The number of sit-ups that a person can do in one minute, both before and after a physical fitness course was recorded. The data is shown below for 10 randomly selected participants. Is there sufficient evidence to conclude that a significant amount of improvement took place? Assume variable is normally distributed. Test using a = 0.01.

**Solution: **We need to test

_{}

where_{}represents the mean difference between after and before the fitness course. This corresponds to a paired-sample t-test. Using Excel we obtain:

The t-statistics is *t* = 3.772, and the one-tailed p-value is *p* = 0.004405. Since the p-value is less than the significance level 0.05, we reject the null hypothesis. This means that there is sufficient evidence to conclude that a significant amount of improvement took place, at the 0.05 significance level.

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