Paired-sample t-test Problem set A

Paired data. Sample 1: 7.4, 5.9, 8.1, 6.3, 7.0, 5.5. Sample 2: 7.9, 6.8, 8.2, 6.6, 7.9, 5.6. A paired t-test is carried out with H0: μ1 = μ2 and H1: μ1 < μ2, at the 10% significance level.

1. (a) Find the differences d = x1 − x2.
2. (b) Find the p-value.
3. (c) State the conclusion of the test.

Paired data. Sample 1: 48, 55, 61, 43, 52, 58, 50, 46. Sample 2: 45, 57, 60, 44, 49, 56, 52, 45. A paired t-test is carried out with H0: μ1 = μ2 and H1: μ1 > μ2, at the 5% significance level.

1. (a) Find the differences d = x1 − x2.
2. (b) Find the p-value.
3. (c) State the conclusion of the test.

A podiatrist tests a new insole. Ten patients give a pain score, out of 10, before and after wearing it for a month. She wants to know whether the insole reduces the mean pain score.

1. (a) State the hypotheses in terms of the mean difference μ_d, saying how d is found.
2. (b) State the assumption needed for a paired t-test.

A doctor measures the systolic blood pressure, in mmHg, of 8 patients before and after a month on a new medicine. Before: 124, 131, 133, 130, 141, 137, 139, 128. After: 118, 125, 131, 122, 140, 128, 135, 121. Assume that the differences are normally distributed. Test, at the 5% significance level, whether the medicine reduces mean blood pressure.

1. (a) Give one advantage of a paired test here.
2. (b) State the hypotheses, defining d.
3. (c) Find the p-value and state the conclusion in context.

A teacher gives 7 students a spelling test before and after a week of a new reading scheme. The marks are: before 33, 41, 31, 37, 43, 36, 38; after 34, 41, 29, 38, 45, 36, 40. The teacher claims that the scheme increases the mean mark.

1. (a) Find the differences, after − before.
2. (b) Test the claim: state H0 and H1, find the p-value and conclude in context.
3. (c) State the assumption you made.

Eight students sit a practice test, then sit a second test after a revision course. First test: 48, 62, 55, 71, 40, 66, 58, 52. Second test: 51, 66, 57, 76, 43, 70, 60, 57. Assume that all the distributions needed are normal and that the two-sample test has equal variances. Test whether the course increases the mean mark.

1. (a) Carry out a two-sample pooled t-test, treating the samples as independent. Find the p-value.
2. (b) Carry out a paired t-test. Find the p-value.
3. (c) Which test is correct here? Explain why the two p-values are so different, and give the conclusion at the 5% significance level.

For each situation say whether to use a paired t-test or a two-sample t-test, and give a reason.

1. (a) The mass of 10 people before and after a diet.
2. (b) The heights of 12 boys and 15 girls chosen at random.
3. (c) The grip strength of the left and the right hand of 9 people.