Question 10
Two researchers each select a random sample of size 50 from the same population, which has an unknown mean $\mu$. Each researcher uses their own sample mean and sample standard deviation to calculate an approximate 95% confidence interval for $\mu$.
Which statement is true?
Not quite. You treated the interval as fixed by the population. It is calculated from the sample, and two random samples almost never give the same $\bar{x}$ and $s$, so the intervals differ.
Not quite. You read 95% confidence as a guarantee. About 5% of intervals made this way miss $\mu$, so either interval could be one of them.
Not quite. You assumed the width depends only on $n$ and the confidence level. The width is $2\times 1.96\times\frac{s}{\sqrt{50}}$, and each researcher uses their own $s$, which varies between samples.
That’s it. Step through the hints below to see why.
This is a simulation, not data from the question. Each round, two new random samples of size 50 are drawn from the same population: researcher 1’s interval (black) above researcher 2’s (blue), with the newest pair at the top. The dot is each sample mean $\bar{x}$. An interval that misses $\mu$ turns orange. In real life $\mu$ is unknown, so the researchers never see this line.
Simulated: the newest pair
(A) identical: no, different centres and widths
(B) both contain :
(C) same width: no, each uses its own
(D) each may or may not contain : true
Both intervals contain $\mu$ this round, but they have different centres and different widths.
This round, researcher 1’s interval misses $\mu$. Neither researcher could tell, because $\mu$ is unknown.
This round, researcher 2’s interval misses $\mu$. Neither researcher could tell, because $\mu$ is unknown.
This round, both intervals miss $\mu$. Neither researcher could tell, because $\mu$ is unknown.
Write down how each interval is built. An approximate 95% confidence interval for $\mu$ from a sample of size $n$ is
$$\left(\bar{x}-1.96\frac{s}{\sqrt{n}},\ \bar{x}+1.96\frac{s}{\sqrt{n}}\right)$$
where $\bar{x}$ is the sample mean and $s$ is the sample standard deviation. Here $n=50$ for both researchers.
Every statement is about the intervals, so start from the formula and see which parts change from sample to sample.
Identify what varies between the two samples. $\bar{x}$ varies (sampling variation of the mean), and $s$ varies too. Only $n=50$ and $1.96$ are the same for both.
The two researchers draw different random samples, so any quantity calculated from the sample will usually differ.
Interpret “95% confidence”. If the procedure were repeated on many random samples, about 95% of the intervals produced would contain $\mu$ and about 5% would not.
Any single interval either contains $\mu$ or it does not, and because $\mu$ is unknown, the researcher cannot tell which.
This is the key idea. The 95% describes the method, not any single interval.
Apply this to each researcher. Each researcher’s interval may contain $\mu$ (the likely case) or may miss it (possible). This is statement (D).
Each interval is one outcome of the procedure.
The 95% describes the method, not any single interval. Each researcher’s interval is one outcome of that method, so each one may contain $\mu$ or may miss it.
Only mark this done when you could do it without help. Reading the solution does not count.
A Tangent Tuition mock exam question. It is not a past QCAA question, and Tangent Tuition is not affiliated with the QCAA.
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