Question

samantha earned a score of 200 on exam a that had a mean of 350 and a standard deviation of 100. she is about to take exam b that has a mean of 46 and a standard deviation of 10. how well must samantha 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. For Samantha to do equivalently well on both exams, her z - scores on Exam A and Exam B must be equal.

First, calculate the z - score for Exam A. For Exam A, $x = 200$, $\mu=350$, $\sigma = 100$.
So, $z_{A}=\frac{200 - 350}{100}=\frac{- 150}{100}=-1.5$

Step2: Use z - score to find score on Exam B

Let the score on Exam B be $x_{B}$. For Exam B, $\mu_{B}=46$, $\sigma_{B}=10$, and $z_{B}=z_{A}=- 1.5$ (since equivalent performance means equal z - scores).
Using the z - score formula $z=\frac{x-\mu}{\sigma}$, we can solve for $x$: $x=\mu + z\sigma$
Substitute $\mu = 46$, $z=-1.5$, and $\sigma = 10$ into the formula:
$x_{B}=46+(-1.5)\times10=46 - 15 = 31$

Answer:

Samantha must score 31 on Exam B.