D.25 Answers: Tests for one mean

Answers to exercises in Sect. 27.13.

Answer to Exercise 27.1: 1. $H_0$: $\mu =7725$; $H_1$: $\mu \ne 7725$ (two tailed). 2. $\overline{x}=6753.64$ and $\text{s.e.}\left(\overline{x}\right)=s/\sqrt{n}=1142.123/\sqrt{11}=344.363$. 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: $\text{s.e.}\left(\overline{x}\right)=23.8/\sqrt{85}=2.581472$. $t$-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: $\text{s.e.}\left(\overline{x}\right)=22.05/\sqrt{22}=4.701076$. $t$-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: $\text{s.e.}\left(\overline{x}\right)=0.60652/\sqrt{43}=0.09249343$. $t$-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 $H_1$: $\mu \ne 1000$, 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).