Suppose that the national average for the math portion of the College Board\'s S

Question

Suppose that the national average for the math portion of the College Board's SAT is 510. The College Board periodically rescales the test scores such that the standard deviation is approximately 100. Answer the following questions using a bell-shaped distribution and the empirical rule for the math test scores.

If required, round your answers to two decimal places.

(a) What percentage of students have an SAT math score greater than 610? % (b) What percentage of students have an SAT math score greater than 710? % (c) What percentage of students have an SAT math score between 410 and 510? % (d) What is the z-score for student with an SAT math score of 625? (e) What is the z-score for a student with an SAT math score of 415?

Explanation / Answer

average for the math portion of the College Board's SAT i.e mean = 510

standard deviation sd= 100

z score = (X-mean)/sd

a)

z score for 610 is (610-510)/100 = 1, correspoding p value using z table is 0.8413

P(X<610) = 0.8413

P(X>610) = 1-0.8413 = 0.1587 or 15.87%

percentage of students have an SAT math score greater than 610 is 15.87%

b)

z score for 710 is (710-510)/100 = 2, correspoding p value using z table is 0.9772

P(X<710) = 0.9772

P(X>710) = 1-0.9772 = 0.0228 or 2.28%

percentage of students have an SAT math score greater than 710 is 2.28%

c)

P(X<mean)=0.5 => P(X<510) = 0.5

z score for 410 is (410-510)/100 = -1, correspoding p value using z table is 0.1587

P(X<410) = 0.1587

P(410<X<510) = P(X<510)-P(X<410) = 0.5- 0.1587 = 0.3413 or 34.13%

percentage of students have an SAT math score between 410 and 510 = 34.13%

d)

z score for 625 is (625-510)/100 = 115/100 = 1.15

z-score for student with an SAT math score of 625 is 1.15

e)

z score for 415 is (415-510)/100 = -95/100 = -0.95

z-score for student with an SAT math score of 415 is -0.95

Note: let me know if you have any doubts in this answer

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