Question 7
The mass of a particular variety of cake is claimed to be normally distributed with a mean of 660 grams. A random sample of five of these cakes is found to have a mean mass of 600 grams.
Which option represents an approximate confidence interval for $\mu$ based on this sample?
Not quite. This interval runs from about $540$ up to $600$, so the sample mean is its right-hand end. A confidence interval is $\bar x\pm E$, which puts $600$ in the middle. In 2021, 0.6% of Queensland students chose (A).
That’s it. Step through the hints below to see why. In 2021, 67.4% of Queensland students chose (B).
Not quite. This interval joins the two numbers in the question, $600$ and $660$. The interval comes from the sample alone, $\bar x\pm E$, so it is centred on $600$. The claimed $660$ is what you compare with the interval afterwards. In 2021, 12.6% of Queensland students chose (C).
Not quite. This interval is centred on $660$, the claimed mean. A confidence interval estimates $\mu$ from the sample, so it is centred on the sample mean, $\bar x=600$. In 2021, 19.3% of Queensland students chose (D).
The question doesn’t give the population standard deviation, so try a few values. The orange bar is the interval and the black dot is its centre.
The interval is centred on the sample mean, $600$. The spread and the confidence level change its width, and its centre stays put. Of the four options, only (B) is centred on $600$.
This is centred on the claimed mean, like option (D). The interval estimates $\mu$ from the sample, so its centre has to be the sample mean, $\bar x=600$.
$660$ is outside this interval, so the sample casts doubt on the claim.
With this much spread, $660$ is inside the interval, so this sample alone would not rule out the claim.
A confidence interval for $\mu$ is built from the sample. It is the sample mean plus or minus a margin of error.
$$\bar x\pm z\,\frac{\sigma}{\sqrt n}$$
Here $\bar x=600$, so every confidence interval from this sample is centred on $600$, whatever the spread or the confidence level. The claimed mean $660$ plays no part in building it.
All four options are about $60$ grams wide, so the width can’t separate them. Look for the one with $600$ in the middle. (A) ends at $600$, (C) starts at $600$, and (D) is centred on $660$.
A confidence interval for $\mu$ is $\bar x\pm z\,\dfrac{\sigma}{\sqrt n}$, so it is centred on the sample mean, $\bar x=600$. Only (B) has $600$ in the middle.
The claimed mean $660$ sits outside that interval, so this sample casts doubt on the claim.
How Queensland students answered in 2021
From the QCAA Specialist Mathematics subject report for 2021. 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 2021 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2021, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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