Question 7
The distribution of a certain sample proportion has a mean of 0.70 and a standard deviation of 0.02.
Determine the sample size.
That’s it. Step through the hints below to see why. In 2023, 77.4% of Queensland students chose (A).
Not quite. 750 is $\dfrac{0.3}{0.02^2}$, which uses only $1-p$. The formula needs the product $p(1-p)=0.7\times0.3=0.21$. In 2023, 11.5% of Queensland students chose (B).
Not quite. 1750 is $\dfrac{0.7}{0.02^2}$, which uses only $p$. Multiply by $1-p=0.3$ as well: $p(1-p)=0.21$. In 2023, 7.2% of Queensland students chose (C).
Not quite. 2500 is $\dfrac{1}{0.02^2}$. It leaves out $p(1-p)=0.21$ altogether. In 2023, 3.2% of Queensland students chose (D).
Too spread out. The standard deviation is , more than 0.02. A bigger sample makes $\hat p$ vary less, so increase $n$.
Too narrow. The standard deviation is , less than 0.02, so the sample is bigger than it needs to be. Decrease $n$.
Very close, but not equal. The standard deviation here is , not 0.02.
This is option (). The standard deviation is only , so the curve is too narrow. That sample size comes from leaving some or all of $p(1-p)$ out.
This is it. With $n=525$ the standard deviation is exactly 0.02, so the bracket meets the blue lines.
The mean of the sample proportion is the population proportion, so $p=0.70$. The standard deviation has its own formula.
$$\sigma_{\hat p}=\sqrt{\frac{p(1-p)}{n}}$$
Substitute $p=0.7$ and the standard deviation, then square both sides.
$$0.02=\sqrt{\frac{0.7\times0.3}{n}}$$
$$0.0004=\frac{0.21}{n}$$
Rearrange for $n$.
$$n=\frac{0.21}{0.0004}=525$$
Check: $\sqrt{0.21\div525}=0.02$. You can also solve the first equation directly on your calculator.
$$\sqrt{\frac{0.7\times0.3}{n}}=0.02$$
$$\frac{0.21}{n}=0.0004$$
$$n=\frac{0.21}{0.0004}=525$$
The mean, 0.70, gives $p$, and the standard deviation formula gives $n$.
How Queensland students answered in 2023
From the QCAA Mathematical Methods subject report for 2023. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2023 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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