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)

1. Find, to 2 decimal places, the standard error of the mean.
2. 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.
3. 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 \(( 29.74,31.86 )\)
4. Find, to 2 decimal places, the standard error of the mean.
5. Hence, or otherwise, find a \(90 \%\) confidence interval for \(\mu\) based on the same sample
   of sheets. (3)
   \includegraphics[max width=\textwidth, alt={}]{0434a6c1-686a-449d-ba16-dbb8e60288e8-28_2646_1824_105_123}

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