## A.4 Answers to Lecture 4 tutorial

### A.4.1 Answers to Sect. 4.6

**1.**Observational: The ‘treatment’ (brand of battery) is not assigned; we simply take measurements from the batteries that exist.

**2.**Units of observation and units of analysis: The individual batteries.

**3.**

*Energizer*: mean: \(7.36\) hours; std dev: \(0.289\).

*Ultracell*: mean: \(7.41\) hours; std dev: \(0.172\). So Ultracell batteries are

*slightly*better (last longer) on average, and more consistent performers.

**4.**

*Energizer*: median: 7.46 hours.

*Ultracell*: median: 7.48 hours. So Ultracell batteries are

*slightly*better (last longer) on average.

**5.**Probably median (outliers?), but mean and median are similar in any case.

**6.**Values are close. Certainly no practically importance difference. Energizer batteries take, on average,

*less*time to reach 1.0 volts, so are ‘worse’ in this regard. They also have a lot more variation. But in practical terms, the difference is minimal. (Based on means, the difference is 0.05 hrs, or 3 mins in over 7 hrs use! Based on medians, the difference is 0.02 hours, or 1.2 mins!)

**The**

*practical*difference is negligible.**7.**Quite possibly, some very low values for both brands.

**8.**Yes: At the time of the study, the Ultracell batteries were substantially cheaper, and hence much better value.

### A.4.2 Answers to Sect. 4.7

FAS here is about 20–25 in general, ranging from 10 to 50 (which are actually the smallest and largest possible scores).
Here’s is a detailed explanation.

**You are not expected to go to this detail!**The FAS is about 25 on average, ranging from about 10 to 50, slightly skewed right with no outliers. The distribution is unimodal. (The heights of the bars are, as best as I can figure, 3; 14; 44; 34; 22; 17; 5; 3.) With \(n=142\), the median is observation number 71.5, which is in the 25–30 bar. The quartiles will have about 35 in them, so \(Q_1\) will be in the 20–25 bar, and \(Q_3\) will be in the 30–35 bar. Histogram is pretty good; bars really should be touching.### Answers to Sect. 4.8

The median temperatures similar; a slight increase from Office A to C.
IQR similar for each office, and range similar for Offices A and C (slightly larger for Office B).

### A.4.3 Answers to Sect. 4.9

**1.**Stacked or side-by-side barcharts; whether bird injured or not, and whether upper or lower are both qualitative variables.

**2.**Boxplots would be OK; one quantitative (length-of-stay) and one qualitative (therapy type) variable.

**3.**Scatterplots; two quantitative variables (frequency; amount wheat consumed).

**4.**Two-way table or stacked/side-by-side barchart, or table; two qualitative variables vars (30 mins of physical activity or not; vigorous physical activity or not).