$11($i$)$ Give a reason why, in carrying out out a statistical investigation, a sample

rather than a complete population may be used.

$($ii$)$ Rose wishes to investigate whether men in her town have a different

life$-$span from the national average of $71\mathrm{.}2$ years. She looks at

government records for her town and takes a random sample of the ages

of $110$ men who have died recently. Their mean age in years was $69\mathrm{.}3$ and

the unbiased estimate of the population variance was $65\mathrm{.}61.$

$($a$)$ Calculate a $90\%$ confidence interval for the population mean and

explain what you understand by this confidence interval.

$($b$)$ State with a reason what conclusion about the life$-$span of men in her

town Rose could draw from this confidence interval.