## 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).