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 10.6. 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 491. the probability that a randomly selected medical student who took the test had a total score that was less than 491 is 0.1977 (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 500 and 512. the probability that a randomly selected medical student who took the test had a total score that was between 500 and 512 is 0.3708 (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 523. the probability that a randomly selected medical student who took the test had a total score that was more than 523 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 \(\mu = 500\), \(\sigma=10.6\), and \(x = 523\).

$$z=\frac{523 - 500}{10.6}=\frac{23}{10.6}\approx2.17$$

Step2: Find the probability

We want to find \(P(X>523)\), which is equivalent to \(1 - P(X\leq523)\). Using the standard normal distribution table (or a calculator with a normal - distribution function), for \(z = 2.17\), \(P(Z\leq2.17)\) is approximately \(0.9850\).

$$P(X>523)=1 - 0.9850=0.0150$$

Answer:

\(0.0150\)