Questions: 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 488. The probability that a randomly selected medical student who took the test had a total score that was less than 488 is 0.1265 . (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 0.4051 . (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 530. The probability that a randomly selected medical student who took the test had a total score that was more than 530 is (Round to four decimal places as needed.)

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 488. The probability that a randomly selected medical student who took the test had a total score that was less than 488 is 0.1265 . (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 0.4051 . (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 530. The probability that a randomly selected medical student who took the test had a total score that was more than 530 is (Round to four decimal places as needed.)
Transcript text: 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 488. The probability that a randomly selected medical student who took the test had a total score that was less than 488 is 0.1265 . (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 0.4051 . (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 530. The probability that a randomly selected medical student who took the test had a total score that was more than 530 is $\square$ (Round to four decimal places as needed.)
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Solution

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Solution Steps

Step 1: Calculate Probability for Score Less Than 488

To find the probability that a randomly selected medical student had a total score less than 488, we use the cumulative distribution function (CDF) of the normal distribution. Given the mean \( \mu = 500 \) and standard deviation \( \sigma = 10.5 \), we calculate:

\[ P(X < 488) = CDF(488) = 0.1265 \]

Step 2: Calculate Probability for Score Between 498 and 510

Next, we find the probability that a randomly selected medical student had a total score between 498 and 510. This is done by calculating the CDF at both points and finding the difference:

\[ P(498 < X < 510) = CDF(510) - CDF(498) = 0.4051 \]

Step 3: Calculate Probability for Score More Than 530

Finally, we determine the probability that a randomly selected medical student had a total score greater than 530. This is calculated by finding the CDF at 530 and subtracting it from 1:

\[ P(X > 530) = 1 - CDF(530) = 0.0021 \]

Final Answer

(a) \( \boxed{0.1265} \)
(b) \( \boxed{0.4051} \)
(c) \( \boxed{0.0021} \)

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