Changing the mean and standard deviation of a Normal distribution by a moderate

Question

Changing the mean and standard deviation of a Normal distribution by a moderate amount can greatly change the percent of observations in the tails. Suppose a college is looking for applicants with SAT math scores 760 and above.

(a) In a certain year, the scores of men on the math SAT followed the N(531, 121) distribution. What percent of men scored 760 or better? (Round your answer to two decimal places.) %

(b) Women's SAT math scores that year had the N(499, 112) distribution. What percent of women scored 760 or better? You see that the percent of men above 760 is almost three times the percent of women with such high scores. Why this is true is controversial. (On the other hand, women score higher than men on the new SAT writing test, though by a smaller amount. Round your answer to two decimal places.) %

Explanation / Answer

Part A

We have:

X= 760

M= 531

SD = 121

We can use following formula to calculate the value of Z:

Z= (X-M)/ SD

= (760 -531)/ 121

= 1.8926

Now probability of getting 760 or better would be:

Probability = 1- area under Z (1.8926)

                      = 1 -0.9708

                      = 2.92%

Part B

We have:

X= 760

M= 499

SD = 112

We can use following formula to calculate the value of Z:

Z= (X-M)/ SD

= (760 -499)/ 112

= 2.33

Now probability of getting 760 or better would be:

Probability = 1- area under Z (2.33)

                      = 1 -0.9901

                      = 1%

This is true. The reason for this is woman scored lower scores in total and also their average scores are lower.

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