D.25 Answers: Tests for one mean

Answers to exercises in Sect. 27.13.

Answer to Exercise 27.1: 1. $$H_0$$: $$\mu=7725$$; $$H_1$$: $$\mu\ne7725$$ (two tailed). 2. $$\bar{x} = 6753.64$$ and $$\text{s.e.}(\bar{x}) = s/\sqrt{n} = 1142.123/\sqrt{11} = 344.363$$. 3. $$t = (6753.64 - 7725)/344.363 = -2.821$$, as in output. This ‘large’; expect small $$P$$-value; software confirms this: two-tailed $$P=0.018$$. 4. Moderate evidence ($$P = 0.018$$) that the mean energy intake is not meeting the recommended daily energy intake (mean: 6753.6kJ; std. dev.: 1142.1kJ).
Answer to Exercise 27.2: $$H_0$$: $$\mu=120$$ and $$H_1$$: $$\mu\ne 120$$ (two-tailed), where $$\mu$$ is the mean time in seconds. Standard error: $$\text{s.e.}(\bar{x}) = 23.8/\sqrt{85} = 2.581472$$. $$t$$-score: $$(60.3 - 120)/2.581472 = -23.13$$, which is huge; $$P$$-value will be really small. Very strong evidence ($$P<0.001$$) that children do not spend 2 minutes (on average) brushing their teeth (mean: 60.3s; std. dev.: 23.8s).
Answer to Exercise 27.3: $$H_0$$: $$\mu=50$$ and $$H_1$$: $$\mu>50$$ (one-tailed), where $$\mu$$ is the mean mental demand. Standard error: $$\text{s.e.}(\bar{x}) = 22.05/\sqrt{22} = 4.701076$$. $$t$$-score: $$(84 - 50)/4.701076 = 7.23$$, which is very large; $$P$$-value will be very small. Very strong evidence ($$P<0.001$$) that the mean mental demand is greater than 50. (Notice we say greater than, because of the RQ and the alternative hypothesis.)
Answer to Exercise 27.4: Physical:* $$t = -1.28$$; Mental:* $$t = 1.80$$. The $$P$$-values both larger than 5%. No evidence that the mean score for patients is different than the general population score.
Answer to Exercise 27.5: $$H_0$$: $$\mu=12$$ and $$H_1$$: $$\mu\ne 12$$ (two-tailed), where $$\mu$$ is the mean weight in grams. Standard error: $$\text{s.e.}(\bar{x}) = 0.60652/\sqrt{43} = 0.09249343$$. $$t$$-score: $$(14.9577 - 12)/0.09249343 = 31.98$$, which is huge; $$P$$-value will be very small. Very strong evidence ($$P<0.001$$) that the mean weight of a Fun Size Cherry Ripe bar is not 12 grams (mean: 14.9577; std. dev.: 0.067g), and they may be larger.
Answer to Exercise 27.6: $$H_0$$: $$\mu=1000$$ and $$H_1$$: $$\mu\ne1000$$, where $$\mu$$ is the population mean guess of the spill volume. Standard error: 46.15526. $$t$$-score: $$(846.4 - 1000)/46.15526 = -3.33$$, which is very large (and negative), so the $$P$$-value will be very small. Very strong evidence that the mean guess of blood volume is not 1000,ml, the actual value. The sample is much larger than 25: the test is statistically valid.

Answer to Exercise 27.7: Hypotheses have the form $$H_0$$: $$\mu=\text{pre-determined target}$$, and $$H_1$$: $$\mu\ne\text{pre-determined target}$$. $$t$$-scores: $$t_1 = 0.318$$, $$t_2 = 2.347$$, $$t_3 = -0.466$$, $$t_4 = -0.726$$. $$P$$-values will be large, except for second test. No evidence that the instruments are dodgy, except perhaps for the first instrument for mid-level LH concentrations. Should be statistically valid.

While assessing the means is useful, how variable the measurements are is also useful (but beyond us).