3 Cholesterol is a lipid (fat) which is manufactured by the liver from the fatty foods that we eat. It plays a vital part in allowing the body to function normally. However, when high levels of cholesterol are present in the blood there is a risk of arterial disease. Among the factors believed to assist with achieving and maintaining low cholesterol levels are weight loss and exercise.
A doctor wishes to test the effectiveness of exercise in lowering cholesterol levels. For a random sample of 12 of her patients, she measures their cholesterol levels before and after they have followed a programme of exercise. The measurements obtained are as follows.
| Patient | A | B | C | D | E | F | G | H | I | J | K | L |
| Before | 5.7 | 5.7 | 4.0 | 6.8 | 7.4 | 5.5 | 6.7 | 6.4 | 7.2 | 7.2 | 7.1 | 4.4 |
| After | 5.8 | 4.0 | 5.2 | 5.7 | 6.0 | 5.0 | 5.8 | 4.2 | 7.3 | 5.2 | 6.4 | 4.1 |
- A \(t\) test is to be used in order to see if, on average, the exercise programme seems to be effective in lowering cholesterol levels. State the distributional assumption necessary for the test, and carry out the test using a \(1 \%\) significance level.
- A second random sample of 12 patients gives a \(95 \%\) confidence interval of \(( - 0.5380,1.4046 )\) for the true mean reduction (before - after) in cholesterol level. Find the mean and standard deviation for this sample. How might the doctor interpret this interval in relation to the exercise programme?