Question

A set of test scores is normally distributed with a mean of 145 and a standard deviation of 11. Seif received a score of 167. What was his percentile score
97.5%

84%

95%

63.5%

Answer 1

Since $145+2\times11=167$, which means the corresponding z-score for Seif's test score is $z=2$. The empirical rule asserts that for any normal distribution, approximately 95% of it lies within two standard deviations of the mean, i.e. $\mathbb P(|Z|$, which leaves 5% outside of that range, and specifically 2.5% to either side.

This means

$\mathbb P(Z$

In other words, a test score of 167, which corresponds to a z-score of 2, marks the 97.5th percentile.