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What Is a Two-Tailed Test?

Understanding a two-tailed test, special considerations, two-tailed vs. one-tailed test.

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What Is a Two-Tailed Test? Definition and Example

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A two-tailed test, in statistics, is a method in which the critical area of a distribution is two-sided and tests whether a sample is greater than or less than a certain range of values. It is used in null-hypothesis testing and testing for statistical significance . If the sample being tested falls into either of the critical areas, the alternative hypothesis is accepted instead of the null hypothesis.

Key Takeaways

• In statistics, a two-tailed test is a method in which the critical area of a distribution is two-sided and tests whether a sample is greater or less than a range of values.
• It is used in null-hypothesis testing and testing for statistical significance.
• If the sample being tested falls into either of the critical areas, the alternative hypothesis is accepted instead of the null hypothesis.
• By convention two-tailed tests are used to determine significance at the 5% level, meaning each side of the distribution is cut at 2.5%.

A basic concept of inferential statistics is hypothesis testing , which determines whether a claim is true or not given a population parameter. A hypothesis test that is designed to show whether the mean of a sample is significantly greater than and significantly less than the mean of a population is referred to as a two-tailed test. The two-tailed test gets its name from testing the area under both tails of a normal distribution , although the test can be used in other non-normal distributions.

A two-tailed test is designed to examine both sides of a specified data range as designated by the probability distribution involved. The probability distribution should represent the likelihood of a specified outcome based on predetermined standards. This requires the setting of a limit designating the highest (or upper) and lowest (or lower) accepted variable values included within the range. Any data point that exists above the upper limit or below the lower limit is considered out of the acceptance range and in an area referred to as the rejection range.

There is no inherent standard about the number of data points that must exist within the acceptance range. In instances where precision is required, such as in the creation of pharmaceutical drugs, a rejection rate of 0.001% or less may be instituted. In instances where precision is less critical, such as the number of food items in a product bag, a rejection rate of 5% may be appropriate.

A two-tailed test can also be used practically during certain production activities in a firm, such as with the production and packaging of candy at a particular facility. If the production facility designates 50 candies per bag as its goal, with an acceptable distribution of 45 to 55 candies, any bag found with an amount below 45 or above 55 is considered within the rejection range.

To confirm the packaging mechanisms are properly calibrated to meet the expected output, random sampling may be taken to confirm accuracy. A simple random sample takes a small, random portion of the entire population to represent the entire data set, where each member has an equal probability of being chosen.

For the packaging mechanisms to be considered accurate, an average of 50 candies per bag with an appropriate distribution is desired. Additionally, the number of bags that fall within the rejection range needs to fall within the probability distribution limit considered acceptable as an error rate. Here, the null hypothesis would be that the mean is 50 while the alternate hypothesis would be that it is not 50.

If, after conducting the two-tailed test, the z-score falls in the rejection region, meaning that the deviation is too far from the desired mean, then adjustments to the facility or associated equipment may be required to correct the error. Regular use of two-tailed testing methods can help ensure production stays within limits over the long term.

Be careful to note if a statistical test is one- or two-tailed as this will greatly influence a model's interpretation.

When a hypothesis test is set up to show that the sample mean would be only higher than the population mean, this is referred to as a  one-tailed test . A formulation of this hypothesis would be, for example, that "the returns on an investment fund would be  at least  x%." One-tailed tests could also be set up to show that the sample mean could be only less than the population mean. The key difference from a two-tailed test is that in a two-tailed test, the sample mean could be different from the population mean by being  either  higher or lower than it.

If the sample being tested falls into the one-sided critical area, the alternative hypothesis will be accepted instead of the null hypothesis. A one-tailed test is also known as a directional hypothesis or directional test.

A two-tailed test, on the other hand, is designed to examine both sides of a specified data range to test whether a sample is greater than or less than the range of values.

Example of a Two-Tailed Test

As a hypothetical example, imagine that a new  stockbroker , named XYZ, claims that their brokerage fees are lower than that of your current stockbroker, ABC) Data available from an independent research firm indicates that the mean and standard deviation of all ABC broker clients are $18 and $6, respectively.

A sample of 100 clients of ABC is taken, and brokerage charges are calculated with the new rates of XYZ broker. If the mean of the sample is $18.75 and the sample standard deviation is $6, can any inference be made about the difference in the average brokerage bill between ABC and XYZ broker?

• H 0 : Null Hypothesis: mean = 18
• H 1 : Alternative Hypothesis: mean <> 18 (This is what we want to prove.)
• Rejection region: Z <= - Z 2.5  and Z>=Z 2.5  (assuming 5% significance level, split 2.5 each on either side).
• Z = (sample mean – mean) / (std-dev / sqrt (no. of samples)) = (18.75 – 18) / (6/(sqrt(100)) = 1.25

This calculated Z value falls between the two limits defined by: - Z 2.5  = -1.96 and Z 2.5  = 1.96.

This concludes that there is insufficient evidence to infer that there is any difference between the rates of your existing broker and the new broker. Therefore, the null hypothesis cannot be rejected. Alternatively, the p-value = P(Z< -1.25)+P(Z >1.25) = 2 * 0.1056 = 0.2112 = 21.12%, which is greater than 0.05 or 5%, leads to the same conclusion.

How Is a Two-Tailed Test Designed?

A two-tailed test is designed to determine whether a claim is true or not given a population parameter. It examines both sides of a specified data range as designated by the probability distribution involved. As such, the probability distribution should represent the likelihood of a specified outcome based on predetermined standards.

What Is the Difference Between a Two-Tailed and One-Tailed Test?

A two-tailed hypothesis test is designed to show whether the sample mean is significantly greater than  or  significantly less than the mean of a population. The two-tailed test gets its name from testing the area under both tails (sides) of a normal distribution. A one-tailed hypothesis test, on the other hand, is set up to show only one test; that the sample mean would be higher than the population mean, or, in a separate test, that the sample mean would be lower than the population mean.

What Is a Z-score?

A Z-score numerically describes a value's relationship to the mean of a group of values and is measured in terms of the number of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean score whereas Z-scores of 1.0 and -1.0 would indicate values one standard deviation above or below the mean. In most large data sets, 99% of values have a Z-score between -3 and 3, meaning they lie within three standard deviations above and below the mean.

San Jose State University. " 6: Introduction to Null Hypothesis Significance Testing ."

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Research Hypothesis In Psychology: Types, & Examples

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Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

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A research hypothesis, in its plural form “hypotheses,” is a specific, testable prediction about the anticipated results of a study, established at its outset. It is a key component of the scientific method .

Hypotheses connect theory to data and guide the research process towards expanding scientific understanding

Some key points about hypotheses:

• A hypothesis expresses an expected pattern or relationship. It connects the variables under investigation.
• It is stated in clear, precise terms before any data collection or analysis occurs. This makes the hypothesis testable.
• A hypothesis must be falsifiable. It should be possible, even if unlikely in practice, to collect data that disconfirms rather than supports the hypothesis.
• Hypotheses guide research. Scientists design studies to explicitly evaluate hypotheses about how nature works.
• For a hypothesis to be valid, it must be testable against empirical evidence. The evidence can then confirm or disprove the testable predictions.
• Hypotheses are informed by background knowledge and observation, but go beyond what is already known to propose an explanation of how or why something occurs.

Predictions typically arise from a thorough knowledge of the research literature, curiosity about real-world problems or implications, and integrating this to advance theory. They build on existing literature while providing new insight.

Types of Research Hypotheses

Alternative hypothesis.

The research hypothesis is often called the alternative or experimental hypothesis in experimental research.

It typically suggests a potential relationship between two key variables: the independent variable, which the researcher manipulates, and the dependent variable, which is measured based on those changes.

The alternative hypothesis states a relationship exists between the two variables being studied (one variable affects the other).

A hypothesis is a testable statement or prediction about the relationship between two or more variables. It is a key component of the scientific method. Some key points about hypotheses:

• Important hypotheses lead to predictions that can be tested empirically. The evidence can then confirm or disprove the testable predictions.

In summary, a hypothesis is a precise, testable statement of what researchers expect to happen in a study and why. Hypotheses connect theory to data and guide the research process towards expanding scientific understanding.

An experimental hypothesis predicts what change(s) will occur in the dependent variable when the independent variable is manipulated.

It states that the results are not due to chance and are significant in supporting the theory being investigated.

The alternative hypothesis can be directional, indicating a specific direction of the effect, or non-directional, suggesting a difference without specifying its nature. It’s what researchers aim to support or demonstrate through their study.

Null Hypothesis

The null hypothesis states no relationship exists between the two variables being studied (one variable does not affect the other). There will be no changes in the dependent variable due to manipulating the independent variable.

It states results are due to chance and are not significant in supporting the idea being investigated.

The null hypothesis, positing no effect or relationship, is a foundational contrast to the research hypothesis in scientific inquiry. It establishes a baseline for statistical testing, promoting objectivity by initiating research from a neutral stance.

Many statistical methods are tailored to test the null hypothesis, determining the likelihood of observed results if no true effect exists.

This dual-hypothesis approach provides clarity, ensuring that research intentions are explicit, and fosters consistency across scientific studies, enhancing the standardization and interpretability of research outcomes.

Nondirectional Hypothesis

A non-directional hypothesis, also known as a two-tailed hypothesis, predicts that there is a difference or relationship between two variables but does not specify the direction of this relationship.

It merely indicates that a change or effect will occur without predicting which group will have higher or lower values.

For example, “There is a difference in performance between Group A and Group B” is a non-directional hypothesis.

Directional Hypothesis

A directional (one-tailed) hypothesis predicts the nature of the effect of the independent variable on the dependent variable. It predicts in which direction the change will take place. (i.e., greater, smaller, less, more)

It specifies whether one variable is greater, lesser, or different from another, rather than just indicating that there’s a difference without specifying its nature.

For example, “Exercise increases weight loss” is a directional hypothesis.

Falsifiability

The Falsification Principle, proposed by Karl Popper , is a way of demarcating science from non-science. It suggests that for a theory or hypothesis to be considered scientific, it must be testable and irrefutable.

Falsifiability emphasizes that scientific claims shouldn’t just be confirmable but should also have the potential to be proven wrong.

It means that there should exist some potential evidence or experiment that could prove the proposition false.

However many confirming instances exist for a theory, it only takes one counter observation to falsify it. For example, the hypothesis that “all swans are white,” can be falsified by observing a black swan.

For Popper, science should attempt to disprove a theory rather than attempt to continually provide evidence to support a research hypothesis.

Can a Hypothesis be Proven?

Hypotheses make probabilistic predictions. They state the expected outcome if a particular relationship exists. However, a study result supporting a hypothesis does not definitively prove it is true.

All studies have limitations. There may be unknown confounding factors or issues that limit the certainty of conclusions. Additional studies may yield different results.

In science, hypotheses can realistically only be supported with some degree of confidence, not proven. The process of science is to incrementally accumulate evidence for and against hypothesized relationships in an ongoing pursuit of better models and explanations that best fit the empirical data. But hypotheses remain open to revision and rejection if that is where the evidence leads.

• Disproving a hypothesis is definitive. Solid disconfirmatory evidence will falsify a hypothesis and require altering or discarding it based on the evidence.
• However, confirming evidence is always open to revision. Other explanations may account for the same results, and additional or contradictory evidence may emerge over time.

We can never 100% prove the alternative hypothesis. Instead, we see if we can disprove, or reject the null hypothesis.

If we reject the null hypothesis, this doesn’t mean that our alternative hypothesis is correct but does support the alternative/experimental hypothesis.

Upon analysis of the results, an alternative hypothesis can be rejected or supported, but it can never be proven to be correct. We must avoid any reference to results proving a theory as this implies 100% certainty, and there is always a chance that evidence may exist which could refute a theory.

How to Write a Hypothesis

• Identify variables . The researcher manipulates the independent variable and the dependent variable is the measured outcome.
• Operationalized the variables being investigated . Operationalization of a hypothesis refers to the process of making the variables physically measurable or testable, e.g. if you are about to study aggression, you might count the number of punches given by participants.
• Decide on a direction for your prediction . If there is evidence in the literature to support a specific effect of the independent variable on the dependent variable, write a directional (one-tailed) hypothesis. If there are limited or ambiguous findings in the literature regarding the effect of the independent variable on the dependent variable, write a non-directional (two-tailed) hypothesis.
• Make it Testable : Ensure your hypothesis can be tested through experimentation or observation. It should be possible to prove it false (principle of falsifiability).
• Clear & concise language . A strong hypothesis is concise (typically one to two sentences long), and formulated using clear and straightforward language, ensuring it’s easily understood and testable.

Consider a hypothesis many teachers might subscribe to: students work better on Monday morning than on Friday afternoon (IV=Day, DV= Standard of work).

Now, if we decide to study this by giving the same group of students a lesson on a Monday morning and a Friday afternoon and then measuring their immediate recall of the material covered in each session, we would end up with the following:

• The alternative hypothesis states that students will recall significantly more information on a Monday morning than on a Friday afternoon.
• The null hypothesis states that there will be no significant difference in the amount recalled on a Monday morning compared to a Friday afternoon. Any difference will be due to chance or confounding factors.

More Examples

• Memory : Participants exposed to classical music during study sessions will recall more items from a list than those who studied in silence.
• Social Psychology : Individuals who frequently engage in social media use will report higher levels of perceived social isolation compared to those who use it infrequently.
• Developmental Psychology : Children who engage in regular imaginative play have better problem-solving skills than those who don’t.
• Clinical Psychology : Cognitive-behavioral therapy will be more effective in reducing symptoms of anxiety over a 6-month period compared to traditional talk therapy.
• Cognitive Psychology : Individuals who multitask between various electronic devices will have shorter attention spans on focused tasks than those who single-task.
• Health Psychology : Patients who practice mindfulness meditation will experience lower levels of chronic pain compared to those who don’t meditate.
• Organizational Psychology : Employees in open-plan offices will report higher levels of stress than those in private offices.
• Behavioral Psychology : Rats rewarded with food after pressing a lever will press it more frequently than rats who receive no reward.

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Statistics and probability

Course: statistics and probability   >   unit 12.

• Hypothesis testing and p-values

One-tailed and two-tailed tests

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• Large sample proportion hypothesis testing

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11.4: One- and Two-Tailed Tests

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Learning Objectives

• Define Type I and Type II errors
• Interpret significant and non-significant differences
• Explain why the null hypothesis should not be accepted when the effect is not significant

In the James Bond case study, Mr. Bond was given \(16\) trials on which he judged whether a martini had been shaken or stirred. He was correct on \(13\) of the trials. From the binomial distribution, we know that the probability of being correct \(13\) or more times out of \(16\) if one is only guessing is \(0.0106\). Figure \(\PageIndex{1}\) shows a graph of the binomial distribution. The red bars show the values greater than or equal to \(13\). As you can see in the figure, the probabilities are calculated for the upper tail of the distribution. A probability calculated in only one tail of the distribution is called a "one-tailed probability."

Binomial Calculator

A slightly different question can be asked of the data: "What is the probability of getting a result as extreme or more extreme than the one observed?" Since the chance expectation is \(8/16\), a result of \(3/16\) is equally as extreme as \(13/16\). Thus, to calculate this probability, we would consider both tails of the distribution. Since the binomial distribution is symmetric when \(\pi =0.5\), this probability is exactly double the probability of \(0.0106\) computed previously. Therefore, \(p = 0.0212\). A probability calculated in both tails of a distribution is called a "two-tailed probability" (see Figure \(\PageIndex{2}\)).

Should the one-tailed or the two-tailed probability be used to assess Mr. Bond's performance? That depends on the way the question is posed. If we are asking whether Mr. Bond can tell the difference between shaken or stirred martinis, then we would conclude he could if he performed either much better than chance or much worse than chance. If he performed much worse than chance, we would conclude that he can tell the difference, but he does not know which is which. Therefore, since we are going to reject the null hypothesis if Mr. Bond does either very well or very poorly, we will use a two-tailed probability.

On the other hand, if our question is whether Mr. Bond is better than chance at determining whether a martini is shaken or stirred, we would use a one-tailed probability. What would the one-tailed probability be if Mr. Bond were correct on only \(3\) of the \(16\) trials? Since the one-tailed probability is the probability of the right-hand tail, it would be the probability of getting \(3\) or more correct out of \(16\). This is a very high probability and the null hypothesis would not be rejected.

The null hypothesis for the two-tailed test is \(\pi =0.5\). By contrast, the null hypothesis for the one-tailed test is \(\pi \leq 0.5\). Accordingly, we reject the two-tailed hypothesis if the sample proportion deviates greatly from \(0.5\) in either direction. The one-tailed hypothesis is rejected only if the sample proportion is much greater than \(0.5\). The alternative hypothesis in the two-tailed test is \(\pi \neq 0.5\). In the one-tailed test it is \(\pi > 0.5\).

You should always decide whether you are going to use a one-tailed or a two-tailed probability before looking at the data. Statistical tests that compute one-tailed probabilities are called one-tailed tests; those that compute two-tailed probabilities are called two-tailed tests. Two-tailed tests are much more common than one-tailed tests in scientific research because an outcome signifying that something other than chance is operating is usually worth noting. One-tailed tests are appropriate when it is not important to distinguish between no effect and an effect in the unexpected direction. For example, consider an experiment designed to test the efficacy of a treatment for the common cold. The researcher would only be interested in whether the treatment was better than a placebo control. It would not be worth distinguishing between the case in which the treatment was worse than a placebo and the case in which it was the same because in both cases the drug would be worthless.

Some have argued that a one-tailed test is justified whenever the researcher predicts the direction of an effect. The problem with this argument is that if the effect comes out strongly in the non-predicted direction, the researcher is not justified in concluding that the effect is not zero. Since this is unrealistic, one-tailed tests are usually viewed skeptically if justified on this basis alone.

Hypothesis Testing for Means & Proportions

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Hypothesis Testing: Upper-, Lower, and Two Tailed Tests

Type i and type ii errors.

All Modules

Z score Table

t score Table

The procedure for hypothesis testing is based on the ideas described above. Specifically, we set up competing hypotheses, select a random sample from the population of interest and compute summary statistics. We then determine whether the sample data supports the null or alternative hypotheses. The procedure can be broken down into the following five steps.  

• Step 1. Set up hypotheses and select the level of significance α.

H 0 : Null hypothesis (no change, no difference);  

H 1 : Research hypothesis (investigator's belief); α =0.05

• Step 2. Select the appropriate test statistic.  

The test statistic is a single number that summarizes the sample information.   An example of a test statistic is the Z statistic computed as follows:

When the sample size is small, we will use t statistics (just as we did when constructing confidence intervals for small samples). As we present each scenario, alternative test statistics are provided along with conditions for their appropriate use.

• Step 3.  Set up decision rule.  

The decision rule is a statement that tells under what circumstances to reject the null hypothesis. The decision rule is based on specific values of the test statistic (e.g., reject H 0 if Z > 1.645). The decision rule for a specific test depends on 3 factors: the research or alternative hypothesis, the test statistic and the level of significance. Each is discussed below.

• The decision rule depends on whether an upper-tailed, lower-tailed, or two-tailed test is proposed. In an upper-tailed test the decision rule has investigators reject H 0 if the test statistic is larger than the critical value. In a lower-tailed test the decision rule has investigators reject H 0 if the test statistic is smaller than the critical value.  In a two-tailed test the decision rule has investigators reject H 0 if the test statistic is extreme, either larger than an upper critical value or smaller than a lower critical value.
• The exact form of the test statistic is also important in determining the decision rule. If the test statistic follows the standard normal distribution (Z), then the decision rule will be based on the standard normal distribution. If the test statistic follows the t distribution, then the decision rule will be based on the t distribution. The appropriate critical value will be selected from the t distribution again depending on the specific alternative hypothesis and the level of significance.  
• The third factor is the level of significance. The level of significance which is selected in Step 1 (e.g., α =0.05) dictates the critical value.   For example, in an upper tailed Z test, if α =0.05 then the critical value is Z=1.645.  

The following figures illustrate the rejection regions defined by the decision rule for upper-, lower- and two-tailed Z tests with α=0.05. Notice that the rejection regions are in the upper, lower and both tails of the curves, respectively. The decision rules are written below each figure.

Rejection Region for Lower-Tailed Z Test (H 1 : μ < μ 0 ) with α =0.05

The decision rule is: Reject H 0 if Z < 1.645.

Rejection Region for Two-Tailed Z Test (H 1 : μ ≠ μ 0 ) with α =0.05

The decision rule is: Reject H 0 if Z < -1.960 or if Z > 1.960.

The complete table of critical values of Z for upper, lower and two-tailed tests can be found in the table of Z values to the right in "Other Resources."

Critical values of t for upper, lower and two-tailed tests can be found in the table of t values in "Other Resources."

• Step 4. Compute the test statistic.  

Here we compute the test statistic by substituting the observed sample data into the test statistic identified in Step 2.

• Step 5. Conclusion.  

The final conclusion is made by comparing the test statistic (which is a summary of the information observed in the sample) to the decision rule. The final conclusion will be either to reject the null hypothesis (because the sample data are very unlikely if the null hypothesis is true) or not to reject the null hypothesis (because the sample data are not very unlikely).  

If the null hypothesis is rejected, then an exact significance level is computed to describe the likelihood of observing the sample data assuming that the null hypothesis is true. The exact level of significance is called the p-value and it will be less than the chosen level of significance if we reject H 0 .

Statistical computing packages provide exact p-values as part of their standard output for hypothesis tests. In fact, when using a statistical computing package, the steps outlined about can be abbreviated. The hypotheses (step 1) should always be set up in advance of any analysis and the significance criterion should also be determined (e.g., α =0.05). Statistical computing packages will produce the test statistic (usually reporting the test statistic as t) and a p-value. The investigator can then determine statistical significance using the following: If p < α then reject H 0 .  

• Step 1. Set up hypotheses and determine level of significance

H 0 : μ = 191 H 1 : μ > 191                 α =0.05

The research hypothesis is that weights have increased, and therefore an upper tailed test is used.

• Step 2. Select the appropriate test statistic.

Because the sample size is large (n > 30) the appropriate test statistic is

• Step 3. Set up decision rule.  

In this example, we are performing an upper tailed test (H 1 : μ> 191), with a Z test statistic and selected α =0.05.   Reject H 0 if Z > 1.645.

We now substitute the sample data into the formula for the test statistic identified in Step 2.  

We reject H 0 because 2.38 > 1.645. We have statistically significant evidence at a =0.05, to show that the mean weight in men in 2006 is more than 191 pounds. Because we rejected the null hypothesis, we now approximate the p-value which is the likelihood of observing the sample data if the null hypothesis is true. An alternative definition of the p-value is the smallest level of significance where we can still reject H 0 . In this example, we observed Z=2.38 and for α=0.05, the critical value was 1.645. Because 2.38 exceeded 1.645 we rejected H 0 . In our conclusion we reported a statistically significant increase in mean weight at a 5% level of significance. Using the table of critical values for upper tailed tests, we can approximate the p-value. If we select α=0.025, the critical value is 1.96, and we still reject H 0 because 2.38 > 1.960. If we select α=0.010 the critical value is 2.326, and we still reject H 0 because 2.38 > 2.326. However, if we select α=0.005, the critical value is 2.576, and we cannot reject H 0 because 2.38 < 2.576. Therefore, the smallest α where we still reject H 0 is 0.010. This is the p-value. A statistical computing package would produce a more precise p-value which would be in between 0.005 and 0.010. Here we are approximating the p-value and would report p < 0.010.                  

In all tests of hypothesis, there are two types of errors that can be committed. The first is called a Type I error and refers to the situation where we incorrectly reject H 0 when in fact it is true. This is also called a false positive result (as we incorrectly conclude that the research hypothesis is true when in fact it is not). When we run a test of hypothesis and decide to reject H 0 (e.g., because the test statistic exceeds the critical value in an upper tailed test) then either we make a correct decision because the research hypothesis is true or we commit a Type I error. The different conclusions are summarized in the table below. Note that we will never know whether the null hypothesis is really true or false (i.e., we will never know which row of the following table reflects reality).

Table - Conclusions in Test of Hypothesis

In the first step of the hypothesis test, we select a level of significance, α, and α= P(Type I error). Because we purposely select a small value for α, we control the probability of committing a Type I error. For example, if we select α=0.05, and our test tells us to reject H 0 , then there is a 5% probability that we commit a Type I error. Most investigators are very comfortable with this and are confident when rejecting H 0 that the research hypothesis is true (as it is the more likely scenario when we reject H 0 ).

When we run a test of hypothesis and decide not to reject H 0 (e.g., because the test statistic is below the critical value in an upper tailed test) then either we make a correct decision because the null hypothesis is true or we commit a Type II error. Beta (β) represents the probability of a Type II error and is defined as follows: β=P(Type II error) = P(Do not Reject H 0 | H 0 is false). Unfortunately, we cannot choose β to be small (e.g., 0.05) to control the probability of committing a Type II error because β depends on several factors including the sample size, α, and the research hypothesis. When we do not reject H 0 , it may be very likely that we are committing a Type II error (i.e., failing to reject H 0 when in fact it is false). Therefore, when tests are run and the null hypothesis is not rejected we often make a weak concluding statement allowing for the possibility that we might be committing a Type II error. If we do not reject H 0 , we conclude that we do not have significant evidence to show that H 1 is true. We do not conclude that H 0 is true.

 The most common reason for a Type II error is a small sample size.

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Content ©2017. All Rights Reserved. Date last modified: November 6, 2017. Wayne W. LaMorte, MD, PhD, MPH

Statistics Tutorial

Descriptive statistics, inferential statistics, stat reference, statistics - hypothesis testing a mean (two tailed).

A population mean is an average of value a population.

Hypothesis tests are used to check a claim about the size of that population mean.

Hypothesis Testing a Mean

The following steps are used for a hypothesis test:

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

For example:

• Population : Nobel Prize winners
• Category : Age when they received the prize.

And we want to check the claim:

"The average age of Nobel Prize winners when they received the prize is not 60"

By taking a sample of 30 randomly selected Nobel Prize winners we could find that:

• The mean age in the sample (\(\bar{x}\)) is 62.1
• The standard deviation of age in the sample (\(s\)) is 13.46

From this sample data we check the claim with the steps below.

1. Checking the Conditions

The conditions for calculating a confidence interval for a proportion are:

• The sample is randomly selected
• The population data is normally distributed
• Sample size is large enough

A moderately large sample size, like 30, is typically large enough.

In the example, the sample size was 30 and it was randomly selected, so the conditions are fulfilled.

Note: Checking if the data is normally distributed can be done with specialized statistical tests.

2. Defining the Claims

We need to define a null hypothesis (\(H_{0}\)) and an alternative hypothesis (\(H_{1}\)) based on the claim we are checking.

The claim was:

In this case, the parameter is the mean age of Nobel Prize winners when they received the prize (\(\mu\)).

The null and alternative hypothesis are then:

Null hypothesis : The average age was 60.

Alternative hypothesis : The average age is not 60.

Which can be expressed with symbols as:

\(H_{0}\): \(\mu = 60 \)

\(H_{1}\): \(\mu \neq 60 \)

This is a ' two-tailed ' test, because the alternative hypothesis claims that the proportion is different from the null hypothesis.

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

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3. Deciding the Significance Level

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

4. Calculating 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.

The formula for the test statistic (TS) of a population mean is:

\(\displaystyle \frac{\bar{x} - \mu}{s} \cdot \sqrt{n} \)

\(\bar{x}-\mu\) is the difference between the sample mean (\(\bar{x}\)) and the claimed population mean (\(\mu\)).

\(s\) is the sample standard deviation .

\(n\) is the sample size.

In our example:

The claimed (\(H_{0}\)) population mean (\(\mu\)) was \( 60 \)

The sample mean (\(\bar{x}\)) was \(62.1\)

The sample standard deviation (\(s\)) was \(13.46\)

The sample size (\(n\)) was \(30\)

So the test statistic (TS) is then:

\(\displaystyle \frac{62.1-60}{13.46} \cdot \sqrt{30} = \frac{2.1}{13.46} \cdot \sqrt{30} \approx 0.156 \cdot 5.477 = \underline{0.855}\)

You can also calculate the test statistic using programming language functions:

With Python use the scipy and math libraries to calculate the test statistic.

With R use built-in math and statistics functions to calculate the test statistic.

5. Concluding

There are two main approaches for making the conclusion of a hypothesis test:

• 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.

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

The Critical Value Approach

For the critical value approach we need to find the critical value (CV) of the significance level (\(\alpha\)).

For a population mean test, the critical value (CV) is a T-value from a student's t-distribution .

This critical T-value (CV) defines the rejection region for the test.

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

Because the claim is that the population proportion is different from 60, the rejection region is split into both the left and right tail:

The student's t-distribution is adjusted for the uncertainty from smaller samples.

This adjustment is called degrees of freedom (df), which is the sample size \((n) - 1\)

In this case the degrees of freedom (df) is: \(30 - 1 = \underline{29} \)

Choosing a significance level (\(\alpha\)) of 0.05, or 5%, we can find the critical T-value from a T-table , or with a programming language function:

Note: Because this is a two-tailed test the tail area (\(\alpha\)) needs to be split in half (divided by 2).

With Python use the Scipy Stats library t.ppf() function find the T-Value for an \(\alpha\)/2 = 0.025 at 29 degrees of freedom (df).

With R use the built-in qt() function to find the t-value for an \(\alpha\)/ = 0.025 at 29 degrees of freedom (df).

Using either method we can find that the critical T-Value is \(\approx \underline{-2.045}\)

For a two-tailed test we need to check if the test statistic (TS) is smaller than the negative critical value (-CV), or bigger than the positive critical value (CV).

If the test statistic is smaller than the negative critical value, the test statistic is in the rejection region .

If the test statistic is bigger than the positive critical value, the test statistic is in the rejection region .

When the test statistic is in the rejection region, we reject the null hypothesis (\(H_{0}\)).

Here, the test statistic (TS) was \(\approx \underline{0.855}\) and the critical value was \(\approx \underline{-2.045}\)

Here is an illustration of this test in a graph:

Since the test statistic is between the critical values we keep the null hypothesis.

This means that the sample data does not support the alternative hypothesis.

And we can summarize the conclusion stating:

The sample data does not support the claim that "The average age of Nobel Prize winners when they received the prize is not 60" at a 5% significance level .

The P-Value Approach

For the P-value approach we need to find the P-value of the test statistic (TS).

If the P-value is smaller than the significance level (\(\alpha\)), we reject the null hypothesis (\(H_{0}\)).

The test statistic was found to be \( \approx \underline{0.855} \)

For a population proportion test, the test statistic is a T-Value from a student's t-distribution .

Because this is a two-tailed test, we need to find the P-value of a T-value bigger than 0.855 and multiply it by 2 .

The student's t-distribution is adjusted according to degrees of freedom (df), which is the sample size \((30) - 1 = \underline{29}\)

We can find the P-value using a T-table , or with a programming language function:

With Python use the Scipy Stats library t.cdf() function find the P-value of a T-value bigger than 0.855 for a two tailed test at 29 degrees of freedom (df):

With R use the built-in pt() function find the P-value of a T-Value bigger than 0.855 for a two tailed test at 29 degrees of freedom (df):

Using either method we can find that the P-value is \(\approx \underline{0.3996}\)

This tells us that the significance level (\(\alpha\)) would need to be smaller 0.3996, or 39.96%, to reject the null hypothesis.

This P-value is bigger than any of the common significance levels (10%, 5%, 1%).

So the null hypothesis is kept at all of these significance levels.

The sample data does not support the claim that "The average age of Nobel Prize winners when they received the prize is not 60" at a 10%, 5%, or 1% significance level .

Calculating a P-Value for a Hypothesis Test with Programming

Many programming languages can calculate the P-value to decide outcome of a hypothesis test.

Using software and programming to calculate statistics is more common for bigger sets of data, as calculating manually becomes difficult.

The P-value calculated here will tell us the lowest possible significance level where the null-hypothesis can be rejected.

With Python use the scipy and math libraries to calculate the P-value for a two tailed hypothesis test for a mean.

Here, the sample size is 30, the sample mean is 62.1, the sample standard deviation is 13.46, and the test is for a mean different from 60.

With R use built-in math and statistics functions find the P-value for a two tailed hypothesis test for a mean.

Left-Tailed and Two-Tailed Tests

This was an example of a left tailed test, where the alternative hypothesis claimed that parameter is smaller than the null hypothesis claim.

You can check out an equivalent step-by-step guide for other types here:

• Right-Tailed Test
• Two-Tailed Test

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One and Two Tailed Tests

Suppose we have a null hypothesis H 0 and an alternative hypothesis H 1 . We consider the distribution given by the null hypothesis and perform a test to determine whether or not the null hypothesis should be rejected in favour of the alternative hypothesis.

There are two different types of tests that can be performed. A one-tailed test looks for an increase or decrease in the parameter whereas a two-tailed test looks for any change in the parameter (which can be any change- increase or decrease).

We can perform the test at any level (usually 1%, 5% or 10%). For example, performing the test at a 5% level means that there is a 5% chance of wrongly rejecting H 0 .

If we perform the test at the 5% level and decide to reject the null hypothesis, we say "there is significant evidence at the 5% level to suggest the hypothesis is false".

One-Tailed Test

We choose a critical region. In a one-tailed test, the critical region will have just one part (the red area below). If our sample value lies in this region, we reject the null hypothesis in favour of the alternative.

Suppose we are looking for a definite decrease. Then the critical region will be to the left. Note, however, that in the one-tailed test the value of the parameter can be as high as you like.

Suppose we are given that X has a Poisson distribution and we want to carry out a hypothesis test on the mean, l, based upon a sample observation of 3.

Suppose the hypotheses are: H 0 : l = 9 H 1 : l < 9

We want to test if it is "reasonable" for the observed value of 3 to have come from a Poisson distribution with parameter 9. So what is the probability that a value as low as 3 has come from a Po(9)?

P(X < 3) = 0.0212 (this has come from a Poisson table)

The probability is less than 0.05, so there is less than a 5% chance that the value has come from a Poisson(3) distribution. We therefore reject the null hypothesis in favour of the alternative at the 5% level.

However, the probability is greater than 0.01, so we would not reject the null hypothesis in favour of the alternative at the 1% level.

Two-Tailed Test

In a two-tailed test, we are looking for either an increase or a decrease. So, for example, H 0 might be that the mean is equal to 9 (as before). This time, however, H 1 would be that the mean is not equal to 9. In this case, therefore, the critical region has two parts:

Lets test the parameter p of a Binomial distribution at the 10% level.

Suppose a coin is tossed 10 times and we get 7 heads. We want to test whether or not the coin is fair. If the coin is fair, p = 0.5 . Put this as the null hypothesis:

H 0 : p = 0.5 H 1 : p =(doesn' equal) 0.5

Now, because the test is 2-tailed, the critical region has two parts. Half of the critical region is to the right and half is to the left. So the critical region contains both the top 5% of the distribution and the bottom 5% of the distribution (since we are testing at the 10% level).

If H 0 is true, X ~ Bin(10, 0.5).

If the null hypothesis is true, what is the probability that X is 7 or above? P(X > 7) = 1 - P(X < 7) = 1 - P(X < 6) = 1 - 0.8281 = 0.1719

Is this in the critical region? No- because the probability that X is at least 7 is not less than 0.05 (5%), which is what we need it to be.

So there is not significant evidence at the 10% level to reject the null hypothesis.

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S.3.1 hypothesis testing (critical value approach).

The critical value approach involves determining "likely" or "unlikely" by determining whether or not the observed test statistic is more extreme than would be expected if the null hypothesis were true. That is, it entails comparing the observed test statistic to some cutoff value, called the " critical value ." If the test statistic is more extreme than the critical value, then the null hypothesis is rejected in favor of the alternative hypothesis. If the test statistic is not as extreme as the critical value, then the null hypothesis is not rejected.

Specifically, the four steps involved in using the critical value approach to conducting any hypothesis test are:

• Specify the null and alternative hypotheses.
• Using the sample data and assuming the null hypothesis is true, calculate the value of the test statistic. To conduct the hypothesis test for the population mean μ , we use the t -statistic \(t^*=\frac{\bar{x}-\mu}{s/\sqrt{n}}\) which follows a t -distribution with n - 1 degrees of freedom.
• Determine the critical value by finding the value of the known distribution of the test statistic such that the probability of making a Type I error — which is denoted \(\alpha\) (greek letter "alpha") and is called the " significance level of the test " — is small (typically 0.01, 0.05, or 0.10).
• Compare the test statistic to the critical value. If the test statistic is more extreme in the direction of the alternative than the critical value, reject the null hypothesis in favor of the alternative hypothesis. If the test statistic is less extreme than the critical value, do not reject the null hypothesis.

Example S.3.1.1

Mean gpa section  .

In our example concerning the mean grade point average, suppose we take a random sample of n = 15 students majoring in mathematics. Since n = 15, our test statistic t * has n - 1 = 14 degrees of freedom. Also, suppose we set our significance level α at 0.05 so that we have only a 5% chance of making a Type I error.

Right-Tailed

The critical value for conducting the right-tailed test H 0 : μ = 3 versus H A : μ > 3 is the t -value, denoted t \(\alpha\) , n - 1 , such that the probability to the right of it is \(\alpha\). It can be shown using either statistical software or a t -table that the critical value t 0.05,14 is 1.7613. That is, we would reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ > 3 if the test statistic t * is greater than 1.7613. Visually, the rejection region is shaded red in the graph.

Left-Tailed

The critical value for conducting the left-tailed test H 0 : μ = 3 versus H A : μ < 3 is the t -value, denoted -t ( \(\alpha\) , n - 1) , such that the probability to the left of it is \(\alpha\). It can be shown using either statistical software or a t -table that the critical value -t 0.05,14 is -1.7613. That is, we would reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ < 3 if the test statistic t * is less than -1.7613. Visually, the rejection region is shaded red in the graph.

There are two critical values for the two-tailed test H 0 : μ = 3 versus H A : μ ≠ 3 — one for the left-tail denoted -t ( \(\alpha\) / 2, n - 1) and one for the right-tail denoted t ( \(\alpha\) / 2, n - 1) . The value - t ( \(\alpha\) /2, n - 1) is the t -value such that the probability to the left of it is \(\alpha\)/2, and the value t ( \(\alpha\) /2, n - 1) is the t -value such that the probability to the right of it is \(\alpha\)/2. It can be shown using either statistical software or a t -table that the critical value -t 0.025,14 is -2.1448 and the critical value t 0.025,14 is 2.1448. That is, we would reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ ≠ 3 if the test statistic t * is less than -2.1448 or greater than 2.1448. Visually, the rejection region is shaded red in the graph.

Two Tailed Hypothesis

In the vast realm of scientific inquiry, the two-tailed hypothesis holds a special place, serving as a compass for researchers exploring possibilities in two opposing directions. Instead of predicting a specific direction of the relationship between variables, it remains open to outcomes on both ends of the spectrum. Understanding how to craft such a hypothesis, enriched with insights and nuances, can elevate the robustness of one’s research. Delve into its world, discover thesis statement examples, learn the art of its formulation, and grasp tips to master its intricacies.

What is Two Tailed Hypothesis? – Definition

A two-tailed hypothesis, also known as a non-directional hypothesis , is a type of hypothesis used in statistical testing that predicts a relationship between variables without specifying the direction of the relationship. In other words, it tests for the possibility of the relationship in both directions. This approach is used when a researcher believes there might be a difference due to the experiment but doesn’t have enough preliminary evidence or basis to predict a specific direction of that difference.

What is an example of a Two Tailed hypothesis statement?

Let’s consider a study on the impact of a new teaching method on student performance:

Hypothesis Statement : The new teaching method will have an effect on student performance.

Notice that the hypothesis doesn’t specify whether the effect will be positive or negative (i.e., whether student performance will improve or decline). It’s open to both possibilities, making it a two-tailed hypothesis.

Two Tailed Hypothesis Statement Examples

The two-tailed hypothesis, an essential tool in research, doesn’t predict a specific directional outcome between variables. Instead, it posits that an effect exists, without specifying its nature. This approach offers flexibility, as it remains open to both positive and negative outcomes. Below are various examples from diverse fields to shed light on this versatile research method. You may also be interested to browse through our other  one-tailed hypothesis .

• Sleep and Cognitive Ability : Sleep duration affects cognitive performance in adults.
• Dietary Fiber and Digestion : Consumption of dietary fiber influences digestion rates.
• Exercise and Stress Levels : Engaging in physical activity impacts stress levels.
• Vitamin C and Immunity : Intake of Vitamin C has an effect on immunity strength.
• Noise Levels and Concentration : Ambient noise levels influence individual concentration ability.
• Artificial Sweeteners and Appetite : Consumption of artificial sweeteners affects appetite.
• UV Light and Skin Health : Exposure to UV light influences skin health.
• Coffee Intake and Sleep Quality : Consuming coffee has an effect on sleep quality.
• Air Pollution and Respiratory Issues : Levels of air pollution impact respiratory health.
• Meditation and Blood Pressure : Practicing meditation affects blood pressure readings.
• Pet Ownership and Loneliness : Having a pet influences feelings of loneliness.
• Green Spaces and Mental Wellbeing : Exposure to green spaces impacts mental health.
• Music Tempo and Heart Rate : Listening to music of varying tempos affects heart rate.
• Chocolate Consumption and Mood : Eating chocolate has an effect on mood.
• Social Media Usage and Self-Esteem : The frequency of social media usage influences self-esteem.
• E-reading and Eye Strain : Using e-readers affects eye strain levels.
• Vegan Diets and Energy Levels : Following a vegan diet influences daily energy levels.
• Carbonated Drinks and Tooth Decay : Consumption of carbonated drinks has an effect on tooth decay rates.
• Distance Learning and Student Engagement : Engaging in distance learning impacts student involvement.
• Organic Foods and Health Perceptions : Consuming organic foods influences perceptions of health.
• Urban Living and Stress Levels : Living in urban environments affects stress levels.
• Plant-Based Diets and Cholesterol : Adopting a plant-based diet impacts cholesterol levels.
• Virtual Reality Training and Skill Acquisition : Using virtual reality for training influences the rate of skill acquisition.
• Video Game Play and Hand-Eye Coordination : Playing video games has an effect on hand-eye coordination.
• Aromatherapy and Sleep Quality : Using aromatherapy impacts the quality of sleep.
• Bilingualism and Cognitive Flexibility : Being bilingual affects cognitive flexibility.
• Microplastics and Marine Health : The presence of microplastics in oceans influences marine organism health.
• Yoga Practice and Joint Health : Engaging in yoga has an effect on joint health.
• Processed Foods and Metabolism : Consuming processed foods impacts metabolic rates.
• Home Schooling and Social Skills : Being homeschooled influences the development of social skills.
• Smartphone Usage and Attention Span : Regular smartphone use affects attention spans.
• E-commerce and Consumer Trust : Engaging with e-commerce platforms influences levels of consumer trust.
• Work-from-Home and Productivity : The practice of working from home has an effect on productivity levels.
• Classical Music and Plant Growth : Exposing plants to classical music impacts their growth rate.
• Public Transport and Community Engagement : Using public transport influences community engagement levels.
• Digital Note-taking and Memory Retention : Taking notes digitally affects memory retention.
• Acoustic Music and Relaxation : Listening to acoustic music impacts feelings of relaxation.
• GMO Foods and Public Perception : Consuming GMO foods influences public perception of food safety.
• LED Lights and Eye Comfort : Using LED lights affects visual comfort.
• Fast Fashion and Consumer Satisfaction : Engaging with fast fashion brands influences consumer satisfaction levels.
• Diverse Teams and Innovation : Working in diverse teams impacts the level of innovation.
• Local Produce and Nutritional Value : Consuming local produce affects its nutritional value.
• Podcasts and Language Acquisition : Listening to podcasts influences the speed of language acquisition.
• Augmented Reality and Learning Efficiency : Using augmented reality in education has an effect on learning efficiency.
• Museums and Historical Interest : Visiting museums impacts interest in history.
• E-books vs. Physical Books and Reading Retention : The type of book, whether e-book or physical, affects memory retention from reading.
• Biophilic Design and Worker Well-being : Implementing biophilic designs in office spaces influences worker well-being.
• Recycled Products and Consumer Preference : Using recycled materials in products impacts consumer preferences.
• Interactive Learning and Critical Thinking : Engaging in interactive learning environments affects the development of critical thinking skills.
• High-Intensity Training and Muscle Growth : Participating in high-intensity training has an effect on muscle growth rate.
• Pet Therapy and Anxiety Levels : Engaging with therapy animals influences anxiety levels.
• 3D Printing and Manufacturing Efficiency : Implementing 3D printing in manufacturing affects production efficiency.
• Electric Cars and Public Adoption Rates : Introducing more electric cars impacts the rate of public adoption.
• Ancient Architectural Study and Modern Design Inspiration : Studying ancient architecture influences modern design inspirations.
• Natural Lighting and Productivity : The amount of natural lighting in a workspace affects worker productivity.
• Streaming Platforms and Traditional TV Viewing : The rise of streaming platforms has an effect on traditional TV viewing habits.
• Handwritten Notes and Conceptual Understanding : Taking notes by hand influences the depth of conceptual understanding.
• Urban Farming and Community Engagement : Implementing urban farming practices impacts levels of community engagement.
• Influencer Marketing and Brand Loyalty : Collaborating with influencers affects brand loyalty among consumers.
• Online Workshops and Skill Enhancement : Participating in online workshops influences skill enhancement.
• Virtual Reality and Empathy Development : Using virtual reality experiences influences the development of empathy.
• Gardening and Mental Well-being : Engaging in gardening activities affects overall mental well-being.
• Drones and Wildlife Observation : The use of drones impacts the accuracy of wildlife observations.
• Artificial Intelligence and Job Markets : The introduction of artificial intelligence in industries has an effect on job availability.
• Online Reviews and Purchase Decisions : Reading online reviews influences purchase decisions for consumers.
• Blockchain Technology and Financial Security : Implementing blockchain technology affects financial transaction security.
• Minimalism and Life Satisfaction : Adopting a minimalist lifestyle influences levels of life satisfaction.
• Microlearning and Long-term Retention : Engaging in microlearning practices impacts long-term information retention.
• Virtual Teams and Communication Efficiency : Operating in virtual teams has an effect on the efficiency of communication.
• Plant Music and Growth Rates : Exposing plants to specific music frequencies influences their growth rates.
• Green Building Practices and Energy Consumption : Implementing green building designs affects overall energy consumption.
• Fermented Foods and Gut Health : Consuming fermented foods impacts gut health.
• Digital Art Platforms and Creative Expression : Using digital art platforms influences levels of creative expression.
• Aquatic Therapy and Physical Rehabilitation : Engaging in aquatic therapy has an effect on the rate of physical rehabilitation.
• Solar Energy and Utility Bills : Adopting solar energy solutions influences monthly utility bills.
• Immersive Theatre and Audience Engagement : Experiencing immersive theatre performances affects audience engagement levels.
• Podcast Popularity and Radio Listening Habits : The rise in podcast popularity impacts traditional radio listening habits.
• Vertical Farming and Crop Yield : Implementing vertical farming techniques has an effect on crop yields.
• DIY Culture and Craftsmanship Appreciation : The rise of DIY culture influences public appreciation for craftsmanship.
• Crowdsourcing and Solution Innovation : Utilizing crowdsourcing methods affects the innovativeness of solutions derived.
• Urban Beekeeping and Local Biodiversity : Introducing urban beekeeping practices impacts local biodiversity levels.
• Digital Nomad Lifestyle and Work-Life Balance : Adopting a digital nomad lifestyle affects perceptions of work-life balance.
• Virtual Tours and Tourism Interest : Offering virtual tours of destinations influences interest in real-life visits.
• Neurofeedback Training and Cognitive Abilities : Engaging in neurofeedback training has an effect on various cognitive abilities.
• Sensory Gardens and Stress Reduction : Visiting sensory gardens impacts levels of stress reduction.
• Subscription Box Services and Consumer Spending : The popularity of subscription box services influences overall consumer spending patterns.
• Makerspaces and Community Collaboration : Introducing makerspaces in communities affects collaboration levels among members.
• Remote Work and Company Loyalty : Adopting long-term remote work policies impacts employee loyalty towards the company.
• Upcycling and Environmental Awareness : Engaging in upcycling activities influences levels of environmental awareness.
• Mixed Reality in Education and Engagement : Implementing mixed reality tools in education affects student engagement.
• Microtransactions in Gaming and Player Commitment : The presence of microtransactions in video games impacts player commitment and longevity.
• Floating Architecture and Sustainable Living : Adopting floating architectural solutions influences perceptions of sustainable living.
• Edible Packaging and Waste Reduction : Introducing edible packaging in markets has an effect on overall waste reduction.
• Space Tourism and Interest in Astronomy : The advent of space tourism influences the general public’s interest in astronomy.
• Urban Green Roofs and Air Quality : Implementing green roofs in urban settings impacts the local air quality.
• Smart Mirrors and Fitness Consistency : Using smart mirrors for workouts affects consistency in fitness routines.
• Open Source Software and Technological Innovation : Promoting open-source software has an effect on the rate of technological innovation.
• Microgreens and Nutrient Intake : Consuming microgreens influences nutrient intake.
• Aquaponics and Sustainable Farming : Implementing aquaponic systems impacts perceptions of sustainable farming.
• Esports Popularity and Physical Sport Engagement : The rise of esports affects engagement in traditional physical sports.

Two Tailed Hypothesis Statement Examples in Research

In academic research, a two-tailed hypothesis is versatile, not pointing to a specific direction of effect but remaining open to outcomes on both ends of the spectrum. Such hypothesis aim to determine if a particular variable affects another, without specifying how. Here are examples tailored to research scenarios.

• Interdisciplinary Collaboration and Innovation : Engaging in interdisciplinary collaborations impacts the degree of innovation in research findings.
• Open Access Journals and Citation Rates : Publishing in open-access journals influences the citation rates of the papers.
• Research Grants and Publication Quality : Receiving larger research grants affects the quality of resulting publications.
• Laboratory Environment and Data Accuracy : The physical conditions of a research laboratory impact the accuracy of experimental data.
• Peer Review Process and Research Integrity : The stringency of the peer review process influences the overall integrity of published research.
• Researcher Mobility and Knowledge Transfer : The mobility of researchers between institutions affects the rate of knowledge transfer.
• Interdisciplinary Conferences and Networking Opportunities : Attending interdisciplinary conferences impacts the depth and breadth of networking opportunities.
• Qualitative Methods and Research Depth : Incorporating qualitative methods in research affects the depth of findings.
• Data Visualization Tools and Research Comprehension : Utilizing advanced data visualization tools influences the comprehension of complex research data.
• Collaborative Tools and Research Efficiency : The adoption of modern collaborative tools impacts research efficiency and productivity.

Two Tailed Testing Hypothesis Statement Examples

In hypothesis testing , a two-tailed test examines the possibility of a relationship in both directions. Unlike one-tailed tests, it doesn’t anticipate a specific direction of the relationship. The following are examples that encapsulate this approach within varied testing scenarios.

• Load Testing and Website Speed : Conducting load testing on a website influences its loading speed.
• A/B Testing and Conversion Rates : Implementing A/B testing affects the conversion rates of a webpage.
• Drug Efficacy Testing and Patient Recovery : Testing a new drug’s efficacy impacts patient recovery rates.
• Usability Testing and User Engagement : Conducting usability testing on an app influences user engagement metrics.
• Genetic Testing and Disease Prediction : Utilizing genetic testing affects the accuracy of disease prediction.
• Water Quality Testing and Contaminant Levels : Performing water quality tests influences our understanding of contaminant levels.
• Battery Life Testing and Device Longevity : Conducting battery life tests impacts claims about device longevity.
• Product Safety Testing and Recall Rates : Implementing rigorous product safety tests affects the rate of product recalls.
• Emissions Testing and Pollution Control : Undertaking emissions testing on vehicles influences pollution control measures.
• Material Strength Testing and Product Durability : Testing the strength of materials affects predictions about product durability.

How do you know if a hypothesis is two-tailed?

To determine if a hypothesis is two-tailed, you must look at the nature of the prediction. A two-tailed hypothesis is neutral concerning the direction of the predicted relationship or difference between groups. It simply predicts a difference or relationship without specifying whether it will be positive, negative, greater, or lesser. The hypothesis tests for effects in both directions.

What is one-tailed and two-tailed Hypothesis test with example?

In hypothesis testing, the choice between a one-tailed and a two-tailed test is determined by the nature of the research question.

One-tailed hypothesis: This tests for a specific direction of the effect. It predicts the direction of the relationship or difference between groups. For example, a one-tailed hypothesis might state: “The new drug will reduce symptoms more effectively than the standard treatment.”

Two-tailed hypothesis: This doesn’t specify the direction. It predicts that there will be a difference, but it doesn’t forecast whether the difference will be positive or negative. For example, a two-tailed hypothesis might state: “The new drug will have a different effect on symptoms compared to the standard treatment.”

What is a two-tailed hypothesis in psychology?

In psychology, a two-tailed hypothesis is frequently used when researchers are exploring new areas or relationships without a strong prior basis to predict the direction of findings. For instance, a psychologist might use a two-tailed hypothesis to explore whether a new therapeutic method has different outcomes than a traditional method, without predicting whether the outcomes will be better or worse.

What does a two-tailed alternative hypothesis look like?

A two-tailed alternative hypothesis is generally framed to show that a parameter is simply different from a certain value, without specifying the direction of the difference. Using mathematical notation, for a population mean (μ) and a proposed value (k), the two-tailed hypothesis would look like: H1: μ ≠ k.

How do you write a Two-Tailed hypothesis statement? – A Step by Step Guide

• Identify the Variables: Start by identifying the independent and dependent variables you want to study.
• Formulate a Relationship: Consider the potential relationship between these variables without setting a direction.
• Avoid Directional Language: Words like “increase”, “decrease”, “more than”, or “less than” should be avoided as they point to a one-tailed hypothesis.
• Keep it Simple: The statement should be clear, concise, and to the point.
• Use Neutral Language: For instance, words like “affects”, “influences”, or “has an impact on” can be used to indicate a relationship without specifying a direction.
• Finalize the Statement: Once the relationship is clear in your mind, form a coherent sentence that describes the relationship between your variables.

Tips for Writing Two Tailed Hypothesis

• Start Broad: Given that you’re not seeking a specific direction, it’s okay to start with a broad idea.
• Be Objective: Avoid letting any biases or expectations shape your hypothesis.
• Stay Informed: Familiarize yourself with existing research on the topic to ensure your hypothesis is novel and not inadvertently directional.
• Seek Feedback: Share your hypothesis with colleagues or mentors to ensure it’s indeed non-directional.
• Revisit and Refine: As with any research process, be open to revisiting and refining your hypothesis as you delve deeper into the literature or collect preliminary data.

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Statistics Made Easy

Two Sample t-test: Definition, Formula, and Example

A two sample t-test is used to determine whether or not two population means are equal.

This tutorial explains the following:

• The motivation for performing a two sample t-test.
• The formula to perform a two sample t-test.
• The assumptions that should be met to perform a two sample t-test.
• An example of how to perform a two sample t-test.

Two Sample t-test: Motivation

Suppose we want to know whether or not the mean weight between two different species of turtles is equal. Since there are thousands of turtles in each population, it would be too time-consuming and costly to go around and weigh each individual turtle.

Instead, we might take a simple random sample of 15 turtles from each population and use the mean weight in each sample to determine if the mean weight is equal between the two populations:

However, it’s virtually guaranteed that the mean weight between the two samples will be at least a little different. The question is whether or not this difference is statistically significant . Fortunately, a two sample t-test allows us to answer this question.

Two Sample t-test: Formula

A two-sample t-test always uses the following null hypothesis:

• H 0 : μ 1  = μ 2 (the two population means are equal)

The alternative hypothesis can be either two-tailed, left-tailed, or right-tailed:

• H 1 (two-tailed): μ 1  ≠ μ 2 (the two population means are not equal)
• H 1 (left-tailed): μ 1  < μ 2  (population 1 mean is less than population 2 mean)
• H 1 (right-tailed):  μ 1 > μ 2  (population 1 mean is greater than population 2 mean)

We use the following formula to calculate the test statistic t:

Test statistic:  ( x 1  –  x 2 )  /  s p (√ 1/n 1  + 1/n 2 )

where  x 1  and  x 2 are the sample means, n 1 and n 2  are the sample sizes, and where s p is calculated as:

s p = √  (n 1 -1)s 1 2  +  (n 2 -1)s 2 2  /  (n 1 +n 2 -2)

where s 1 2  and s 2 2  are the sample variances.

If the p-value that corresponds to the test statistic t with (n 1 +n 2 -1) degrees of freedom is less than your chosen significance level (common choices are 0.10, 0.05, and 0.01) then you can reject the null hypothesis.

Two Sample t-test: Assumptions

For the results of a two sample t-test to be valid, the following assumptions should be met:

• The observations in one sample should be independent of the observations in the other sample.
• The data should be approximately normally distributed.
• The two samples should have approximately the same variance. If this assumption is not met, you should instead perform Welch’s t-test .
• The data in both samples was obtained using a random sampling method .

Two Sample t-test : Example

Suppose we want to know whether or not the mean weight between two different species of turtles is equal. To test this, will perform a two sample t-test at significance level α = 0.05 using the following steps:

Step 1: Gather the sample data.

Suppose we collect a random sample of turtles from each population with the following information:

• Sample size n 1 = 40
• Sample mean weight  x 1  = 300
• Sample standard deviation s 1 = 18.5
• Sample size n 2 = 38
• Sample mean weight  x 2  = 305
• Sample standard deviation s 2 = 16.7

Step 2: Define the hypotheses.

We will perform the two sample t-test with the following hypotheses:

• H 0 :  μ 1  = μ 2 (the two population means are equal)
• H 1 :  μ 1  ≠ μ 2 (the two population means are not equal)

Step 3: Calculate the test statistic  t .

First, we will calculate the pooled standard deviation s p :

s p = √  (n 1 -1)s 1 2  +  (n 2 -1)s 2 2  /  (n 1 +n 2 -2)  = √  (40-1)18.5 2  +  (38-1)16.7 2  /  (40+38-2)  = 17.647

Next, we will calculate the test statistic  t :

t = ( x 1  –  x 2 )  /  s p (√ 1/n 1  + 1/n 2 ) =  (300-305) / 17.647(√ 1/40 + 1/38 ) =  -1.2508

Step 4: Calculate the p-value of the test statistic  t .

According to the T Score to P Value Calculator , the p-value associated with t = -1.2508 and degrees of freedom = n 1 +n 2 -2 = 40+38-2 = 76 is  0.21484 .

Step 5: Draw a conclusion.

Since this p-value is not less than our significance level α = 0.05, we fail to reject the null hypothesis. We do not have sufficient evidence to say that the mean weight of turtles between these two populations is different.

Note:  You can also perform this entire two sample t-test by simply using the Two Sample t-test Calculator .

Additional Resources

The following tutorials explain how to perform a two-sample t-test using different statistical programs:

How to Perform a Two Sample t-test in Excel How to Perform a Two Sample t-test in SPSS How to Perform a Two Sample t-test in Stata How to Perform a Two Sample t-test in R How to Perform a Two Sample t-test in Python How to Perform a Two Sample t-test on a TI-84 Calculator

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

2 Replies to “Two Sample t-test: Definition, Formula, and Example”

I like the detailed information and simplified in the way I can understand and relate easily. Thank you

It seems a couple of parenthesis is missed at the pooled standard deviation formula. Under square root you have (n1-1)s12 + (n2-1)s22 / (n1+n2-2) but it should be [(n1-1)s12 + (n2-1)s22] / (n1+n2-2) I used square bracket

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