7 A swimming coach believes that times recorded by people using stopwatches are on average 0.2 seconds faster than those recorded by an electronic timing system. In order to test this, the coach takes a random sample of 40 competitors' times recorded by both methods, and finds the differences between the times recorded by the two methods. The mean difference in the times (electronic time minus stopwatch time) is 0.1442 s and the standard deviation of the differences is 0.2580 s .

1. Find a 95\% confidence interval for the mean difference between electronic and stopwatch times.
2. Explain whether there is evidence to suggest that the coach’s belief is correct.
3. Explain how you can calculate the confidence interval in part (a) even though you do not know the distribution of the parent population of differences.
4. If the coach wanted to produce a \(95 \%\) confidence interval of width no more than 0.12 s , what is the minimum sample size that would be needed, assuming that the standard deviation remains the same?