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 10.3. Answer parts (a below d )

(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 (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 496 and 611 The probability that a randomly selected medical student who took the test had a total score that was between 400 and 511 (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 622 is (Round to four decimafplaces as needed.)

Answer 1

To find the probability that a randomly selected medical student who took the test had a total score that was less than 488, you would calculate the z-score for 488 and look it up in the standard normal table. To find the probability that a randomly selected medical student who took the test had a total score that was between 496 and 611, you would calculate the z-scores for both scores and calculate the difference between their probabilities. To find the probability that a randomly selected medical student who took the test had a total score that was more than 522, you would calculate the z-score for 522 and look up the probability of a z-score greater than that in the standard normal table.

Step-by-step explanation:

To find the probability that a randomly selected medical student who took the test had a total score that was less than 488, we need to calculate the z-score for 488 and then use the standard normal table to find the corresponding probability. The z-score is calculated as (X - mean) / standard deviation, so in this case, it would be (488 - 500) / 10.3 = -1.165. Looking up the z-score in the standard normal table, we find that the corresponding probability is approximately 0.1211.

To find the probability that a randomly selected medical student who took the test had a total score that was between 496 and 611, we need to calculate the z-scores for both scores and then calculate the difference between their probabilities. The z-score for 496 is (496 - 500) / 10.3 = -0.388, and the z-score for 611 is (611 - 500) / 10.3 = 10.78. Using the standard normal table, we find that the probability for a z-score of -0.388 is approximately 0.35, and the probability for a z-score of 10.78 is approximately 1. So, the probability of the score being between 496 and 611 is 1 - 0.35 = 0.65.

To find the probability that a randomly selected medical student who took the test had a total score that was more than 522, we need to calculate the z-score for 522 and then use the standard normal table to find the probability of a z-score greater than that. The z-score for 522 is (522 - 500) / 10.3 = 2.136. Looking up the z-score in the standard normal table, we find that the probability of a z-score greater than 2.136 is approximately 0.0161.