Two-sample tests for means Problem set B One question at a time

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)?

   • Let μ1 and μ2 be the population mean time to charge of model P (group 1) and time to charge of model Q (group 2).
   • H0: μ1 = μ2
   • H1: μ1 ≠ μ2 (different, so two-tailed)

2. (b) A pharmacist thinks that tablet A (group 1) takes less time to work, on average, than tablet B (group 2).

   • Let μ1 and μ2 be the population mean time for tablet A to work (group 1) and time for tablet B to work (group 2).
   • H0: μ1 = μ2
   • H1: μ1 < μ2 (group 1 is less)

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.

   • H0: μ1 = μ2, H1: μ1 < μ2
   • Two-sample z-test, because both σ are known.
   • From the calculator: z = −1.38, p = 0.0842

2. (b) σ1 = 18, n1 = 40, x̄1 = 215; σ2 = 21, n2 = 35, x̄2 = 224.

   • H0: μ1 = μ2, H1: μ1 < μ2
   • Two-sample z-test, because both σ are known.
   • From the calculator: z = −1.98, p = 0.0240

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.

• H0: μ1 = μ2, H1: μ1 > μ2
• From the calculator, two-sample z-test with lists, σ1 = 3.5, σ2 = 4.0: z = 2.63
• p = 0.00424

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.

   • H0: μ1 = μ2, H1: μ1 ≠ μ2
   • Two-sample t-test, pooled, two-tailed.
   • From the calculator: t = 1.70, p = 0.102

2. (b) x̄1 = 19.4, s1 = 2.1, n1 = 8; x̄2 = 17.9, s2 = 1.7, n2 = 12.

   • H0: μ1 = μ2, H1: μ1 ≠ μ2
   • Two-sample t-test, pooled, two-tailed.
   • From the calculator: t = 1.76, p = 0.0952

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.

• H0: μ1 = μ2, H1: μ1 < μ2
• Two-sample t-test from lists, pooled, lower tail.
• From the calculator: t = −3.56, p = 0.00223

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.

   • μ1 and μ2 are the population mean height of a sunflower from nursery 1 and height of a sunflower from nursery 2.
   • H0: μ1 = μ2, H1: μ1 ≠ μ2

2. (b) Find the p-value.

   • Both σ are known: two-sample z-test, two-tailed.
   • From the calculator: z = 2.48, p = 0.0133

3. (c) State the conclusion in context.

   • 0.0133 > 0.01, so do not reject H0. There is insufficient evidence, at the 1% significance level, that the nurseries grow sunflowers of different mean heights.

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.

   • μ1 and μ2 are the population mean daily steps (thousands) with the app and daily steps (thousands) without it.
   • H0: μ1 = μ2, H1: μ1 > μ2

2. (b) Find the p-value.

   • Unknown σ, assumed equal: two-sample pooled t-test, upper tail.
   • From the calculator, with lists: t = 2.81, p = 0.00841

3. (c) State the conclusion in context.

   • 0.00841 < 0.05, so reject H0. There is sufficient evidence, at the 5% significance level, that the app increases the mean number of daily steps.

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.

   • The standard deviations are known, so use a two-sample z-test. μ1 and μ2 are the population mean mass of an apple from orchard 1 and mass of an apple from orchard 2.
   • H0: μ1 = μ2, H1: μ1 ≠ μ2

2. (b) Find the p-value.

   • From the calculator, with lists: z = −2.08, p = 0.0373

3. (c) State the conclusion in context.

   • 0.0373 < 0.1, so reject H0. There is sufficient evidence, at the 10% significance level, that the mean masses of apples from the two orchards differ.

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.

   • H0: μ1 = μ2, H1: μ1 ≠ μ2. Both σ are known, and they are different, which does not matter for the z-test.
   • From the calculator: z = 2.50, p = 0.0123

2. (b) State the conclusion at the 5% significance level.

   • 0.0123 < 0.05, so reject H0. There is sufficient evidence, at the 5% significance level, that the mean masses of bags from the two suppliers differ.

3. (c) State the conclusion at the 1% significance level.

   • 0.0123 > 0.01, so do not reject H0. There is insufficient evidence, at the 1% significance level, that the mean masses of bags from the two suppliers differ.