## A.3 Answer: TW 3 tutorial

### Answers for Sect. 3.2

- Some of the significant figures look ridiculous.
Mean: \(990.0791\), or \(990.1\) tonnes.
Standard deviation: \(1588.514579\), or \(1588.5\) tonnes.
(Specifically,
**not**\(1485.919327\).) - Median: \(180.5\) tonnes.
- Splitting the data into two parts of four observations each: \(Q_1\) is half-way between \(8.0001\) and \(29.4\) tonnes: \(Q_1 = 18.7005\). \(Q_3\) is half-way between \(676.2\) and \(2547.3\) tonnes: \(Q_1 = 1611.75\). The IQR is \(1611.75 - 18.7005 = 1593.05\), or about \(1593\) tonnes.

### Answers for Sect. 3.3

Answers implied by H5P.
Median largest: **Class D**.
Median smallest: **Class C**.
Standard deviation largest: **Class A**.
Standard deviation smallest: **Class D**.

### Answers for Sect. 3.4

Answers implied by H5P.

- Volume of drinks in \(375\) ml:
**Graph D**. (Students sometimes say Graph A, thinking that most cans have very similar amounts.) - Time in exam for
**short**or**easy**exam:**Graph C**. - Time in exam for
**long**or**hard**exam:**Graph B**. - The heights of females UniSC students:
**Graph D**. - The
*starting*salaries:**Graph C**. (Students sometimes say Graph B, thinking that salaries rise over time.)

### Answers for Sect. 3.6

- Female Weddell: about \(260\)--\(270\) cm long, vary from about \(200\) to \(310\) cm. Slight negative skewness; possible small outlier at about \(170\)--\(180\) cm.
- Dotchart. (Too much data for a useful stem-and-leaf plot probably.)
- Better with a title; some more labelled tick marks would be better.

### Answers for Sect. 3.7

- Mean: \(168.5714\), or \(168.57\) m.
Standard deviation: \(6.966279\), or \(6.97\) m.
(Specifically,
**not**\(6.44952\).) The median: \(167.9\) m. - \(\bar{x} = 102.62\) MPa; \(s = 5.356\) MPa. The median is \(101.1\) MPa.
- Mean: \(13.88778\), or \(13.888\)%. Standard deviation: \(2.617402\), or \(2.617\)%. Median: \(13.35\)%.

### Answers for Sect. 3.8.1

- Observational: the 'treatment' (brand of battery) is not assigned; we simply take measurements from the batteries that exist.
- Units of observation and units of analysis: The individual batteries.
*Energizer*: mean: \(7.36\) hours; std dev: \(0.289\) hours.*Ultracell*: mean: \(7.41\) hours; std dev: \(0.172\) hours. So Ultracell batteries are*slightly*better (last longer) on average, and more consistent performers.*Energizer*: median: \(7.46\) hours.*Ultracell*: median: \(7.48\) hours. So Ultracell batteries are*slightly*better (last longer) on average.- Probably median (outliers?), but mean and median are similar in any case.
- 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 \(0.02\) hrs, or \(1.2\) mins!)**The***practical*difference is negligible. - Quite possibly, some very low values for both brands.
- Yes: at the time of the study, the Ultracell batteries were substantially cheaper, and hence much better value.

### Answers for Sect. 3.8.2

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.

### Answer for Sect. 3.8.3

**Centre**: About \(11\).4 inches;
**Variation**: Most are between \(11.0\) and \(12.3\) inches;
**Shape**: Slightly skewed right (apart from that outlier...);
A small outlier (near \(10.5\) inches)