Question 14
At a particular game at a local sporting venue, 60% of spectators support the home team and the remainder support the away team.
A researcher asked six groups of 10 spectators which team they supported. Each spectator was recorded as either H (supports home team) or A (supports away team). The results were:
The researcher would like to see the distribution of the sample proportions of home team supporters obtained by entering the information into a column graph. The sample proportion for group 1 is shown.
Complete the column graph by including the sample proportions for the remaining five groups.
If the researcher had interviewed more groups, each containing 100 spectators, describe two ways the distribution of the resulting sample proportions would be expected to differ from the distribution shown in Question 14a).
[2 marks]Each group becomes one block on the graph
Watch six groups turn into six sample proportions, then six blocks. Then see what groups of 100 would look like instead.
A column’s height counts groups, so 0.2 and 0.5 are two blocks high. Bigger groups pull every sample proportion in towards 0.6.
More groups, or bigger groups?
Part b) changes two things at once: more groups, and 100 spectators in each. Change one at a time and watch which one makes the graph narrower. The groups are simulated from a population where 60% support the home team.
This is the completed graph from part a). Six groups of 10 spread from 0.2 to 0.9, nothing landed on 0.6, and the shape is not symmetrical about 0.6.
A new six groups of 10 give a different graph. Six blocks are too few to show a shape, and they can land a long way from 0.6.
More groups fill in the shape, but they don’t make it narrower. Each sample proportion still comes from only 10 people, so the SD stays around 0.15. Now try 100 spectators.
Groups of 100 land much closer to 0.6. Six blocks still can’t show a shape, so try more groups.
This is part b). The graph is roughly symmetrical about 0.6, and much narrower. The SD is near 0.05 instead of 0.15, so there are fewer very high or very low sample proportions.
With 100 spectators the proportions go up in steps of 0.01, so the columns are thinner.
Work through the solution one mark at a time
Try each step yourself before you reveal it. Part a) gives a mark for each new column, and part b) a mark for each way the graph changes.
Each sample proportion is the number of H in the group, out of 10. Group 1 has 8 H, which is why its column is at 0.8. Groups 2 and 6 each have 2 H.
$$\begin{aligned}\text{Group 2: }\hat{p}&=\frac{2}{10}=0.2\\[6pt]\text{Group 6: }\hat{p}&=\frac{2}{10}=0.2\end{aligned}$$
Two groups give 0.2, so draw the 0.2 column up to a frequency of 2.
Marker: correctly sketches the sample proportion 0.2
Groups 3 and 5 each have 5 H and 5 A.
$$\begin{aligned}\text{Group 3: }\hat{p}&=\frac{5}{10}=0.5\\[6pt]\text{Group 5: }\hat{p}&=\frac{5}{10}=0.5\end{aligned}$$
Two groups again, so the 0.5 column also has a frequency of 2.
Marker: correctly sketches the sample proportion 0.5
Group 4 has 9 H and only one A.
$$\text{Group 4: }\hat{p}=\frac{9}{10}=0.9$$
One group, so the 0.9 column has a frequency of 1. That accounts for all six groups.
For each mark, the column needs the correct frequency, not just the right place. Count H, not A. Counting A gives the five values 0.8, 0.5, 0.1, 0.5 and 0.8 instead.
Marker: correctly sketches the sample proportion 0.9
For large samples the distribution of $\hat{p}$ is approximately normal, centred on the population proportion $p=0.6$. So the first change is the shape.
One way: the graph would be more symmetrical about 0.6. The graph in part a) is not symmetrical about 0.6.
Other ways of saying the graph is closer to normal are accepted too. For example, the sample proportions would be centred on (or cluster around) the mean of 0.6, 0.6 would be the expected mode, or the graph would look like a bell-shaped curve.
Marker: correctly identifies one change to the distribution of $\hat{p}$ as it more closely approximates normality due to the larger samples – a more symmetrical graph
The second change is the spread. The standard deviation of $\hat{p}$ gets smaller as the sample size grows.
Groups of 10:
$$\sigma_{\hat{p}}=\sqrt{\frac{0.6(0.4)}{10}}\approx0.155$$
Groups of 100:
$$\sigma_{\hat{p}}=\sqrt{\frac{0.6(0.4)}{100}}\approx0.049$$
You don’t need these numbers for the mark. They just show why the spread shrinks.
2. There would be less variability about 0.6, so the graph would look narrower, with fewer very high or very low sample proportions.
Marker: correctly identifies a second change to the distribution of $\hat{p}$ as it more closely approximates normality due to the larger samples – there would be less variability and the appearance of the graph would look narrower or ‘skinnier’
Each phrase of the question gave you something
Part a) is counting and drawing, with the first column done for you as a model. Part b) is the standard result about sample proportions: as the sample size grows, the distribution of $\hat{p}$ gets closer to normal, centred on $p$, with a smaller standard deviation.
If I had to guess where you might lose marks, it would be part b). “Narrower” and “less spread out” are the same change, so writing both only earns one mark. Give one point about shape and one about spread. Don’t just say the results would be “more accurate” either, because that doesn’t describe the distribution. In part a), draw the 0.2 and 0.5 columns two high, not as two separate blocks of height 1 somewhere else.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording and marking-guide steps are from the 2024 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2024, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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