Basics of One-sample mean test



You might have learned about hypothesis testing. In this article, you will learn how to perform the one-sample mean test. In this type of hypothesis testing, the population mean (µ) is the parameter of interest. We collect the sampled data from the population and make inferences and conclusions based on this sample. Before going to perform one-sample mean testing you, need to understand the assumptions that should be satisfied to carry out hypothesis testing.


Assumptions for one-sample mean testing

  1. The data should be continuous.
  2. The observations in the data should be independent.
  3. The data should come from the normal distribution.


If the data follow these conditions, then one can go to test the claim about population mean (µ). You should check if the population standard deviation (σ) is known before going to test the claim. If you know, the population standard deviation in advance then uses a one-sample Z test. Moreover, if you do not know the population standard deviation then use a one-sample t-test.

Let us see what the steps are involved in the one-sample mean test one by one.


Step 1: Fix the null and alternative hypotheses

The null and alternative hypotheses for a Z test are as follows.


Step 2: Chose the correct formula to calculate the test statistics

Use the following formula to find the test statistic for the Z test.



Use the following formula to find the test statistic for the t-test.



In the above formulas, M is the sample mean; µ0 is the value of mean for which we perform the test; σ is the population standard deviation, s is the sample standard deviation and n is the sample size.


Step 3: Fix the significance level (α)

Most of the time the value of α is given in the question. If you do not know the value of α then just take α=0.05.


Step 4: Find the critical value/s or p-value

The critical values or p-value are essential in making the decision, whether to reject or fail to reject the null hypothesis. You can use either the statistical tables or technology like excel or TI84 calculator or R-Studio, etc. to find critical values as well as p-value.


Step 5: Make the decisions based on critical value/ p-value

There are basically two ways to make decisions for the hypothesis testing based on critical values or p-value.

Based on critical values

Using critical values, you can decide the rejection region. If the value of test statistic found in step 2 lies in the rejection region, then reject the null hypothesis at α level of significance, otherwise fail to reject the null hypothesis.

Based on p-value

If the p-value ≤ α then reject the null hypothesis otherwise fail to reject the null hypothesis.


Step 6: Conclusion based on the decision

If you reject the null hypothesis then write the conclusion is as follows.

There is sufficient evidence to support the claim the population mean is different from µ0 (OR the population mean is less than µ0 OR the population mean is greater than µ0).

If you fail to reject the null hypothesis then write the conclusion is as follows.

There is insufficient evidence to support the claim the population mean is different from µ0 (OR the population mean is less than µ0 OR the population mean is greater than µ0).