\begin{enumerate}
  \item A random sample of 8 three-month-old golden retriever dogs is taken.
\end{enumerate}

The heights of the golden retrievers are recorded.\\
Using this sample, a 95\% confidence interval for the mean height, in cm, of three-month-old golden retrievers is found to be $( 45.72,53.88 )$\\
(a) Find a 99\% confidence interval for the mean height.

You may assume that the heights are normally distributed with known population standard deviation.

Some summary statistics for the weights, $x \mathrm {~kg}$, of this sample are given below.

$$\sum x = 91.2 \quad \sum x ^ { 2 } = 1145.16 \quad n = 8$$

(b) Calculate unbiased estimates of the mean and the variance of the weights of three-month-old golden retrievers.

A further random sample of 24 three-month-old golden retrievers is taken. The unbiased estimates of the mean and the variance of the weights, in kg , from this sample are found to be 10.8 and 17.64 respectively.\\
(c) Estimate the standard error of the mean weight for the combined sample of 32 three-month-old golden retrievers.

\hfill \mbox{\textit{Edexcel S3 2024 Q6 [15]}}