Educational worksheet on relative frequency from CensusAtSchool, demonstrating how to estimate probability using sample data and a line graph.

Let’s solve part (a) step by step.

We are given a table with sample sizes and the number of people who chose PE/sport. We need to calculate the relative frequency for each row using the formula:

> Relative frequency = number of successes ÷ sample size

Then, we’ll imagine plotting these on a line graph (like the one shown), and comment on what happens as the sample size increases.

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Step 1: Calculate relative frequencies

We already have some values filled in — let’s compute the missing ones.

- Sample size 5 → 1 chose PE → 1/5 = 0.2 ✔ (already given)
- Sample size 10 → 3 chose PE → 3/10 = 0.3
- Sample size 20 → 4 chose PE → 4/20 = 0.2
- Sample size 30 → 6 chose PE → 6/30 = 0.2
- Sample size 50 → 9 chose PE → 9/50 = 0.18
- Sample size 100 → 20 chose PE → 20/100 = 0.2
- Sample size 200 → 46 chose PE → 46/200 = 0.23
- Sample size 500 → 111 chose PE → 111/500 = 0.222
- Sample size 1000 → 213 chose PE → 213/1000 = 0.213

Let me double-check those calculations:

- 3 ÷ 10 = 0.3 ✔️
- 4 ÷ 20 = 0.2 ✔️
- 6 ÷ 30 = 0.2 ✔️
- 9 ÷ 50 = 0.18 ✔️
- 20 ÷ 100 = 0.2 ✔️
- 46 ÷ 200 = 0.23 ✔️
- 111 ÷ 500 = 0.222 ✔️
- 213 ÷ 1000 = 0.213 ✔️

All correct.

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Step 2: Think about the graph

The example graph shows “Relative Frequency of sport” vs “Sample size”, with sample size on a logarithmic scale (so 1, 10, 100, 1000 are equally spaced).

If we plot our calculated relative frequencies against sample size, we’d see:

- At small sample sizes (like 5, 10, 20), the relative frequency jumps around: 0.2, 0.3, 0.2
- As sample size gets bigger (100, 200, 500, 1000), it starts to settle near 0.21–0.23

This makes sense! The more data you collect (larger sample), the closer your estimate gets to the true probability. It doesn’t jump around as much.

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Step 3: Comment on results

As the sample size increases, the relative frequency becomes more stable and less likely to change wildly. This is because larger samples give better estimates of the real probability. In this case, it seems like the true probability of someone choosing PE/sport as their favorite subject is probably around 0.21 or 0.22, since that’s where the values are settling when we have 500 or 1000 people.

Also, notice how at sample size 1000, we got 0.213 — very close to 0.21. That’s a good sign our estimate is getting accurate.

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Final Answer:
The completed relative frequencies are:
- 10 → 0.3
- 20 → 0.2
- 30 → 0.2
- 50 → 0.18
- 100 → 0.2
- 200 → 0.23
- 500 → 0.222
- 1000 → 0.213

As sample size increases, the relative frequency stabilizes around 0.21–0.23, showing that larger samples give more reliable estimates of probability.