D.28 Answers: Tests for two means

Answers to exercises in Sect. 30.12.

Answer to Exercise 30.1: \(H_0\): \(\mu_S - \mu_{NS} = 0\) and \(H_1\): \(\mu_S - \mu_{NS} \ne 0\). From output: \(t = 5.478\) and \(P<0.001\). Very strong evidence to support \(H_1\).

Answer to Exercise 30.2: 1. Table D.7. 2. \(H_0\): \(\mu_C - \mu_{SO} = 0\) and \(H_1\): \(\mu_C - \mu_{SO} \ne 0\). Then \(t=( (51 - 56) - 0 )/3.3044 = -1.513\); \(P\)-value larger than 5%. Sample size are small; test may not be statistically valid. 3. \(H_0\): \(\mu_C - \mu_{SO} = 0\) and \(H_1\): \(\mu_C - \mu_{SO} \ne 0\). Then \(t=( (36 - 47) - 0 )/4.0689 = -2.70\); \(P\)-value smaller than 5%. Sample size are small; the test may not be statistically valid.

TABLE D.7: The physical profile of conventional and special operation paramedics in Western Australia

	Conventional	Special Operations
Sample size	11	11
Grip strength (in kg)
Mean	51	56
Standard deviation	8	9
Standard error	1.86	2.71
Push-ups (per minute)
Mean	36	47
Standard deviation	10	11
Standard error	2.36	3.3

Answer to Exercise 30.3: \(H_0\): \(\mu_M - \mu_{F} = 0\) and \(H_1\): \(\mu_M - \mu_{F} \ne 0\). From output, \(t = -2.285\); (two-tailed) \(P\)-value is 0.024 Moderate evidence to support \(H_1\): Moderate evidence (\(P=0.024\)) that the mean internal body temperature is different for females (mean: \(36.886^{\circ}\text{C}\)) and males (mean: \(36.725^{\circ}\text{C}\)).

The difference between the means, of 0.16 of a degree, is hardly of any practical importance in everyday use.

Answer to Exercise 30.4: 1. \(H_0\): The means are equal: \(\mu_I = \mu_{NI}\) or \(\mu_I - \mu_{NI} = 0\). \(H_1\): The means are not equal: \(\mu_I \ne \mu_{NI}\) or \(\mu_I - \mu_{NI} \ne 0\). 2. CI from -22.54 to -11.95: the mean sugar consumption between 11.95 and 22.54 kg/person/year greater in industrialised countries. 3. Very strong evidence in the sample (\(P<0.001\)) that the mean annual sugar consumption per person is different for industrialised (mean: 41.8 kg/person/year) and non-industrialised (mean: 24.6 kg/person/year) countries (95% CI for the difference 11.95 to 22.54).