Two-sample tests for means Problem set A

State the null and alternative hypotheses for each test. Define μ1 and μ2.

1. (a) Does the mean fuel use differ between petrol cars (group 1) and hybrid cars (group 2)?
2. (b) A gardener thinks plants given fertiliser X (group 1) grow taller on average than plants given fertiliser Y (group 2).

Use a two-sample z-test with H0: μ1 = μ2 and H1: μ1 < μ2 to find the p-value, from these summary statistics for two normal populations with known standard deviations.

1. (a) σ1 = 2.4, n1 = 25, x̄1 = 14.8; σ2 = 3.1, n2 = 30, x̄2 = 16.1.
2. (b) σ1 = 6.5, n1 = 20, x̄1 = 84.2; σ2 = 8.0, n2 = 24, x̄2 = 87.9.

Two normal populations have known standard deviations σ1 = 0.7 and σ2 = 0.7. Sample 1: 8.4, 9.1, 7.9, 9.6, 8.8, 9.3, 8.5. Sample 2: 7.8, 8.2, 8.9, 7.5, 8.1, 8.6. Use a two-sample z-test with H0: μ1 = μ2 and H1: μ1 > μ2 to find the p-value.

Use a two-sample pooled t-test with H0: μ1 = μ2 and H1: μ1 ≠ μ2 to find the p-value. Assume that the populations are normal with equal variances. Each s is a sample standard deviation.

1. (a) x̄1 = 72.5, s1 = 8.1, n1 = 12; x̄2 = 66.8, s2 = 9.4, n2 = 15.
2. (b) x̄1 = 5.6, s1 = 1.2, n1 = 9; x̄2 = 6.9, s2 = 1.5, n2 = 11.

Sample 1: 23.1, 25.4, 22.8, 24.0, 21.9, 23.5. Sample 2: 25.9, 27.2, 24.8, 26.5, 25.1, 27.8, 26.0. Use a two-sample pooled t-test with H0: μ1 = μ2 and H1: μ1 < μ2 to find the p-value. Assume that the populations are normal with equal variances.

Two machines fill bags of rice. The masses, in grams, are normally distributed with known standard deviations of 4.2 g for machine 1 and 5.1 g for machine 2. A random sample of 30 bags from machine 1 has mean mass 502.8 g. A random sample of 25 bags from machine 2 has mean mass 499.6 g. Test, at the 5% significance level, whether the machines fill bags to different mean masses.

1. (a) State the hypotheses, defining μ1 and μ2.
2. (b) Find the p-value.
3. (c) State the conclusion in context.

A teacher tests whether a memory technique helps. Eight students used the technique (group 1) and seven different students did not (group 2). The numbers of words recalled were: technique: 14, 17, 15, 18, 16, 13, 17, 19; no technique: 12, 15, 13, 16, 14, 12, 15. Assume that the numbers are normally distributed with equal variances. Test, at the 5% significance level, whether the technique increases the mean number of words recalled.

1. (a) State the hypotheses, defining μ1 and μ2.
2. (b) Find the p-value.
3. (c) State the conclusion in context.

Two classes sit the same test. The marks in each class are normally distributed with standard deviation 6.5. Class 1 marks: 68, 74, 71, 66, 79, 72, 70, 77. Class 2 marks: 64, 70, 66, 69, 62, 71, 67. Test, at the 10% significance level, whether the mean marks of the two classes differ.

1. (a) Choose the test and state the hypotheses, defining μ1 and μ2.
2. (b) Find the p-value.
3. (c) State the conclusion in context.

Two suppliers sell steel cables. The breaking loads, in kN, are normally distributed with known standard deviations of 12 kN (supplier 1) and 15 kN (supplier 2). A sample of 40 cables from supplier 1 has mean breaking load 95.5 kN. A sample of 35 cables from supplier 2 has mean breaking load 88.0 kN. A test is carried out of whether the mean breaking loads differ.

1. (a) Find the p-value.
2. (b) State the conclusion at the 5% significance level.
3. (c) State the conclusion at the 1% significance level.