## 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.

*variable*the measurements are is also useful (but beyond us).