A manufacturer of women$$s clothing wants to know the mean height of the

women in a town $($in order to plan what proportion of garments should be of

each size$).$ She knows that the standard deviation of their heights is $5$ cm$.$ She

selects a random sample of $50$ women from the town and finds their mean

height to be $165\mathrm{.}2$ cm$.$

$($i$)$ Use the available information to estimate the proportion of women in the

town who were

$($a$)$ over $170$ cm tall

$($b$)$ less than $155$ cm tall.

$($ii$)$ Construct a $95\%$ confidence interval for the mean height of women in the

town.

$($iii$)$ Another manufacturer in the same town wants to know the mean height

of women in the town to within $0\mathrm{.}5$ cm with $95\%$ confidence. What is the

minimum sample size that would ensure this?