Two-sample tests for means Problem set B

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

1. (a) Does the mean time to charge differ between phone model P (group 1) and phone model Q (group 2)?
2. (b) A pharmacist thinks that tablet A (group 1) takes less time to work, on average, than tablet B (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 = 4.1, n1 = 18, x̄1 = 33.5; σ2 = 3.6, n2 = 22, x̄2 = 35.2.
2. (b) σ1 = 18, n1 = 40, x̄1 = 215; σ2 = 21, n2 = 35, x̄2 = 224.

Two normal populations have known standard deviations σ1 = 3.5 and σ2 = 4.0. Sample 1: 52, 58, 49, 61, 55, 57. Sample 2: 48, 53, 50, 46, 51, 49, 52. 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 = 41.2, s1 = 5.3, n1 = 14; x̄2 = 37.6, s2 = 4.8, n2 = 10.
2. (b) x̄1 = 19.4, s1 = 2.1, n1 = 8; x̄2 = 17.9, s2 = 1.7, n2 = 12.

Sample 1: 64, 71, 58, 69, 62, 66, 60. Sample 2: 72, 75, 68, 78, 70, 73. 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 nurseries grow sunflowers. The heights, in cm, are normally distributed with known standard deviations of 9.5 cm for nursery 1 and 7.8 cm for nursery 2. A random sample of 22 sunflowers from nursery 1 has mean height 171.4 cm. A random sample of 28 sunflowers from nursery 2 has mean height 165.2 cm. Test, at the 1% significance level, whether the nurseries grow sunflowers of different mean heights.

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

A health club tests whether a phone app increases daily activity. Six members used the app (group 1) and seven different members did not (group 2). Their mean daily steps, in thousands, were: app: 8.1, 9.4, 7.8, 10.2, 8.9, 9.6; no app: 7.2, 8.0, 7.5, 8.8, 7.1, 8.3, 7.7. Assume that the numbers are normally distributed with equal variances. Test, at the 5% significance level, whether the app increases the mean number of daily steps.

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

Apples are picked from two orchards. The masses, in grams, in each orchard are normally distributed with standard deviation 12 g. Orchard 1: 182, 195, 171, 188, 201, 177. Orchard 2: 190, 204, 196, 210, 199, 205, 193. Test, at the 10% significance level, whether the mean masses of apples from the two orchards 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 bags of compost. The masses, in kg, of the bags are normally distributed with known standard deviations of 0.8 kg (supplier 1) and 1.1 kg (supplier 2). A sample of 30 bags from supplier 1 has mean mass 5.35 kg. A sample of 45 bags from supplier 2 has mean mass 4.80 kg. A test is carried out of whether the mean masses 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.