Ok I am stuck on the second part of this question.
(a) what percentage of the population is within ± $0.5$ standard deviations of the mean.
(b) what percentage of the population is more than 1 standard deviation above the mean?
For part (a), I got $38.3$% using $P_r(-0.5<z<0.5)$
However, I am stuck on part (b). Is it similar to part (a)?
The empirical rule tells use that $68.2\%$ of the population is within $1$ standard deviation of the mean. We know that the normal distribution is symmetrical about the mean; thus, we can deduce that $34.1\%$ of the population is between $0$ and $1$ standard deviations above the mean. Thus, by symmetry, the percentage of the population that is over $1$ standard deviation above the mean is
$50\% - 34.1\% = 15.9\%$
The empirical rule is very useful for reasoning about normal distributions and is something you should commit to memory. If you need an overview check out this page.