Question 4
The mass of biscuit packets produced by a company is normally distributed with a mean of 250 g and a standard deviation of 1.5 g. The distribution of the sample mean mass of these biscuit packets is formed using repeated random sampling of size 5.
The mean and standard deviation of this distribution of sample means are
Not quite. The mean doesn’t get divided by the sample size, and the standard deviation is divided by $\sqrt5$, not 5. In 2024, 2.8% of Queensland students chose (A).
Not quite. The standard deviation is right, but the mean of the sample means is the population mean, 250 g. In 2024, 4.7% of Queensland students chose (B).
Not quite. The mean is right, but the standard deviation is divided by $\sqrt5$, not 5. In 2024, 3.2% of Queensland students chose (C).
That’s it. Step through the hints below to see why. In 2024, 89.1% of Queensland students chose (D).
Sample means are centred on the population mean. Averaging doesn’t move the centre, so the mean of the sample means is 250 g.
$$\mu_{\bar X}=\mu=250\text{ g}$$
Averaging does reduce the spread. The standard deviation of the sample means is the population standard deviation divided by the square root of the sample size.
$$\sigma_{\bar X}=\frac{\sigma}{\sqrt n}$$
Substitute $\sigma=1.5$ and $n=5$.
$$\sigma_{\bar X}=\frac{1.5}{\sqrt5}\approx0.67\text{ g}$$
Two of the wrong options divide the mean by 5, and two divide the standard deviation by 5 instead of $\sqrt5$.
$$\mu_{\bar X}=\mu=250\text{ g}$$
$$\sigma_{\bar X}=\frac{1.5}{\sqrt5}\approx0.67\text{ g}$$
How Queensland students answered in 2024
From the QCAA Specialist Mathematics subject report for 2024. The percentages can add to less than 100 because some students didn’t answer.
Prefer to watch? Watch me work through it on YouTube (opens in a new tab)
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2024 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2024, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
Spot a mistake on this page? Report it