## E.4 Answers to Lecture 4 tutorial

### E.4.1 Answers to Sect. 4.1

1. The smallest jellyfish has a breadth of (approx.) 6mm. A median of 4mm is silly!

With 22 observations, the median is halfway between the 11th and 12th ordered observations (observation number $$(22+1)/2=11.5$$): both be in the second bar, so the median is somewhere between 8 and 10mm (we cannot be sure based on the histogram, as information is lost). If they understand the histogram, they should also see that the smallest breadth is about 6mm, so a median of 4mm makes no sense.

If you have the time: Ask how to explain why they are wrong, and ask approximately what the median would be**.

2. A typical Dangar Island jellyfish has a breadth of about 10mm, but the variation is from about 6 to 16mm. The data are slightly skewed to the right (most jellyfish have smaller breadths, but some have larger breadths), but the shape is a bit funny (more data would smooth it out).

3. Site A is Dangar Island.

Jellyfish at Salamander Bay generally have a larger breadth: The median breadth at Salamander Bay (about 16mm) is greater than almost all the jellyfish at Dangar Island.

Jellyfish at Salamander Bay are a little less variable in terms of breadth. The distributions look slightly skewed right at both sites.

### E.4.2 Answers to Sect. 4.2

You may like to suggest that students search on YouTube for some tutorials on using their calculator’s Statistics Mode.

Students need to know how to use their calculator’s statistics mode, including for the exam. The Course Outline indicates that a calculator is needed. Help students as much as you can, but you cannot be expected to know how to work every type of calculator that is out there. You can perhaps direct students to work with other students having similar calculators.
1. Descriptive RQ.

2. $$\bar{x}=102.62$$MPa; $$s=5.356$$MPa.

If they use the wrong std dev button, they will get $$\sigma=4.79$$MPa in error. This is one important outcome of this question: That students know what button to press on their calculator to get the sample standard deviation.

3. The median is $$101.1$$MPa. The range is from $$97.6$$ to $$111.2$$, or $$13.6$$MPa.

4. Report none based on five values! Too few data points!

### E.4.3 Answers to Sect. 4.3

You can have groups answer each part separately, then share their explanations with the class.

You may wish to quiz the students about means and IQRs too.

For your info only (note the means and medians are very similar), see Table E.1.
TABLE E.1: Some statistics from the graphs
A B C D
Std devs 7.04 5.90 5.10 2.84
IQRs 12.10 7.96 5.82 4.71
Medians 27.13 25.19 20.07 45.03
Means 27.15 25.26 21.49 44.95
1. The largest median is for D.
2. The smallest median is for C.
3. Largest standard deviation is for A: the values are far more diverse.
4. Smallest standard deviation is for D, as the observations all have very similar values (all are about 22) and none are very far from about 22.

### E.4.4 Answers to Sect. 4.4

1. A few issues…
• Five decimal places is to the nearest 0.01 of a mm!
• The standard deviation of the difference is not the difference between the individual standard deviations.
• A standard deviation cannot be negative. (Same applies to standard errors, but we aren’t there yet.)
• Note that there is a sample size of 0 for the difference!
2. A few issues…
• Five decimal places: That’s accuracy to 0.00001 of a millimetre per second. I don’t think so…
• There is no numerical measures of the most important thing and the thing the RQ (presumably) concerns:
• The differences between the two brands.

### E.4.5 Answers to Sect. 4.5

1. Five variables. (‘Participants’ would not be summarised, it is technically an identifier and not a variable, as each person has a unique value).
2. Age; Height; Weight.
3. Gender (nominal; two levels); GMFCS (ordinal; three levels)
4. As follows:
• ‘Gender’: Percentages (or number) F and M
• ‘Age’: Mean/median; standard deviation/IQR
• ‘Height’: Mean/median; standard deviation/IQR
• ‘Weight’: Mean/median; standard deviation/IQR
• ‘GMFCS’: Percentages (or numbers) in each group
5. As follows:
• ‘Gender’: Barchart (not really needed)
• ‘Age’: Histogram/stemplot
• ‘Height’: Histogram/stemplot
• ‘Weight’: Histogram/stemplot
• ‘GMFCS’: Barchart/piechart
6. As follows:
• Between Gender and Height: Boxplot
• Between Gender and GMFCS: Side-by-side or stacked bar chart.