Challenging +1.8 This question requires understanding of confidence intervals for normal distributions, working backwards from an interval to find sample statistics, then performing a two-sample hypothesis test with pooled variance. While conceptually demanding with multiple steps (finding sample mean and variance from CI, computing test statistic, critical value comparison), each individual technique is standard A-level Further Maths material. The reverse-engineering aspect and the two-sample test elevate it above routine questions but don't require novel mathematical insight.
Petra is studying a particular species of bird. She takes a random sample of 12 birds from nature reserve \(A\) and measures the wing span, \(x \mathrm {~cm}\), for each bird. She then calculates a \(95 \%\) confidence interval for the population mean wing span, \(\mu \mathrm { cm }\), for birds of this species, assuming that wing spans are normally distributed. Later, she is not able to find the summary of the results for the sample, but she knows that the \(95 \%\) confidence interval is \(25.17 \leqslant \mu \leqslant 26.83\). Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) for this sample.
Petra also measures the wing spans of a random sample of 7 birds from nature reserve \(B\). Their wing spans, \(y \mathrm {~cm}\), are as follows.
$$\begin{array} { l l l l l l l }
23.2 & 22.4 & 27.6 & 25.3 & 28.4 & 26.5 & 23.6
\end{array}$$
She believes that the mean wing span of birds found in nature reserve \(A\) is greater than the mean wing span of birds found in nature reserve \(B\). Assuming that this second sample also comes from a normal distribution, with variance the same as the first distribution, test, at the \(10 \%\) significance level, whether there is evidence to support Petra's belief.
Petra is studying a particular species of bird. She takes a random sample of 12 birds from nature reserve $A$ and measures the wing span, $x \mathrm {~cm}$, for each bird. She then calculates a $95 \%$ confidence interval for the population mean wing span, $\mu \mathrm { cm }$, for birds of this species, assuming that wing spans are normally distributed. Later, she is not able to find the summary of the results for the sample, but she knows that the $95 \%$ confidence interval is $25.17 \leqslant \mu \leqslant 26.83$. Find the values of $\sum x$ and $\sum x ^ { 2 }$ for this sample.
Petra also measures the wing spans of a random sample of 7 birds from nature reserve $B$. Their wing spans, $y \mathrm {~cm}$, are as follows.
$$\begin{array} { l l l l l l l }
23.2 & 22.4 & 27.6 & 25.3 & 28.4 & 26.5 & 23.6
\end{array}$$
She believes that the mean wing span of birds found in nature reserve $A$ is greater than the mean wing span of birds found in nature reserve $B$. Assuming that this second sample also comes from a normal distribution, with variance the same as the first distribution, test, at the $10 \%$ significance level, whether there is evidence to support Petra's belief.
\hfill \mbox{\textit{CAIE FP2 2016 Q11 OR}}