Question

annabelle earned a score of 288 on exam a that had a mean of 300 and a standard deviation of 40. she is about to take exam b that has a mean of 450 and a standard deviation of 20. how well must annabelle score on exam b in order to do equivalently well as she did on exam a? assume that scores on each exam are normally distributed.

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the raw score, $\mu$ is the mean, and $\sigma$ is the standard deviation. We first find the z - score for Annabelle's score on Exam A.
For Exam A: $x = 288$, $\mu=300$, $\sigma = 40$.
Substitute into the z - score formula: $z=\frac{288 - 300}{40}=\frac{- 12}{40}=-0.3$

Step2: Use the same z - score for Exam B

For Exam B, we know that $z=-0.3$ (since we want the same relative performance), $\mu = 450$, $\sigma=20$. We need to find $x$ (the score on Exam B).
From the z - score formula $z=\frac{x-\mu}{\sigma}$, we can solve for $x$:
$x=\mu+z\times\sigma$
Substitute $\mu = 450$, $z=-0.3$, and $\sigma = 20$ into the formula:
$x=450+(-0.3)\times20$
$x = 450-6$
$x = 444$

Answer:

Annabelle must score 444 on Exam B.