Question

in a recent year, the total scores for a certain standardized test were normally distributed, with a mean of 500 and a standard deviation of 10.5. answer parts (a)-(d) below.
(a) find the probability that a randomly selected medical student who took the test had a total score that was less than 487.
the probability that a randomly selected medical student who took the test had a total score that was less than 487 is 0.1075. (round to four decimal places as needed.)
(b) find the probability that a randomly selected medical student who took the test had a total score that was between 498 and 510.
the probability that a randomly selected medical student who took the test had a total score that was between 498 and 510 is 0.4045. (round to four decimal places as needed.)
(c) find the probability that a randomly selected medical student who took the test had a total score that was more than 522.
the probability that a randomly selected medical student who took the test had a total score that was more than 522 is
(round to four decimal places as needed.)

Explanation:

Step1: Calculate the z - score

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the data set, $\mu$ is the mean and $\sigma$ is the standard deviation. Here, $\mu = 500$, $\sigma=10.5$, and $x = 522$. So, $z=\frac{522 - 500}{10.5}=\frac{22}{10.5}\approx2.0952$.

Step2: Find the probability

We want $P(X>522)$. Since the total area under the normal - distribution curve is 1, and the standard normal distribution table gives $P(Z\leq z)$. So, $P(X > 522)=1 - P(Z\leq2.0952)$. Looking up the value of $P(Z\leq2.0952)$ in the standard normal distribution table (or using a calculator with a normal - distribution function), we find that $P(Z\leq2.0952)\approx0.9819$. Then $P(X > 522)=1 - 0.9819 = 0.0181$.

Answer:

0.0181