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).