Question 4 - A-Level Maths

The random variable $X$ is normally distributed with mean $\mu$ and variance $\sigma ^ { 2 }$.
1. If $2 \mu = 3 \sigma$, find $\mathrm { P } ( X < 2 \mu )$.
2. If, instead, $\mathrm { P } ( X < 3 \mu ) = 0 \cdot 86$,
  1. find $\mu$ in terms of $\sigma$,
  2. calculate $\mathrm { P } ( X > 0 )$.
3. The stem-and-leaf diagram shows the values taken by two variables $A$ and $B$.
$A$ $B$
$8,7,4,1,0$ 1 $1,1,2,5,6,8,9$
$9,8,7,6,6,5,2$ 2 $0,3,4,6,7,7,9$
$9,7,6,4,2,1,0$ 3 $1,4,5,5,8$
$8,6,3,2,2$ 4 $0,2,6,6,9,9$
$6,4,0$ 5 $2,3,5,7$
$5,3,1$ 6 0,1
Key : 3| 1 | 2 means $$A = 13 , B = 12$$
For each set of data, calculate estimates of the median and the quartiles.
Calculate the 42nd percentile for $A$.
On graph paper, indicating your scale clearly, construct box and whisker plots for both sets of data.
Describe the skewness of the distribution of $A$ and of $B$.