In a reading test for eight$-$year$-$old children, it is found that a reading score X

is normally distributed with mean $5\mathrm{.}0$ and standard deviation $2\mathrm{.}0.$

$($i$)$ What proportion of children would you expect to score between $4\mathrm{.}5$

and $6\mathrm{.}0?$

$($ii$)$ There are about $700000$ eight$-$year$-$olds in the country. How many

would you expect to have a reading score of more than twice the mean?

$($iii$)$ Why might educationalists refer to the reading score X as a $$score out of $10?$

The reading score is often reported, after scaling, as a value Y which is

normally distributed, with mean $100$ and standard deviation $15.$ Values of Y

are usually given to the nearest integer.

$($iv$)$ Find the probability that a randomly chosen eight$-$year$-$old gets a score,

after scaling, of $103.$

$($v$)$ What range of Y scores would you expect to be attained by the best $20\%$

of readers?