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 . (round to four decimal places as needed.)

Explanation:

Step1: Calculate z - scores

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu = 500$ (mean), $\sigma=10.5$ (standard deviation).
For $x = 498$, $z_1=\frac{498 - 500}{10.5}=\frac{- 2}{10.5}\approx - 0.1905$.
For $x = 510$, $z_2=\frac{510 - 500}{10.5}=\frac{10}{10.5}\approx0.9524$.

Step2: Use z - table

We want $P(498From the standard normal table, $P(Z < 0.9524)\approx0.8292$ and $P(Z<-0.1905)\approx0.4247$.

Step3: Calculate probability

$P(-0.1905$P(-0.1905

Answer:

$0.4045$