Paired-sample t-test Problem set B Worked solutions

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.

   • d = x1 − x2, pair by pair: −0.2, 0.2, −0.4, 0.1, −0.4, −0.1

2. (b) Find the p-value.

   • H0: μ_d = 0, H1: μ_d < 0
   • From the calculator, t-test on d, μ0 = 0: t = −1.30, p = 0.124

3. (c) State the conclusion of the test.

   • 0.124 > 0.05, so do not reject H0.

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.

   • d = x1 − x2, pair by pair: 3, 5, 1, 5, 0, 8, 2, 4

2. (b) Find the p-value.

   • H0: μ_d = 0, H1: μ_d > 0
   • From the calculator, t-test on d, μ0 = 0: t = 3.86, p = 0.00310

3. (c) State the conclusion of the test.

   • 0.00310 < 0.1, so reject H0.

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.

   • d = with the drink − without the drink; μ_d is the population mean of d.
   • H0: μ_d = 0 (no change)
   • H1: μ_d > 0 (the drink increases the speed, so d is positive)

2. (b) State the assumption needed for a paired t-test.

   • The paired test is a one-sample t-test on the differences.
   • So the differences must come from a normal population.

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.

   • A paired test removes the differences between services, so it detects only the change caused by the new system.

2. (b) State the hypotheses, defining d.

   • d = after − before: −1.6, −1.2, −1.3, 0.3, −1.9, −1.7, −1.6
   • μ_d is the population mean change. H0: μ_d = 0, H1: μ_d < 0

3. (c) Find the p-value and state the conclusion in context.

   • From the calculator, t-test on d, μ0 = 0: t = −4.61, p = 0.00183
   • 0.00183 < 0.05, so reject H0. There is sufficient evidence, at the 5% significance level, that the system reduces the mean delay.

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.

   • d = after − before: 5, 0, 5, 3, 4, 5, 1, 6

2. (b) Test the claim: state H0 and H1, find the p-value and conclude in context.

   • H0: μ_d = 0, H1: μ_d > 0
   • From the calculator, t-test on d, μ0 = 0: t = 4.80, p = 0.000978
   • 0.000978 < 0.05, so reject H0. There is sufficient evidence, at the 5% significance level, that the course increases the mean number of press-ups.

3. (c) State the assumption you made.

   • The test is a t-test on the differences, which must come from a normal population.

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.

   • H0: μ1 = μ2, H1: μ1 > μ2 (after > before)
   • From the calculator, two-sample pooled t-test: t = 0.940, p = 0.182

2. (b) Carry out a paired t-test. Find the p-value.

   • d = after − before: 4, 4, 3, 6, 3, 4, 2, 5
   • From the calculator, t-test on d, μ0 = 0, μ > μ0: t = 8.79, p = 0.0000248

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.

   • The same people are measured twice, so the data are paired.
   • Individuals differ a lot from one another, which hides the change in the two-sample test. The differences vary much less.
   • 0.0000248 < 0.05: reject H0. There is sufficient evidence, at the 5% significance level, that the interval training increases the mean score.

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.

   • Each driver has two reaction times, so the data are paired.

2. (b) The battery lives of 10 phones of brand X and 12 phones of brand Y.

   • The phones are different, with no pairing, and the sample sizes differ.

3. (c) The marks of 15 students in mathematics and in physics.

   • Each student has a mathematics and a physics mark, so they form a pair.