Question

SAT scores are normally distributed with a mean of 500 points and a standard deviation of 100 points. suppose you take the SAT. several weeks later you receive your results which show you reached the 80th percentile for the math portion recalling that SAT scores are always expressed as multiples of 10, how many points did you get on the test
a. 690
b. 600
c. 580
d. 85
e. 550

Answer 1

The 80th percentile is the test score $k$ such that

$\mathbb P(X\le k)=0.8$

where $X$ denotes a random variable for SAT test scores. Converting to a standard normal distribution, you have

$\mathbb P\left(\dfrac{X-500}{100}\le\dfrac{k-500}{100}\right)=\mathbb P\left(Z\le\dfrac{k-500}{100}\right)=0.8$

where $Z$ follows the standard normal distribution. A probability of 0.8 roughly corresponds to a z-score of about $z\approx0.8416$.

So if $\mathbb P(Z\le z)=0.8$, the 80th percentile in terms of $X$ would be

$\dfrac{k-500}{100}=0.8416\implies k=584.16$

Rounding to the nearest ten suggests you scored 580 on the SAT.