Question

Suppose that the national average for the math portion of the College Board's SAT is 540. The College Board periodically rescales the test scores such that the standard deviation is approximately 50. 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. If your answer is negative use "minus sign".

(a) What percentage of students have an SAT math score greater than 615?

(b) What percentage of students have an SAT math score greater than 690?

(c) What percentage of students have an SAT math score between 465 and 540?

(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?

Answer 1

a) 6.68%

b) 0.135%

c) 43.32%

d) 1.7

e) -2.5

Explanation:

a)

P(X>615)=P(z>(615-540)/50)

P(X>615)=P(z>1.5)

P(X>615)=P(0<z<∞)-P(0<z<1.5)

P(X>615)=0.5-0.4332

P(X>615)=0.0668

The percentage of students have an SAT math score greater than 615 is 6.68%

b)

P(X>690)=P(z>(690-540)/50)

P(X>690)=P(z>3)

P(X>690)=P(0<z<∞)-P(0<z<3)

P(X>690)=0.5-0.49865

P(X>690)=0.00135

The percentage of students have an SAT math score greater than 690 is 0.135%

c)

P(465<X<540)=P((465-540)/50<z<(540-540)/50))

P(465<X<540)=P(-1.5<z<0)

P(465<X<540)=0.4332

The percentage of students have an SAT math score between 465 and 540 is 43.32%

d)

z=(625-540)/50

z=85/50

z=1.7

The z-score for student with an SAT math score of 625 is 1.7.

e)

z=(415-540)/50

z=-125/50

z=-2.5

The z-score for a student with an SAT math score of 415 is -2.5.