Paired-sample t-test Problem set A One question at a time

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.

   • d = x1 − x2, pair by pair: −0.5, −0.9, −0.1, −0.3, −0.9, −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 = −3.11, p = 0.0132

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

   • 0.0132 < 0.1, so reject H0.

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.

   • d = x1 − x2, pair by pair: 3, −2, 1, −1, 3, 2, −2, 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 = 0.856, p = 0.210

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

   • 0.210 > 0.05, so do not reject H0.

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.

   • d = after − before; μ_d is the population mean of d.
   • H0: μ_d = 0 (no change)
   • H1: μ_d < 0 (the insole reduces the pain score, so d is negative)

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 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.

   • A paired test removes the differences between patients, so it detects only the change caused by the medicine.

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

   • d = after − before: −6, −6, −2, −8, −1, −9, −4, −7
   • 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 = −5.38, p = 0.000515
   • 0.000515 < 0.05, so reject H0. There is sufficient evidence, at the 5% significance level, that the medicine reduces mean blood pressure.

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.

   • d = after − before: 1, 0, −2, 1, 2, 0, 2

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 = 1.08, p = 0.160
   • 0.160 > 0.05, so do not reject H0. There is insufficient evidence, at the 5% significance level, that the scheme increases the mean mark.

3. (c) State the assumption you made.

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

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.

   • H0: μ1 = μ2, H1: μ1 > μ2 (after > before)
   • From the calculator, two-sample pooled t-test: t = 0.681, p = 0.254

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

   • d = after − before: 3, 4, 2, 5, 3, 4, 2, 5
   • From the calculator, t-test on d, μ0 = 0, μ > μ0: t = 8.28, p = 0.0000365

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.
   • 0.0000365 < 0.05: reject H0. There is sufficient evidence, at the 5% significance level, that the revision course increases the mean mark.

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.

   • Each person has a 'before' and an 'after' mass, so the data are paired.

2. (b) The heights of 12 boys and 15 girls chosen at random.

   • The boys and the girls are different people, with no natural pairing, and the sample sizes differ.

3. (c) The grip strength of the left and the right hand of 9 people.

   • Each person gives a left and a right measurement, so they form a pair.