Question 1
Repeated random samples will be used to calculate a large number of 90% confidence intervals for a population mean $\mu$.
Which statement best describes the possible outcomes?
That’s it. Step through the hints below to see why. In 2024, 72.7% of Queensland students chose (A).
Not quite. Each interval either contains $\mu$ or misses it, and about 10% of them miss. Nothing pushes the count above 90%. In 2024, 4.3% of Queensland students chose (B).
Not quite. The method is built so that about 90% of the intervals catch $\mu$. Nothing pushes the count below that in the long run. In 2024, 2.8% of Queensland students chose (C).
Not quite. The samples are random, so the number of intervals that contain $\mu$ changes from one batch to the next. Draw a few batches below and watch the count move. In 2024, 19.7% of Queensland students chose (D).
A 90% confidence interval comes from a method that catches $\mu$ for about 90% of all possible samples. The 90% describes the method over many samples. It doesn’t describe any single interval.
Each sample is random, so each batch of intervals is random too. Out of a large number of intervals you can expect close to 90% to contain $\mu$, but you can’t say exactly 90%.
Draw a few batches of 40. The count for one batch jumps around, sometimes above 36 and sometimes below. The running total over all your draws settles close to 90%.
In the long run, 90% of intervals built this way contain $\mu$. With a large but finite number of random samples, the proportion will be close to 90%, but it is unlikely to be exactly 90%.
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.
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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