## A.12 Answer: TW 12 tutorial

1. Both students are incorrect as they used $$s$$ rather than $$\text{s.e.}(\bar{x})$$.
• Student A: This is an approximate 68% CI for the values of individual concrete samples, not for the mean, assuming the individual first-crack strengths have an approximate normal distribution.
• Student B: This is an approximate 95% CI for the values of individual concrete samples, not for the mean, assuming the individual first-crack strengths are approximately normally distributed.
2. We need to use the standard error because the question asks about a sample mean... and standard errors tells us how much the sample means are likely to vary. $$12.4\pm(2\times 2.8\div\sqrt{6})$$, or $$12.4\pm 2.286$$ MPa, or from $$10.11$$ to $$14.69$$ MPa.

• Research question: Use mean distance. So more directly:

For casual golfers, is the mean distance travelled by a golf ball the same when hit by a wooden or metal club?

• Summaries: There is no mention of the distance travelled by the ball. It really doesn't make sense to say 'the iron golf club' is 'moderately higher than the wooden golf club'. That sounds like the height of the clubs are different!
• Results (1): Spelling/grammar errors ('verse'; 'differes'). Is two decimal places suitable? Certainly no more. The CI is for the difference between the means of hitting distances.
• Results (2): The results just say there is a difference between metal and wooden clubs (of course!); there is no mention of hitting distance.

• Null hypothesis: The mean heart rates.
• Results: A $$\chi^2$$ test is inappropriate for comparing means.

• Research question: Spelling error ('lily pily'); mean size. It sounds odd, too, to ask if they 'become' longer. We are just comparing the mean length of leaves of eastern and western sides.

• Null hypothesis: Spelling again; and the hypothesis should be explicitly comparing the means: "For weeping lilly pilly trees at USC, it the mean leaf length the same for leaves on the western and eastern sides of the tree?"

• Results: Cannot find the mean of a qualitative variable ('Side of tree'); the table should give the mean, standard deviation and standard error for leaves taken from the west side (one row) and the east side (second row), plus information about the difference.

• Methods: There are only five units of analysis. Leaves taken from the same tree are likely to share a lot in common: genetics; sunlight, watering regime, soil condition, etc. The students should have used 50 trees.

Alternatively, I suppose they could have used one tree (so the RQ was about what happens on one tree only), and compared 50 leaves on either side. While this is kinda OK, I do not recommend it.

• Research question: Cumbersome wording...; mean estimates.
• Null hypothesis: Spelling/grammar errors; mean again; where does 'increased level of perception' come from? We are just comparing estimation distances; perception is much more than just that.
• Results: This doesn't answer the question. This is comparing the estimates from both groups---and yes, there may be a difference. But the RQ was about which group could estimate closer (on average) to the actual true width of the path. We still don't know: The RQ was not answered.
• Discussion: You don't prove anything like this... Again, what was tested doesn't help answer the RQ.

$$t$$-test: Age as it is the only quantitative variable. The rest would use a chi-square test. Between the 'alive' and 'dead' groups, there is evidence that the proportion with diabetes is different and whether or not a (prosthetic) limb was fitted.
1. P: school age children with ambulatory cerebral palsy; O: In this case, the hand-writing legibility; C: Between scores made when using standard and specialty furniture; I: The furniture configurations imposed. 2. True experiment, as treatments were imposed (experiment), and the researchers allocated children to the groups (true experiment). 3. Unit of observation: Each students. Unit of analysis: Each student, as the two measurements from each student are related to the same person. The students are compared. So this is a paired analysis. A paired-$$t$$ test. 4. Blinding not used, as students would know the desk configurations were different. No blinding, so double blinding is not used. However: The assessors are blinded to the intervention, as assessor unaware of furniture configuration, so assessor does not influence outcomes (unintentionally). 5. Sample mean: $$30.7$$; $$s=3.3$$. 6. Standard errors $$\text{s.e.}(\bar{x})=3.3\div\sqrt{30} = 0.60249$$; the approximate CI is $$30.7\pm(2\times0.60249)$$, or $$30.7\pm 1.2$$; from $$29.5$$ to $$31.9$$. 7. The sample provides no evidence (paired $$t$$: $$-0.30$$; $$\text{df}=29$$; two-tailed $$P=0.77$$) of a population mean difference between the handwriting legibility when performed using standard ($$30.7$$; standard deviation: $$3.3$$) and speciality furniture configurations ($$30.6$$; $$3.3$$; 95% CI for the difference from $$-0.8$$ to $$0.6$$). 8. No.