A sample of seven students who were given a GMAT test before and after they took instructional review classes had the following scores. These data are also in the accompanying spreadsheet.

You would like to be able to assert that the instructional review classes were beneficial.

a. What is the null hypothesis and what is the alternative hypothesis?

b. State which test should be used and explain why it is the appropriate test for this problem.

c. At the 0.05 level of significance, can we conclude that the instructional review classes were helpful? Show your work.

**Solution: a. **First, we have to compute a table to compute the difference between the results.

We define _{}to the mean difference between the scores after and before

The null and alternative hypotheses are:

_{}

**b. **We have to use *the one-tailed paired t-test*, because the samples are not independent. As a matter of fact we compare results of the same individuals after and before.

**c. **From the table we use EXCEL to compute the standard deviation of the sample to get that _{}. The paired t-statistic is

_{}

The 0.05 significance one-tailed critical t-value with 6 degrees of freedom is 1.943. Since the statistic is greater than the critical value, we reject the null hypothesis of_{}, and *we can conclude that the instructional review classes were helpful.*

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