Question 4
The distribution for a sample proportion $\hat{p}$ has a mean of 0.15 and a standard deviation of 0.0345.
The sample size is
Not quite. 10 is $\dfrac{\sqrt{0.15\times0.85}}{0.0345}\approx10.3$. The sample size sits under the square root, so square both sides before you rearrange. In 2022, 7.4% of Queensland students chose (A).
Not quite. With $n=14$, the standard deviation is $\sqrt{0.1275\div14}\approx0.0954$, nearly three times 0.0345. Check that you squared the standard deviation and used $p(1-p)$. In 2022, 8.9% of Queensland students chose (B).
That’s it. Step through the hints below to see why. In 2022, 79.4% of Queensland students chose (C).
Not quite. With $n=116$, the standard deviation is $\sqrt{0.1275\div116}\approx0.0332$, a little less than 0.0345. Solving $\dfrac{0.1275}{n}=0.0345^2$ gives $n\approx107.1$. In 2022, 3.5% of Queensland students chose (D).
Too spread out. The standard deviation is , more than 0.0345. A bigger sample makes $\hat p$ vary less, so increase $n$.
Too narrow. The standard deviation is , less than 0.0345, so the sample is bigger than it needs to be. Decrease $n$.
Very close, but not equal. The standard deviation here is , not 0.0345.
This is option (A). The standard deviation is , more than three times 0.0345, so the curve is far too spread out. 10 is what you get if you don’t square: $\sqrt{0.1275}\div0.0345\approx10.3$.
This is option (). The standard deviation is , not 0.0345, so the bracket misses the blue lines.
This is it. With $n=107$ the standard deviation rounds to 0.0345, so the bracket meets the blue lines.
The mean of the sample proportion is the population proportion, so $p=0.15$. The standard deviation has its own formula, with $n$ inside it.
$$\sigma_{\hat p}=\sqrt{\frac{p(1-p)}{n}}$$
Substitute $p=0.15$ and the standard deviation, then square both sides.
$$0.0345=\sqrt{\frac{0.15\times0.85}{n}}$$
$$0.00119025=\frac{0.1275}{n}$$
Rearrange for $n$. A sample size is a whole number, so round.
$\displaystyle n=\frac{0.1275}{0.00119025}$$\displaystyle \approx107.1\approx107$
Check: $\sqrt{0.1275\div107}\approx0.0345$. You can also solve the first equation directly on your calculator.
$$\sqrt{\frac{0.15\times0.85}{n}}=0.0345$$
$$n=\frac{0.1275}{0.0345^2}\approx107$$
The mean, 0.15, gives $p$, and the standard deviation formula gives $n$.
How Queensland students answered in 2022
From the QCAA Mathematical Methods subject report for 2022. 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 2022 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2022, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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