Question 13
The wait time for customers put on hold when calling complaint departments is assumed to be normally distributed. A company claims that the mean wait time for their customers is 7.6 minutes.
The following data represents the wait time (minutes) from a random sample of 12 customers who called the complaint department of this company.
Determine the mean of this data.
[1 mark]The standard deviation of this data is calculated to be 2.384 minutes.
Use an approximate 95% confidence interval for the mean to evaluate the reasonableness of the company’s claim. Justify your decision using mathematical reasoning.
[3 marks]Build the interval from the sample, then test the claim
The interval is centred on the sample mean, not on the claim. Watch where 7.6 minutes ends up.
The claim is 0.28 minutes below the lower end of the interval. A mean as low as 7.6 doesn’t fit this sample.
Test other claims against the same interval
The interval comes from the sample, so it stays put when the claim changes. Move the claimed mean and see which claims this sample supports. You can also switch to $z\approx2$, which the QCAA accepted.
Work through the solution one mark at a time
Try each step yourself before you reveal it. The steps follow the QCAA sample response.
Enter the data into a list on your GDC and use one-variable statistics. By hand, the twelve values add to $110.8$.
$$\bar{x}=\frac{110.8}{12}\approx9.23$$
The QCAA also accept $9.2$.
Marker: correctly determines the mean of the data
Write down what you have. The standard deviation comes from the sample, so it is $s$, and you use it in place of $\sigma$. For 95% confidence, $z=1.96$.
$$n=12,\quad\bar{x}\approx9.23,\quad s=2.384,\quad z=1.96$$
$$\left(\bar{x}-z\frac{s}{\sqrt{n}},\ \bar{x}+z\frac{s}{\sqrt{n}}\right)=\left(9.23-1.96\times\frac{2.384}{\sqrt{12}},\ 9.23+1.96\times\frac{2.384}{\sqrt{12}}\right)$$
Your GDC’s one-sample $z$ interval gives this directly, with $2.384$ entered as the standard deviation. Using $z\approx2$ gives $(7.9,\ 10.6)$, which is also accepted.
Marker: calculates 95% confidence interval
Make a decision about the claim.
Saying the claim is unlikely, should be rejected, or gives reason to be suspicious is also accepted.
Marker: states an appropriate evaluation of the reasonableness of the claim
Justify the decision by saying where the claim falls against the interval.
The claimed population mean of 7.6 minutes lies outside the confidence interval, because $7.6\lt7.88$.
If you made a slip in part a) or your interval, you can still earn this mark when your reasoning follows from your own interval.
Marker: uses mathematical reasoning to justify the decision
Each phrase of the question gave you something
Finding a mean, building a 95% confidence interval and judging a claim against it is a standard Paper 2 sequence. The standard deviation is given, so the GDC does the rest.
If I had to guess where you might lose marks, it would be centring the interval on the claim of 7.6 instead of the sample mean. The other is writing “not reasonable” without saying that 7.6 is below the lower end of the interval.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording and marking-guide steps are from the 2023 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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