Question

Amy scores an 82% on her math test with Ms. Smith. The average score for her class is a 75% with a standard deviation of 2%. Amy’s friend Karina is taking the same test with Mr. Adams. His class average is a 73% and a standard deviation of 3%. What is the lowest score Karina needs to score higher than Amy relative to the class distributions?

Answer 1

Assuming scores are normally distributed, a score of 82% on Ms. Smith's test corresponds to the $p$-th percentile, i.e.

$P(X_S\le82)=p$

where $X_S$ is a random variable denoting scores on Ms. Smith's test.

Transform $X_S$ to $Z$, which follows the standard normal distribution:

$P(X_S\le82)=P\left(\dfrac{X_S-75}2\le\dfrac{82-75}2\right)=P(Z\le3.5)\approx0.9998$

which means Amy scored at the 99.98th percentile.

This makes it so that Karina needs to score $X_A=x$ on Mr. Adams' test so that

$P(X_A\le x)=0.9998$

Their test scores have the same $z$ score computed above, so

$\dfrac{x-73}3=3.5\implies x=83.5$

so Karina needs to get a test score of at least 83.5%.