Two sample mean tests

Introduction

You might have learned about one-sample mean testing. In this article, you will learn how to perform the two-sample mean test. Consider the following scenarios.

While comparing IQ scores of females and males, in the study of quality of crops produced using two different methods, in the comparison of effects of two health products, etc. we need to compare the population means. In such situations, the population means are the parameter of interest.

Suppose there are two populations under study, and we need to compare the population means of these populations then we use a two-sample mean test. There are some assumptions that should be satisfied to carry out hypothesis testing.

 

Assumptions for two-sample mean testing

1. The data should be continuous.
2. The observations in the data should be independent.
3. The two samples are independent of each other.
4. Both the samples should become from the normal distribution.

If the data follow these conditions, then one can go to compare two population means. You should check if the population standard deviations are known before going to test the claim. If you know, both the population standard deviations in advance then use the two-sample Z test. Moreover, if you do not know either of the two population standard deviations then use a two-sample t-test.

Let us see what the steps are involved in the two-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 statistic

Use the following formula to find the test statistic for the two-sample Z test.

In the two-sample t-test, if we assume population standard deviations are unequal then we use the below formula to calculate the test statistic.

Where the test statistic follows a t distribution with v is the degrees of freedom under the null hypothesis.

In the two-sample t-test, if we assume population standard deviations are equal then we calculate pooled sample standard deviation and use the below formula to calculate the test statistic.

Where the test statistic follows a t distribution with (n₁+n₂-2) are the degrees of freedom under the null hypothesis. Use the following formula to calculate pooled sample standard deviation (s).

 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

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 rejects 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 means are unequal (OR µ₁ < µ₂ OR µ₁ > µ₂).

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 means are unequal (OR µ₁ < µ₂ OR µ₁ > µ₂).