Moderate -0.3 This is a straightforward application of confidence interval formulas requiring students to work backwards from the interval to find the sample mean (midpoint) and use the t-distribution critical value with the interval width to find the sample variance. While it requires understanding of the relationship between confidence intervals and their parameters, it's a standard textbook exercise with clear mechanical steps and no conceptual surprises.
A normally distributed variable, \(X\), has unknown mean \(\mu\) and unknown standard deviation \(\sigma\).
A sample of 10 values of \(X\) was taken. From these 10 values, a 95% confidence interval for \(\mu\) was calculated to be
$$(30.47, 32.93)$$
Use this confidence interval to find unbiased estimates for \(\mu\) and \(\sigma^2\).
[4 marks]
A normally distributed variable, $X$, has unknown mean $\mu$ and unknown standard deviation $\sigma$.
A sample of 10 values of $X$ was taken. From these 10 values, a 95% confidence interval for $\mu$ was calculated to be
$$(30.47, 32.93)$$
Use this confidence interval to find unbiased estimates for $\mu$ and $\sigma^2$.
[4 marks]
\hfill \mbox{\textit{AQA S2 2016 Q2 [4]}}