One hundred men from that country, selected at random, had their heights measured.

The mean of this sample was $185$ cm. Calculate a $95\%$ confidence interval for the mean height of the population.

A second random sample of size $n$ is taken from the same population. Find the minimum value of $n$ needed for the width of a $95\%$ confidence interval to be less than $3$ cm.

valid attempt to use $\bar x \pm z\frac{\sigma }{{\sqrt n }}$     (M1)

$[182,{\text{ }}188]$     A1A1

Note:     Accept answers that round to the correct $3$ sf.

[3 marks]

$1.96 \times \frac{{15.0}}{{\sqrt n }} < 1.5$     M1A1

$n > {\left( {\frac{{15.0}}{{1.5}} \times 1.96} \right)^2}$     (M1)

Note:     Award M1 for attempting to solve the inequality.

Note:     Allow the use of $=$.

minimum value $n = 385$     A1

[4 marks]

Total [7 marks]