n \sigma is unknown

If you know the population standard deviation σ \sigma σ and you are confident that the statistic used in your hypothesis test is normally distributed, then you can use a Z-test.

As with all test statistics, you should only use a Z-test or a t-test when your data is from a randomly and independently drawn sample.

We use test statistics together with critical values, p-values, and significance levels to determine whether to reject or not a null hypothesis.

A critical value is a value of a test statistic that marks a cutoff point. If a test statistic is more extreme than the critical value—greater than the critical value in the right tail of a distribution or less than the critical value in the left tail of a distribution—the null hypothesis is rejected.

Critical values are determined by the significance level (or alpha level) of a hypothesis test. The significance level you use is up to you, but the most commonly used significance level is 0.05 ( α \alpha α =0.05).

A significance level of 0.05 means that if the probability of observing a sample statistic at least as extreme as the one you observed is less than 0.05 (or 5%), you should reject your null hypothesis. In a one-sided hypothesis test that uses a Z-test statistic, a significance level of 0.05 is associated with a critical value of 1.645 when you conduct the test in the right tail and a value of -1.645 when you conduct the test in the left tail.

A p-value is the probability associated with your test statistic’s value. Let’s say you calculate a Z-test statistic that maps to the standard normal distribution. You find that the test statistic is equal to 1.75. For this ‌value of a Z-test statistic, the associated p-value is 0.04 or 4%—you can find p-values using tables or statistical software.

A p-value of 0.04 means that the probability of observing a sample statistic at least as extreme as the one you found from your sample data is 4%. If you choose a significance level of 0.05 for your test, we would reject the null hypothesis, since the p-value of 0.04 is less than the significance level of 0.05.

It can be easy to confuse test statistics, critical values, significance levels, and p-values. Remember, these are all different measures involved in determining whether to reject or fail to reject a null hypothesis.

Critical values and significance levels provide cut-offs for your test. The difference between a critical value and a significance level is that the critical value is a point on the distribution, and the significance level is a probability represented by an area under the distribution.

You can compare the test statistic and the p-value against the critical value and the significance level.

If the test statistic is more extreme than the critical value, you reject the null hypothesis.

If the p-value is less than the significance level, you reject the null hypothesis.

If the test statistic is less extreme than the critical value, you fail to reject the null hypothesis.

If the p-value is greater than the significance level, you reject the null hypothesis.

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• How to Conduct Hypothesis Testing in Statistics

The Fundamentals of Hypothesis Testing: What Every Student Should Know

Hypothesis testing is a fundamental statistical technique used to make inferences about populations based on sample data. This blog will guide you through the process of hypothesis testing, helping you understand and apply the concepts to solve similar assignments efficiently. By following this structured approach, you'll be able to solve your hypothesis testing homework problem with confidence.

Understanding the Basics of Hypothesis Testing

Hypothesis testing involves making a decision about the validity of a hypothesis based on sample data. It comprises four key steps: defining hypotheses, calculating the test statistic, determining the p-value, and drawing conclusions. Let's explore each of these steps in detail.

Defining Hypotheses

The first step in hypothesis testing is to define the null and alternative hypotheses. These hypotheses represent the statements we want to test.

Null Hypothesis (H0)

The null hypothesis (H0) is a statement that there is no effect or difference. It serves as the default assumption that we aim to test against.

Alternative Hypothesis (Ha or H1)

The alternative hypothesis (Ha or H1) is a statement that indicates the presence of an effect or difference. It represents what we want to prove.

Types of Tests

Depending on the direction of the hypothesis, we have three types of tests: left-tailed, right-tailed, and two-tailed tests.

Left-Tailed Test

A left-tailed test is used when we want to determine if the population mean is less than a specified value.

Right-Tailed Test

A right-tailed test is used when we want to determine if the population mean is greater than a specified value.

Two-Tailed Test

A two-tailed test is used when we want to determine if the population mean is different from a specified value, either higher or lower.

Example Scenario

Consider a scenario where we want to test if the average vehicle price from a sample is less than $27,000. We would set up our hypotheses as follows:

• Null Hypothesis (H0): μ = 27,000
• Alternative Hypothesis (Ha): μ < 27,000

Calculating the Test Statistic

Once the hypotheses are defined, the next step is to calculate the test statistic. The test statistic helps us determine the likelihood of observing the sample data under the null hypothesis.

Formula for the T-Test Statistic

The t-test statistic is calculated using the formula:

[ t = \frac{\bar{X} - \mu}{S / \sqrt{n}} ]

• (\bar{X}) is the sample mean
• (S) is the sample standard deviation
• (n) is the sample size
• (\mu) is the population mean defined in the null hypothesis

Standard Error

The denominator of the t-test formula, (S / \sqrt{n}), is known as the standard error (SE). It measures the variability of the sample mean.

Example Calculation

Let's calculate the test statistic for our vehicle price example. Given:

• Sample mean ((\bar{X})) = 25,650
• Sample standard deviation (S) = 3,488
• Sample size (n) = 10
• Population mean ((\mu)) = 27,000

First, we calculate the standard error (SE):

[ SE = \frac{S}{\sqrt{n}} = \frac{3488}{\sqrt{10}} \approx 1103 ]

Next, we calculate the test statistic (t):

[ t = \frac{25650 - 27000}{1103} \approx -1.2238 ]

Determining the P-Value

The p-value is a critical component of hypothesis testing. It indicates the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true.

Calculating the P-Value

The method to calculate the p-value depends on the type of test (left-tailed, right-tailed, or two-tailed) and the direction of the alternative hypothesis.

For a left-tailed test, the p-value is calculated using the T.DIST() function in Excel.

For a right-tailed test, the p-value is calculated using the T.DIST.RT() function in Excel.

For a two-tailed test, the p-value is calculated using the T.DIST.2T() function in Excel. When the test statistic is negative, use the absolute value function (ABS()) to remove the negative sign before calculating the p-value.

For our vehicle price example with a left-tailed test, we calculate the p-value using the T.DIST() function in Excel:

[ \text{p-value} = T.DIST(-1.2238, 9, TRUE) \approx 0.1261 ]

Drawing Conclusions

The final step in hypothesis testing is to draw a conclusion based on the p-value and a pre-determined significance level ((\alpha)).

Significance Level ((\alpha))

The significance level ((\alpha)) is the threshold for deciding whether to reject the null hypothesis. Common values for (\alpha) are 0.05, 0.01, 0.10, and 0.005.

Decision Rule

• If the p-value is less than (\alpha), we reject the null hypothesis.
• If the p-value is greater than (\alpha), we fail to reject the null hypothesis.

Example Conclusion

For our vehicle price example with (\alpha = 0.05):

• p-value = 0.1261
• (\alpha) = 0.05

Since 0.1261 > 0.05, we fail to reject the null hypothesis. There is not enough evidence to suggest that the average vehicle price is less than $27,000.

Practical Examples of Hypothesis Testing

To further illustrate hypothesis testing, let's explore three different scenarios: left-tailed test, right-tailed test, and two-tailed test.

Left-Tailed Test Example

In this example, we test if the average vehicle price is less than $27,000.

Step-by-Step Process

Define Hypotheses:

Calculate Test Statistic:

• Standard error (SE) = 1103
• Test statistic (t) = -1.2238

Determine P-Value:

Draw Conclusion:

• Since 0.1261 > 0.05, fail to reject the null hypothesis.
• Conclusion: There is not enough evidence to suggest that the average vehicle price is less than $27,000.

Right-Tailed Test Example

In this example, we test if the average vehicle price is greater than $23,500.

• Null Hypothesis (H0): μ = 23,500
• Alternative Hypothesis (Ha): μ > 23,500
• Population mean ((\mu)) = 23,500
• Test statistic (t) = 1.9490
• p-value = 0.0416
• Since 0.0416 < 0.05, reject the null hypothesis.
• Conclusion: There is enough evidence to suggest that the average vehicle price is greater than $23,500.

Two-Tailed Test Example

In this example, we test if the average vehicle price is different from $23,500.

• Alternative Hypothesis (Ha): μ ≠ 23,500
• p-value = 0.0831
• Since 0.0831 > 0.05, fail to reject the null hypothesis.
• Conclusion: There is not enough evidence to suggest that the average vehicle price is different from $23,500.

Tips for Conducting Hypothesis Testing

Successfully conducting hypothesis testing involves several critical steps. Here are some tips to help you perform hypothesis testing effectively.

Proper Data Collection

Accurate and reliable data collection is crucial for hypothesis testing. Ensure that your sample is representative of the population and collected using appropriate methods.

Random Sampling

Use random sampling techniques to avoid bias and ensure that your sample accurately represents the population.

Sample Size

Ensure that your sample size is large enough to provide reliable results. Larger sample sizes reduce the margin of error and increase the power of the test.

Verify Assumptions

Hypothesis tests often rely on certain assumptions about the data. Verify these assumptions before proceeding with the test.

Many hypothesis tests, including the t-test, assume that the data follows a normal distribution. Use graphical methods (e.g., histograms, Q-Q plots) or statistical tests (e.g., Shapiro-Wilk test) to check for normality.

Independence

Ensure that the observations in your sample are independent of each other. Independence is a key assumption for most hypothesis tests.

Utilize Software Tools

Software tools like Excel , R , and SPSS can simplify the calculations involved in hypothesis testing and reduce the risk of errors.

Excel provides several functions for hypothesis testing, such as T.DIST(), T.DIST.RT(), and T.DIST.2T(). Use these functions to calculate p-values and make decisions based on your test statistics.

R is a powerful statistical software that offers various packages for hypothesis testing. Use functions like t.test() to perform t-tests and obtain p-values and confidence intervals.

Interpret Results Carefully

Proper interpretation of the results is crucial for drawing accurate conclusions from hypothesis testing.

Statistical Significance

A statistically significant result (p-value < (\alpha)) indicates that there is strong evidence against the null hypothesis. However, it does not imply practical significance. Consider the context and the practical implications of the results.

Type I and Type II Errors

Be aware of the potential for Type I and Type II errors. A Type I error occurs when the null hypothesis is incorrectly rejected, while a Type II error occurs when the null hypothesis is not rejected despite being false. The significance level ((\alpha)) affects the probability of Type I errors, while the sample size and effect size influence the probability of Type II errors.

Report Results Transparently

When reporting the results of hypothesis testing, include all relevant information to ensure transparency and reproducibility.

Detailed Description

Provide a detailed description of the hypotheses, test statistic, p-value, significance level, and the conclusion. This information helps others understand and evaluate your analysis.

Confidence Intervals

Include confidence intervals for the estimated parameters. Confidence intervals provide a range of plausible values for the population parameter and offer additional context for interpreting the results.

Common Pitfalls in Hypothesis Testing

Hypothesis testing is a powerful tool, but it is essential to be aware of common pitfalls to avoid incorrect conclusions.

Misinterpreting P-Values

P-values indicate the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true. A small p-value suggests strong evidence against the null hypothesis, but it does not provide a measure of the effect size or practical significance.

P-Value Misconceptions

Avoid common misconceptions about p-values, such as believing that a p-value of 0.05 means there is a 5% chance that the null hypothesis is true. P-values do not measure the probability that the null hypothesis is true or false.

Ignoring Assumptions

Ignoring the assumptions underlying hypothesis tests can lead to incorrect conclusions. Always verify the assumptions before proceeding with the test.

Assumption Violations

If the assumptions are violated, consider using alternative tests that do not rely on those assumptions. For example, if the data is not normally distributed, use non-parametric tests like the Wilcoxon rank-sum test or the Mann-Whitney U test.

Overemphasizing Statistical Significance

Statistical significance does not imply practical significance. A result can be statistically significant but have a negligible practical effect. Always consider the context and practical implications of the results.

Effect Size

Report and interpret effect sizes alongside p-values. Effect sizes provide a measure of the magnitude of the observed effect and offer valuable context for interpreting the results.

Hypothesis testing is a critical tool in statistics for making inferences about populations based on sample data. By understanding the steps involved—defining hypotheses, calculating the test statistic, determining the p-value, and drawing conclusions—you can approach hypothesis testing with confidence.

Ensure proper data collection, verify assumptions, utilize software tools, interpret results carefully, and report findings transparently to enhance the reliability and validity of your hypothesis tests. By avoiding common pitfalls and considering both statistical and practical significance, you'll be well-equipped to tackle statistics homework and research projects effectively.

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Statistics By Jim

Making statistics intuitive

How t-Tests Work: t-Values, t-Distributions, and Probabilities

By Jim Frost 12 Comments

T-tests are statistical hypothesis tests that you use to analyze one or two sample means. Depending on the t-test that you use, you can compare a sample mean to a hypothesized value, the means of two independent samples, or the difference between paired samples. In this post, I show you how t-tests use t-values and t-distributions to calculate probabilities and test hypotheses.

As usual, I’ll provide clear explanations of t-values and t-distributions using concepts and graphs rather than formulas! If you need a primer on the basics, read my hypothesis testing overview .

What Are t-Values?

The term “t-test” refers to the fact that these hypothesis tests use t-values to evaluate your sample data. T-values are a type of test statistic. Hypothesis tests use the test statistic that is calculated from your sample to compare your sample to the null hypothesis. If the test statistic is extreme enough, this indicates that your data are so incompatible with the null hypothesis that you can reject the null. Learn more about Test Statistics .

Don’t worry. I find these technical definitions of statistical terms are easier to explain with graphs, and we’ll get to that!

When you analyze your data with any t-test, the procedure reduces your entire sample to a single value, the t-value. These calculations factor in your sample size and the variation in your data. Then, the t-test compares your sample means(s) to the null hypothesis condition in the following manner:

• If the sample data equals the null hypothesis precisely, the t-test produces a t-value of 0.
• As the sample data become progressively dissimilar from the null hypothesis, the absolute value of the t-value increases.

Read the companion post where I explain how t-tests calculate t-values .

The tricky thing about t-values is that they are a unitless statistic, which makes them difficult to interpret on their own. Imagine that we performed a t-test, and it produced a t-value of 2. What does this t-value mean exactly? We know that the sample mean doesn’t equal the null hypothesis value because this t-value doesn’t equal zero. However, we don’t know how exceptional our value is if the null hypothesis is correct.

To be able to interpret individual t-values, we have to place them in a larger context. T-distributions provide this broader context so we can determine the unusualness of an individual t-value.

What Are t-Distributions?

A single t-test produces a single t-value. Now, imagine the following process. First, let’s assume that the null hypothesis is true for the population. Now, suppose we repeat our study many times by drawing many random samples of the same size from this population. Next, we perform t-tests on all of the samples and plot the distribution of the t-values. This distribution is known as a sampling distribution, which is a type of probability distribution.

Related posts : Sampling Distributions and Understanding Probability Distributions

If we follow this procedure, we produce a graph that displays the distribution of t-values that we obtain from a population where the null hypothesis is true. We use sampling distributions to calculate probabilities for how unusual our sample statistic is if the null hypothesis is true.

Luckily, we don’t need to go through the hassle of collecting numerous random samples to create this graph! Statisticians understand the properties of t-distributions so we can estimate the sampling distribution using the t-distribution and our sample size.

The degrees of freedom (DF) for the statistical design define the t-distribution for a particular study. The DF are closely related to the sample size. For t-tests, there is a different t-distribution for each sample size.

Related posts : Degrees of Freedom in Statistics and T Distribution: Definition and Uses .

Use the t-Distribution to Compare Your Sample Results to the Null Hypothesis

T-distributions assume that the null hypothesis is correct for the population from which you draw your random samples. To evaluate how compatible your sample data are with the null hypothesis, place your study’s t-value in the t-distribution and determine how unusual it is.

The sampling distribution below displays a t-distribution with 20 degrees of freedom, which equates to a sample size of 21 for a 1-sample t-test. The t-distribution centers on zero because it assumes that the null hypothesis is true. When the null is true, your study is most likely to obtain a t-value near zero and less liable to produce t-values further from zero in either direction.

On the graph, I’ve displayed the t-value of 2 from our hypothetical study to see how our sample data compares to the null hypothesis. Under the assumption that the null is true, the t-distribution indicates that our t-value is not the most likely value. However, there still appears to be a realistic chance of observing t-values from -2 to +2.

We know that our t-value of 2 is rare when the null hypothesis is true. How rare is it exactly? Our final goal is to evaluate whether our sample t-value is so rare that it justifies rejecting the null hypothesis for the entire population based on our sample data. To proceed, we need to quantify the probability of observing our t-value.

Related post : What are Critical Values?

t-Tests Use t-Values and t-Distributions to Calculate Probabilities

Hypothesis tests work by taking the observed test statistic from a sample and using the sampling distribution to calculate the probability of obtaining that test statistic if the null hypothesis is correct. In the context of how t-tests work, you assess the likelihood of a t-value using the t-distribution. If a t-value is sufficiently improbable when the null hypothesis is true, you can reject the null hypothesis.

I have two crucial points to explain before we calculate the probability linked to our t-value of 2.

Because I’m showing the results of a two-tailed test, we’ll use the t-values of +2 and -2. Two-tailed tests allow you to assess whether the sample mean is greater than or less than the target value in a 1-sample t-test. A one-tailed hypothesis test can only determine statistical significance for one or the other.

Additionally, it is possible to calculate a probability only for a range of t-values. On a probability distribution plot, probabilities are represented by the shaded area under a distribution curve. Without a range of values, there is no area under the curve and, hence, no probability.

Related posts : One-Tailed and Two-Tailed Tests Explained and T-Distribution Table of Critical Values

t-Test Results for Our Hypothetical Study

Considering these points, the graph below finds the probability associated with t-values less than -2 and greater than +2 using the area under the curve. This graph is specific to our t-test design (1-sample t-test with N = 21).

The probability distribution plot indicates that each of the two shaded regions has a probability of 0.02963—for a total of 0.05926. This graph shows that t-values fall within these areas almost 6% of the time when the null hypothesis is true.

There is a chance that you’ve heard of this type of probability before—it’s the P value! While the likelihood of t-values falling within these regions seems small, it’s not quite unlikely enough to justify rejecting the null under the standard significance level of 0.05.

Learn how to interpret the P value correctly and avoid a common mistake!

Related posts : How to Find the P value: Process and Calculations and Types of Errors in Hypothesis Testing

t-Distributions and Sample Size

The sample size for a t-test determines the degrees of freedom (DF) for that test, which specifies the t-distribution. The overall effect is that as the sample size decreases, the tails of the t-distribution become thicker. Thicker tails indicate that t-values are more likely to be far from zero even when the null hypothesis is correct. The changing shapes are how t-distributions factor in the greater uncertainty when you have a smaller sample.

You can see this effect in the probability distribution plot below that displays t-distributions for 5 and 30 DF.

Sample means from smaller samples tend to be less precise. In other words, with a smaller sample, it’s less surprising to have an extreme t-value, which affects the probabilities and p-values. A t-value of 2 has a P value of 10.2% and 5.4% for 5 and 30 DF, respectively. Use larger samples!

Click here for step-by-step instructions for how to do t-tests in Excel !

If you like this approach and want to learn about other hypothesis tests, read my posts about:

• How the F-test Works in ANOVA .
• How Chi-Squared Tests of Independence Work

To see an alternative to traditional hypothesis testing that does not use probability distributions and test statistics, learn about bootstrapping in statistics !

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Reader Interactions

May 25, 2021 at 10:42 pm

what statistical tools, is recommended for measuring the level of satisfaction

May 26, 2021 at 3:55 pm

Hi McKienze,

The correct analysis depends on the nature of the data you have and what you want to learn. You don’t provide enough information to be able to answer the question. However, read my hypothesis testing overview to learn about the options.

August 23, 2020 at 1:33 am

Hi Jim, I want to ask about standardizing data before the t test.. For example I have USD prices of a big Mac across the world and this varies by quite a bit. Doing the t-test here would be misleading since some countries would have a higher mean… Should the approach be standardizing all the usd values? Or perhaps even local values?

August 24, 2020 at 12:37 am

Yes, that makes complete sense. I don’t know what method is best. If you can find a common scale to use for all prices, I’d do that. You’re basically using a data transformation before analysis, which is totally acceptable when you have a good reason.

April 3, 2020 at 4:47 am

Hey Jim. Your blog is one of the only few ones where everything is explained in a simple and well structured manner, in a way that both an absolute beginner and a geek can equally benefit from your writing. Both this article as well as your article on one tailed and two tailed hypothesis tests have been super helpful. Thank you for this post

March 6, 2020 at 11:04 am

Thank you, Jim, for sharing your knowledge with us.

I have a 2 part question. I am testing the difference in walking distance within a busy environment compared with a simple environment. I am also testing walking time within the 2 environments. I am using the same individuals for both scenarios. I was planning to do a paired ttest for distance difference between busy and simple environments and a 2nd paired ttest for time difference between the environments.

My question(s) for you is: 1. Do you feel that a paired ttest is the best choice for these? 2. Do you feel that, because there are 2 tests, I should do a bonferroni correction or do you believe that because the data is completely different (distance as opposed to time), it is okay not to do a multiple comparison test?

August 13, 2019 at 12:43 pm

thank you very eye opening on the use of two or one tailed test

April 19, 2019 at 3:49 pm

Hi Mr. Frost,

Thanks for the breakdown. I have a question … if I wanted to run a test to show that the medical professionals could use more training with data set consisting of questions which in your opinion would be my best route?

January 14, 2019 at 2:22 pm

Hello Jim, I find this statement in this excellent write up contradicting : 1)This graph shows that t-values fall within these areas almost 6% of the time when the null hypothesis is true I mean if this is true the t-value =0 hypothesis is rejected.

January 14, 2019 at 2:51 pm

I can see how that statement sounds contradictory, but I can assure that it is quite accurate. It’s often forgotten but the underlying assumption for the calculations surrounding hypothesis testing, significance levels, and p-values is that the null hypothesis is true.

So, the probabilities shown in the graph that you refer to are based on the assumption that the null hypothesis is true. Further, t-values for this study design have a 6% chance of falling in those critical areas assuming the null is true (a false positive).

Significance levels are defined as the maximum acceptable probability of a false positive. Usually, we set that as 5%. In the example, there’s a large probability of a false positive (6%), so we fail to reject the null hypothesis. In other words, we fail to reject the null because false positives will happen too frequently–where the significance level defines the cutoff point for too frequently.

Keep in mind that when you have statistically significant results, you’re really saying that the results you obtained are improbable enough assuming that the null is true that you can reject the notion that the null is true. But, the math and probabilities are all based on the assumption that the null is true because you need to determine how unlikely your results are under the null hypothesis.

Even the p-value is defined in terms of assuming the null hypothesis is true. You can read about that in my post about interpreting p-values correctly .

I hope this clarifies things!

November 9, 2018 at 2:36 am

Jim …I was involved in in a free SAT/ACT tutoring program that I need to analyze for effectiveness .

I have pre test scores of a number of students and the post test scores after they were tutored (treatment ).

Glenn dowell

November 9, 2018 at 9:05 am

It sounds like you need to perform a paired t-test assuming.

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