**PAIRED SAMPLE t-TEST**

A paired t-test is used to compare two population means where the observations in one sample can be paired with observations in the other sample. It is generally used when:

- The measurements are taken from the same subject before and after a particular course of action.
- The number of observations in each data set is the same, and they are related in pairs.
We are interested in testing the equality of two population means (μ

_{1}and μ_{2}). The hypotheses for the comparison of the means in a two-sample paired*t-*test are as follows:H

_{0}: μ_{1}= μ_{2}H

_{1}: μ_{1}≠ μ_{2}**(Two tailed test)**or H_{1}: μ_{2}> or (<) μ_{1 }**(One tailed test)**

**CASE ANALYSIS-1 **

** ****PROBLEM**

The marketing head of company wants to compare the performance of two sales executives based on their average sales in last 6 weeks before and after they were trained. The average sales of salesmen A and B are given in the following table.

**Table-1:** Sample Data

The hypotheses for the analysis are:

Null hypothesis-H_{0}: There is no significant difference between the average sales of A and B.

(μ_{1} = μ_{2})

Alternative Hypothesis- H_{1}: There is a significant difference between the average sales of A and B. (μ_{1} μ_{2})

** ****Input Data**

**Table-2:** Input Data

**Performing the Analysis with SPSS**

For SPSS Version 11, click on **Analyze ⇒ ****Compare Means ⇒ Paired-Samples T Test .**This will bring up the SPSS screen dialogue box as shown below

After clicking **Paired-Samples T test, **this will bring up the following SPSS screen dialogue box

the variables and move them to **Paired variables** box.

Click **Option **and select confidence interval 95% (5% level of significance) and then **Continue**.

This will bring the **Paired-Samples T Test** dialogue box. Finally click OK.

**SPSS Output**

The SPSS outputs of the analysis are depicted in table-3 and table-5

**T-test**

**Table-3:** Paired Samples Statistics

The average sales by the salesman A is 22.5 units and 22 units by B.

**Table-4:** Paired Samples Correlations

The correlation between A and B is 0.624

**Table-5:** Paired Samples Test

**From the output, ****t ****= 0.165 with 5 degrees of freedom**

**DECISION**

Reject the null hypothesis if p-value (Sig. value) ≤ 0.05

**INTERPRETATION**

The p-value is 0.875 and it is more than 0.05 (**5% level of significance**), so we accept the null hypothesis and reject the alternative hypothesis at 5% level of significance. It is concluded that the average sales by two salesmen are equal.