Standard +0.8 This is a multi-part Further Maths statistics question requiring calculation of summary statistics from raw data, construction of confidence intervals, and a two-sample t-test. While the individual techniques are standard, the question demands careful handling of pooled variance assumptions, correct degrees of freedom, and interpretation across three related parts. The computational burden and need to state assumptions appropriately places it moderately above average difficulty.
Fish of a certain species live in two separate lakes, \(A\) and \(B\). A zoologist claims that the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). To test his claim, he catches a random sample of 8 fish from \(A\) and a random sample of 6 fish from \(B\). The lengths of the 8 fish from \(A\), in appropriate units, are as follows.
$$\begin{array} { l l l l l l l l }
15.3 & 12.0 & 15.1 & 11.2 & 14.4 & 13.8 & 12.4 & 11.8
\end{array}$$
Assuming a normal distribution, find a \(95 \%\) confidence interval for the mean length of fish in \(A\).
The lengths of the 6 fish from \(B\), in the same units, are as follows.
$$\begin{array} { l l l l l l }
15.0 & 10.7 & 13.6 & 12.4 & 11.6 & 12.6
\end{array}$$
Stating any assumptions that you make, test at the \(5 \%\) significance level whether the mean length of fish in \(A\) is greater than the mean length of fish in \(B\).
Calculate a 95\% confidence interval for the difference in the mean lengths of fish from \(A\) and from \(B\).
Fish of a certain species live in two separate lakes, $A$ and $B$. A zoologist claims that the mean length of fish in $A$ is greater than the mean length of fish in $B$. To test his claim, he catches a random sample of 8 fish from $A$ and a random sample of 6 fish from $B$. The lengths of the 8 fish from $A$, in appropriate units, are as follows.
$$\begin{array} { l l l l l l l l }
15.3 & 12.0 & 15.1 & 11.2 & 14.4 & 13.8 & 12.4 & 11.8
\end{array}$$
Assuming a normal distribution, find a $95 \%$ confidence interval for the mean length of fish in $A$.
The lengths of the 6 fish from $B$, in the same units, are as follows.
$$\begin{array} { l l l l l l }
15.0 & 10.7 & 13.6 & 12.4 & 11.6 & 12.6
\end{array}$$
Stating any assumptions that you make, test at the $5 \%$ significance level whether the mean length of fish in $A$ is greater than the mean length of fish in $B$.
Calculate a 95\% confidence interval for the difference in the mean lengths of fish from $A$ and from $B$.
\hfill \mbox{\textit{CAIE FP2 2014 Q11 OR}}