4 The lifetimes, \(X\) hours, of a random sample of 50 batteries of a certain kind were found. The results are summarised by \(\Sigma x = 420\) and \(\Sigma x ^ { 2 } = 27530\).
- Calculate an unbiased estimate of the population mean of \(X\) and show that an unbiased estimate of the population variance is 490 , correct to 3 significant figures.
- The lifetimes of a further large sample of \(n\) batteries of this kind were noted, and the sample mean, \(\bar { X }\), was found. Use your estimates from part (i) to find the value of \(n\) such that \(\mathrm { P } ( \bar { X } > 5 ) = 0.9377\).
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