Mathematics 9709 · AS & A Level · Continuous random variables
Continuous random variables — practice question
The middle-finger lengths, measured in centimetres, of women in Raneland follow a normal distribution with mean $\mu$ and standard deviation $\sigma$. It is given that $25\%$ of these women have fingers longer than $8.8\text{ cm}$, while $17.5\%$ have fingers shorter than $7.7\text{ cm}$.
(a(i))[5]
Determine the values of $\mu$ and $\sigma$.
(a(ii))[5]
The middle-finger lengths, in centimetres, of women in Snoland are normally distributed with mean $7.9$ and standard deviation $0.44$. A random sample of $5$ women from Snoland is selected. Find the probability that exactly $3$ of these women have middle fingers shorter than $8.2\text{ cm}$.
(b)[3]
The random variable $X$ follows a normal distribution whose mean is equal to its standard deviation. Find the probability that a given value of $X$ is less than $1.5$ times the mean.
Worked solution & mark scheme
This 13-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Form equations using $z=0.674$ and $z=-0.935$, for example $0.674=\dfrac{8.8-\mu}{\sigma}$” …