Question 3
A streaming service surveyed a random sample of 100 customers, and 60 customers said that they prefer to watch movies rather than miniseries.
Based on these results, the 95% confidence interval for the proportion of customers who prefer to watch movies was $(0.5,\ 0.7)$.
Which statement shows the correct interpretation of this outcome?
That’s it. Step through the hints below to see why. In 2025, 85.6% of Queensland students chose (A).
Not quite. The interval estimates a proportion of customers. It isn’t a probability, and the survey never asked how often anyone watches movies, so “95% of the time” has no meaning here. In 2025, 6.8% of Queensland students chose (B).
Not quite. 60% is the sample proportion, but $(0.5,\ 0.7)$ isn’t about how often people watch movies. It estimates the proportion of all customers who prefer movies. In 2025, 5.6% of Queensland students chose (C).
Not quite. 60 customers in the sample said they prefer movies, but 95% is the confidence level of the method. It says nothing about how often those 60 people watch movies. In 2025, 1.8% of Queensland students chose (D).
At 95%, about 19 in every 20 intervals built this way contain $p$. That long-run success rate is what “95% confident” means. The survey’s interval is one of those intervals, and it either contains $p$ or it doesn’t.
At % the intervals are than at 95%, and in the long run about in 100 of them miss $p$. The confidence level is the method’s long-run success rate.
The sample gives $\hat p=\dfrac{60}{100}=0.6$. You already know that number exactly, so there’s nothing to estimate about the sample.
The interval $(0.5,\ 0.7)$ estimates $p$, the proportion of all the service’s customers who prefer movies.
The 95% describes the method. If you took many random samples and built an interval from each, about 95% of those intervals would contain $p$.
That’s why you say you are “95% confident” that $p$ is between 0.5 and 0.7, that is, between 50% and 70%.
Check the interval comes from this sample. With $z=1.96$ for 95%:
$\displaystyle 0.6\pm1.96\sqrt{\frac{0.6\times0.4}{100}}$$\displaystyle =0.6\pm0.096$$\displaystyle \approx(0.5,\ 0.7)$
Options (B), (C) and (D) all talk about how often people watch movies. The survey only asked which they prefer.
The interval $(0.5,\ 0.7)$ estimates the proportion of all customers who prefer movies. The 95% is how often intervals built this way contain that true proportion. So the streaming service can be 95% confident that the proportion of customers who prefer to watch movies is between 50% and 70%.
How Queensland students answered in 2025
From the QCAA Mathematical Methods subject report for 2025. 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 2025 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2025, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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