Question 10
A random variable is drawn from a population with the distribution shown in the histogram.
A number of samples of size 10 were randomly selected from this distribution and the sample means, $\bar{x}$, were recorded. The histogram that most likely represents the distribution of the sample means is
Not quite. This has the same shape and spread as the population. It is what one large sample of single values looks like, not the means of samples of $10$. In 2023, 33.5% of Queensland students chose (A).
That’s it. Step through the hints below to see why. In 2023, 37.6% of Queensland students chose (B).
Not quite. It is narrow and roughly bell-shaped, but it is centred near $11$. Sample means are centred on the population mean, and most of this population is below $8$. In 2023, 18.7% of Queensland students chose (C).
Not quite. It is centred near $11$ and spread almost as widely as the population. Sample means are centred on the population mean and spread less. In 2023, 9.8% of Queensland students chose (D).
Each bar counts sample means from 500 samples drawn from the population above. The blue line is the population mean.
With samples of size $1$, each sample mean is just one value, so the histogram looks like the population, as in option (A).
With samples of size $10$, the means pile up near $\mu$ and spread much less than the population. Of the four options, only (B) looks like this.
The centre stays near $\mu$ for every sample size. The spread shrinks as the sample size grows, and the shape gets closer to a normal curve.
The sample means are centred on the population mean, $\mu$. Most of this population is below $8$, so $\mu$ is small, around $4$ or $5$. That rules out (C) and (D), which are centred near $11$.
The standard deviation of the sample means is $\frac{\sigma}{\sqrt{n}}$. With $n=10$, that is about a third of $\sigma$, so the sample means are much less spread out than the population.
$$\frac{\sigma}{\sqrt{10}}\approx0.32\sigma$$
(A) has the same spread and shape as the population, so it can’t be the sample means. (B) is narrow and sits at the low end of the axis.
The distribution of $\bar{x}$ is centred on $\mu$, which is small for this population, and its standard deviation is $\frac{\sigma}{\sqrt{10}}$, much smaller than $\sigma$.
Only (B) is narrow and sits at the low end of the axis, near the population mean.
How Queensland students answered in 2023
From the QCAA Specialist Mathematics 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 Specialist Mathematics 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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