The distributions of the heights of $1000$ women and of $1000$ men both produce normal curves, as shown. The mean height of the women is $160 \mathrm{cm}$ and the mean height of the men is $180 \mathrm{cm}.$ The heights of these women and men are now combined to form a new set of data. Assuming that the combined heights also produce a normal curve, copy the graph opposite and sketch onto it the curve for the combined heights of the $2000$ women and men.

Given that $Z~\mathrm{N}(0,1),$ find the following probability correct to $3$ significant figures.

$P (Z>-1.53)$

The random variable $Z$ is normally distributed with mean $0$ and variance $1$. Find the following probability, correct to $3$ significant figures.

$P(1.645<Z<2.326)$

Given that $Z~\mathrm{N}(0,1),$ find the value of $k,$ given that:

$P(Z>k)=0.9296$

(Correct the answer up to three decimal place)

Find the value of $c$ in the following where $Z$ has a normal distribution with $\mu =0$ and ${\sigma }^2=1$.

$P(c<Z<2.878)=0.4968$

(Write answer up to $3$ decimal places)

Calculate the required probabilities correct to $3$ significant figures.

Find $P(X<13.5)$ and $P(X\ge 13.5),$ given that $X~\mathrm{N}(20, 15)$.

Calculate the required probabilities correct to $3$ significant figures.

Find $P(26\le X\le 28),$ given that $X~\mathrm{N}(25, 6)$.

(Round answer up to $3$ decimal places)

Colleen exercises at home every day. The length of time she does this is normally distributed with mean $12.8$ minutes and standard deviation $\sigma .$ She exercises for more than $15$ minutes on $42$ days in a year of $365$ days.

Calculate the value of $\sigma .$[Use ${\Phi }^{-1}(0.885)=1.2$]