## 9.7Optional questions

These questions are optional; e.g., if you need more practice, or you are studying for the exam. (Answers are available in Sect. A.9.)

### 9.7.1Optional: Tests for one mean

This question has a video solution in the online book, so you can hear and see the solution.

Shortly after metric units were introduced to Australia in 1977, a lecturer wondered how accurately students could estimate lengths using the metric measurements .

The aim of the study was to determine if, on average, students could correctly guess the width of the hall (which was $$13.1$$ m).

To answer the RQ, a group of $$n = 44$$ students were asked to estimate the width of the lecture hall to the nearest metre (data provided by Prof. Lewis). The data are given in Table 9.1.

The data were entered into software, producing the output in Fig. 9.9 (jamovi), and Fig. 9.10 (SPSS).

 8 9 10 10 10 10 10 10 11 11 11 11 12 12 13 13 13 14 14 14 15 15 15 15 15 15 15 15 16 16 16 17 17 17 17 18 18 20 22 25 27 35 38 40
1. What are the values of $$s$$ and $$\text{s.e.}(\bar{x})$$? Explain the difference in the meaning of the two terms.
2. Perform a hypothesis test to determine if the mean guess is the actual distance of $$13.1$$ m. (The CI was computed in Sect. 7.8.1.)
3. Is it reasonable to assume the conditions are satisfied? The histogram in Fig. 9.8 may (or may not) help.
4. Do you think students were very good at estimating the width of the hall using metric units?

### 9.7.3Optional: $$P$$-values

An ecologist is studying two different grasses to help combat soil salinity, by comparing to a new grass (Grass A) to a native grass (Grass B). She uses $$50$$ different sites, allocating the two grasses at random to the sites ($$25$$ sites for each grass).

After $$12$$ months, the ecologist records whether the soil salinity at each site has improved, and hence computes the odds that each grass will improve the salinity. She finds a statistically significant difference between the odds in the two groups. Which of these statements is consistent with this conclusion?

1. The $$\text{OR} = 4.1$$ and $$P = 0.36$$
2. The $$\text{OR} = 4.1$$ and $$P = 0.0001$$
3. The $$\text{OR} = 0.91$$ and $$P = 0.36$$
4. The $$\text{OR} = 0.91$$ and $$P = 0.0001$$

How would the other statements be interpreted then?

### 9.7.4Optional: Tests for paired samples

This question has a video solution in the online book, so you can hear and see the solution.

After a number of runners collapsed near the finish of the Tyneside annual Great North Run, researchers decided to study the role of $$\beta$$ endorphins as a factor in the collapses. ($$\beta$$ endorphins are a peptide which suppress pain.)

A study examined what happened with plasma $$\beta$$ endorphins during fun runs: does the concentration of plasma $$\beta$$ endorphins increase, on average, during fun runs.

The researchers recorded the plasma $$\beta$$ concentrations in fun runners (participating in the Tyneside Great North Run). Eleven runners (who did not collapse) had their plasma $$\beta$$ concentrations measured (in pmol/litre) before and after the race (Table 9.2.) A 'typical' value for $$\beta$$ endorphines is usually under about 11 pmol/litre.

1. Explain why these data should be analysed as a paired means situation.

2. Which of the following is the correct null hypothesis for this question? Why are the others incorrect? Is the test one- or two-tailed?

1. $$\mu_{\text{Before}} = \mu_{\text{After}}$$
2. $$\mu_{\text{Difference}} = 0$$
3. $$\mu_{\text{Before}} = 0$$
4. $$\mu_{\text{After}} = 0$$
5. $$\mu_{\text{Before}} < \mu_{\text{After}}$$
6. $$\mu_{\text{Difference}} > 0$$
7. $$\mu_{\text{Before}} < 0$$
8. $$\mu_{\text{After}} > 0$$
3. Write down the mean, standard deviation and standard error, using Fig. 9.11 (jamovi) or Fig. 9.12 (SPSS).

4. Write down the value of the $$t$$-statistic, then estimate the $$P$$-value (using the $$68$$--$$95$$--$$99.7$$ rule).

5. Write a statement that communicates the result of the test.

6. What conditions must be met for this test to be valid?

7. Is it reasonable to assume the assumptions are satisfied? Use the histogram of the data in Fig. 9.13 to help, if necessary.

TABLE 9.2: Plasma-beta concentration for runners, before and after the race
Before After Difference
4.3 29.6 25.3
4.6 25.1 20.5
5.2 15.5 10.3
5.2 29.6 24.4
6.6 24.1 17.5
7.2 37.8 30.6
8.4 20.1 11.7
9.0 21.9 12.9
10.4 14.2 3.8
14.0 34.6 20.6
17.8 46.2 28.4

### References

Dale G, Fleetwood JA, Weddell Ann, Ellis RD, Sainsbury JR. $$\beta$$ endorphin: A factor in “fun run” collapse? British Medical Journal. 1987;294:1004.
Hand DJ, Daly F, Lunn AD, McConway KY, Ostrowski E. A handbook of small data sets. London: Chapman; Hall; 1996.