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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 A Answers

Skills

1. (a) H0: μ1 = μ2, H1: μ1 ≠ μ2
2. (b) p = 0.0224
3. (c) There is sufficient evidence, at the 5% significance level, that the mean masses of trout in the two lakes are different.
1. (a) The population variance is not known: it is estimated from the sample.
2. (b) H0: μ = 250, H1: μ < 250
3. (c) p = 0.0229
4. (d) There is sufficient evidence, at the 5% significance level, that the mean volume of a takeaway coffee is less than 250 ml.
1. (a) −1.4, 0.5, −1.5, −1.5, −0.3, −1.2, −0.7, −0.8
2. (b) mean = −0.863, s = 0.699
1. (a) H0: μ = 72, H1: μ < 72
2. (b) \(\bar X \sim N(72, 6.4^2/25)\)
3. (c) \(\bar X < 69.0\)
4. (d) There is insufficient evidence, at the 1% significance level, that the mean resting heart rate of members has decreased.

1. (a) \(\bar X \sim N(500, 4.2^2/36)\)
2. (b) \(\bar X < 498.6\) or \(\bar X > 501.4\)
3. (c) There is sufficient evidence, at the 5% significance level, that the mean volume of a bottle 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 = 0.00118
4. (d) There is sufficient evidence, at the 5% significance level, that listening to music reduces the mean reaction time.
1. (a) Paired test: each student has a mark in both tests.
2. (b) Two-sample test: the boys and girls are different people.
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 = 4.95, s2² = 4.29
2. (b) H0: μ1 − μ2 = 2, H1: μ1 − μ2 > 2; sp² = 4.58, t = 1.56, p = 0.0682
3. (c) There is insufficient evidence, at the 5% significance level, that the mean time of group 1 is more than 2 seconds greater than that of group 2.

1. (a) The samples are independent, the populations are normal and both standard deviations are known.
2. (b) N(4, 4.45)
3. (c) p = 0.109
4. (d) There is insufficient evidence, at the 5% significance level, that the mean mark with exam board X is more than 4 marks greater than with exam board Y.
1. (a) H0: μD = 0, H1: μD > 0
2. (b) Var(D) = 2.88
3. (c) \(\bar D \sim N(0, 0.2880)\)
4. (d) \(\bar D > 0.883\)
5. (e) There is sufficient evidence, at the 5% significance level, that the new feed raises the mean milk yield.
1. (a) n is large, so by the central limit theorem the sample mean is approximately normal.
2. (b) p = 0.0453
3. (c) There is sufficient evidence, at the 5% significance level, that the new system has reduced the mean waiting time.
1. (a) mean = 122, s = 7.04
2. (b) p = 0.0109
3. (c) A small p-value is evidence, not proof: if the mean really were 115 g, a sample like this would be unlikely, but it could still happen.

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