Question 8
$T$ is a random variable. A random sample of four values of $T$ is collected and used to produce an approximate confidence interval for the population mean of $(3.3,\ 4.1)$.
Given that three of the sample values are 3.4, 3.6 and 3.9, the remaining sample value is
Not quite. With 3.6 the sample mean is 3.625, but the interval is centred on 3.7. In 2024, 4.1% of Queensland students chose (A).
Not quite. 3.7 is the centre of the interval, which is the sample mean. The missing value has to bring the mean of all four to 3.7. In 2024, 17.1% of Queensland students chose (B).
Not quite. With 3.8 the sample mean is 3.675, just short of the centre of the interval, 3.7. In 2024, 8.9% of Queensland students chose (C).
That’s it. Step through the hints below to see why. In 2024, 69.3% of Queensland students chose (D).
A confidence interval is centred on the sample mean. Find the centre of the interval.
$$\bar x=\frac{3.3+4.1}{2}=3.7$$
Four values with a mean of 3.7 must add to $4\times3.7$.
$$3.4+3.6+3.9+t=14.8$$
Solve for the missing value, then move the slider on the left to check that the dashed mean line lands on the centre.
$$t=14.8-10.9$$
$$\bar x=\frac{3.3+4.1}{2}=3.7$$
$$3.4+3.6+3.9+t=4\times3.7=14.8\ \Rightarrow\ t=3.9$$
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.
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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.
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