Recover parameters from given CI

7. The times taken to travel to school by sixth form students are normally distributed. A head teacher records the times taken to travel to school, in minutes, of a random sample of 10 sixth form students from her school. Based on this sample, the \(95 \%\) confidence interval for the mean time taken to travel to school for sixth form students from her school is
[0pt] [28.5, 48.7] Calculate a 99\% confidence interval for the variance of the time taken to travel to school for sixth form students from her school.
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1. A nutritionist studied the levels of cholesterol, \(X \mathrm { mg } / \mathrm { cm } ^ { 3 }\), of male students at a large college. She assumed that \(X\) was distributed \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\) and examined a random sample of 25 male students. Using this sample she obtained unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\) as \(\hat { \mu }\) and \(\hat { \sigma } ^ { 2 }\)

A \(95 \%\) confidence interval for \(\mu\) was found to be \(( 1.128,2.232 )\)

1. Show that \(\hat { \sigma } ^ { 2 } = 1.79\) (correct to 3 significant figures)
2. Obtain a \(95 \%\) confidence interval for \(\sigma ^ { 2 }\)