Paired-sample t-test Problem set B

Paired data. Sample 1: 3.2, 4.8, 2.9, 5.1, 3.8, 4.4. Sample 2: 3.4, 4.6, 3.3, 5.0, 4.2, 4.5. 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.

Paired data. Sample 1: 83, 76, 91, 68, 79, 88, 72, 85. Sample 2: 80, 71, 90, 63, 79, 80, 70, 81. 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.

A sports scientist tests an energy drink. Twelve cyclists complete a time trial without the drink and again with it. She wants to know whether the drink increases the mean speed.

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 rail company records the delay, in minutes, of 7 services on one day before and after a new signalling system. Before: 24.0, 26.3, 21.1, 27.0, 25.9, 23.3, 27.8. After: 22.4, 25.1, 19.8, 27.3, 24.0, 21.6, 26.2. Assume that the differences are normally distributed. Test, at the 5% significance level, whether the system reduces the mean delay.

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 gym measures the number of press-ups that 8 members can do in one minute, before and after a six-week course. Before: 57, 58, 66, 63, 50, 64, 59, 58. After: 62, 58, 71, 66, 54, 69, 60, 64. The gym claims that the course increases the mean number of press-ups.

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 cyclists ride a time trial, then ride it again after a month of interval training. Scores, in points, from the trial: first ride 84, 91, 78, 96, 73, 93, 82, 85; second ride 88, 95, 81, 102, 76, 97, 84, 90. Assume that all the distributions needed are normal and that the two-sample test has equal variances. Test whether the training increases the mean score.

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 reaction times of 8 drivers, with and without a night's sleep.
2. (b) The battery lives of 10 phones of brand X and 12 phones of brand Y.
3. (c) The marks of 15 students in mathematics and in physics.