Challenging +1.2 This question requires working backwards from a confidence interval to find summary statistics, involving manipulation of the t-distribution formula and solving simultaneous equations. While the concepts (confidence intervals, t-distribution) are standard Further Maths content, the reverse-engineering aspect and algebraic manipulation of two equations in two unknowns elevates it slightly above routine application, but it remains a structured problem with clear methodology.
2 Shane is studying the lengths of the tails of male red kangaroos. He takes a random sample of 14 male red kangaroos and measures the length of the tail, \(x \mathrm {~m}\), for each kangaroo. He then calculates a \(90 \%\) confidence interval for the population mean tail length, \(\mu \mathrm { m }\), of male red kangaroos. He assumes that the tail lengths are normally distributed and finds that \(1.11 \leqslant \mu \leqslant 1.14\).
Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) for this sample.
2 Shane is studying the lengths of the tails of male red kangaroos. He takes a random sample of 14 male red kangaroos and measures the length of the tail, $x \mathrm {~m}$, for each kangaroo. He then calculates a $90 \%$ confidence interval for the population mean tail length, $\mu \mathrm { m }$, of male red kangaroos. He assumes that the tail lengths are normally distributed and finds that $1.11 \leqslant \mu \leqslant 1.14$.
Find the values of $\sum x$ and $\sum x ^ { 2 }$ for this sample.\\
\hfill \mbox{\textit{CAIE Further Paper 4 2023 Q2 [6]}}