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IB Mathematics: Applications and Interpretation HL · 13.4: Hypothesis Testing · 13.4.14

Hypothesis tests for a mean: mixed practice Problem set B Answers

Skills

1. (a) H0: μ1 = μ2, H1: μ1 ≠ μ2
2. (b) p = 0.00388
3. (c) There is sufficient evidence, at the 10% significance level, that the mean times of the two groups are different.
1. (a) The population variance is not known: it is estimated from the sample.
2. (b) H0: μ = 35, H1: μ > 35
3. (c) p = 0.0275
4. (d) There is sufficient evidence, at the 5% significance level, that the mean journey time of the school bus is more than 35 minutes.
1. (a) 5, 3, 7, 3, 1, 6, 4, 6
2. (b) mean = 4.38, s = 2.00
1. (a) H0: μ = 55, H1: μ > 55
2. (b) \(\bar X \sim N(55, 7^2/30)\)
3. (c) \(\bar X > 57.1\)
4. (d) There is sufficient evidence, at the 5% significance level, that the mean drying time has increased.

1. (a) \(\bar X \sim N(120, 0.9^2/25)\)
2. (b) \(\bar X < 119.5\) or \(\bar X > 120.5\)
3. (c) There is sufficient evidence, at the 1% significance level, that the mean length of a rod has changed.
1. (a) The same people are measured twice, so the two samples are not independent.
2. (b) H0: μD = 0, H1: μD > 0
3. (c) p = 0.00123
4. (d) There is sufficient evidence, at the 5% significance level, that the walk raises the mean pulse rate.
1. (a) Paired test: each athlete is measured twice.
2. (b) Two-sample test: the two groups are different athletes.
3. (c) The differences are normally distributed.
4. (d) Both populations are normally distributed and have equal variances.
5. (e) It removes the variation between individuals, so a real difference is easier to detect.
1. (a) s1² = 18.3, s2² = 16.0
2. (b) H0: μ1 − μ2 = 3, H1: μ1 − μ2 > 3; sp² = 17.3, t = 1.29, p = 0.106
3. (c) There is insufficient evidence, at the 10% significance level, that the mean mark with the old method is more than 3 marks greater than with the new method.

1. (a) The samples are independent, the populations are normal and both standard deviations are known.
2. (b) N(15, 7.82)
3. (c) p = 0.0160
4. (d) There is sufficient evidence, at the 10% significance level, that the mean daily sales at shop M are more than 15 greater than at shop N.
1. (a) H0: μD = 0, H1: μD > 0
2. (b) Var(D) = 1.28
3. (c) \(\bar D \sim N(0, 0.1067)\)
4. (d) \(\bar D > 0.419\)
5. (e) There is sufficient evidence, at the 10% significance level, that the new layout shortens the mean queue time.
1. (a) n is large, so by the central limit theorem the sample mean is approximately normal.
2. (b) p = 0.0255
3. (c) There is sufficient evidence, at the 5% significance level, that the mean daily rainfall has increased.
1. (a) mean = 48.6, s = 3.91
2. (b) p = 0.150
3. (c) A large p-value means there is insufficient evidence that the mean is below 50 minutes. It does not prove the mean is 50.

jmaths.xyz · AI HL · 13.4.14 Hypothesis tests for a mean: mixed practice