8. A factory produces steel sheets whose weights $X \mathrm {~kg}$, are such that $X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$

A random sample of these sheets is taken and a $95 \%$ confidence interval for $\mu$ is found to be (29.74, 31.86)
\begin{enumerate}[label=(\alph*)]
\item Find, to 2 decimal places, the standard error of the mean.
\item Hence, or otherwise, find a $90 \%$ confidence interval for $\mu$ based on the same sample of sheets.

Using four different random samples, four $90 \%$ confidence intervals for $\mu$ are to be found.
\item Calculate the probability that at least 3 of these intervals will contain $\mu$.

\section*{8. A factory produces steel sheets whose weights $X \mathrm { gg }$, are such $X \sim N ( \mu , \sigma ) ^ { 2 }$}
A. A.

A random sample of these sheets is taken and a $95 \%$ confidence interval for $\mu$ is found to\\
be $( 29.74,31.86 )$\\
(a) Find, to 2 decimal places, the standard error of the mean.\\
(b) Hence, or otherwise, find a $90 \%$ confidence interval for $\mu$ based on the same sample\\
of sheets. (3)

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VIIIV SIHI NI JIIIM ION OC & VIUV SIHI NI JIIAM ION OO & VEXV SIHI NI JIIIM IONOO \\
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VIIIV SIHI NI IIIYM ION OC & VIIV SIHI NI JIIIM ION OC & VI4V SIHI NI JIIIM I ION OO \\
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\end{enumerate}

\hfill \mbox{\textit{Edexcel S3 2018 Q8 [9]}}