**Question**

A sample of 20 students were selected and given a diagnostic module prior to studying for a test. And then they were given the test again after completing the module. .

The result of the students scores in the test before and after the test is tabulated below.

We want to see if there is significant improvement in the student's performance due to this teaching method

**Solution Steps**

As we always say, you need to clearly understand the question. In this very case, we want to compare two means, therefore we use the paired-sample t-test.

**Step 1: Set up the null and the alternate hypothesis**

From the question

**Step 2: Extend the table to include the mean difference**

Watch a video to see how to do this using Excel

The table gives us the mean difference between each pair of observation and also the sum of the deviation from the mean.

mean difference = -2.05 ( it could also be positive)

**Step 3: Calculate the Standard Deviation**

You could calculate the Standard deviation using the formular

The gives us a standard deviation, S

_{d}

**Step 4: Calculate the Standard Error,**

*SE(d)*You could calculate the Standard Error using the formular below

Substituting the real values into this equation, we would have a value of

*0.634*

**Step 5: Calculate the t values**

Calculate the t statistic using the formular

This would give us a value of 3.231

**Step 6: Look up the t-value from Statistical table**

From th statistical table, we get the value of t as 0.004

**Step 7: Compare the calculated t and tabulated t and draw conclusion**

Calculated value of t is much larger than this value. We can see that there is significant difference in the means. Therefore we conclude that that there is much improvements in the test scores after the exams.