Question

Suppose that the national average for the math portion of the College Board's SAT is 548. 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 648?
	%
(b)	What percentage of students have an SAT math score greater than 748?
	%
(c)	What percentage of students have an SAT math score between 448 and 548?
	%
(d)	What is the z-score for student with an SAT math score of 630?
(e)	What is the z-score for a student with an SAT math score of 395?

Answer 1

This is a normal distribution question with

a) x = 648

P(x > 648.0)=?

The z-score at x = 648.0 is,

z = 1.0

This implies that

P(x > 648.0) = P(z > 1.0) = 1 - 0.8413447460685429

b) x = 748

P(x > 748.0)=?

The z-score at x = 748.0 is,

z = 2.0

This implies that

P(x > 748.0) = P(z > 2.0) = 1 - 0.9772498680518208

c) x1 = 448

x2 = 548

P(448.0 < x < 548.0)=?

This implies that

P(448.0 < x < 548.0) = P(-1.0 < z < 0.0) = P(Z < 0.0) - P(Z < -1.0)

P(448.0 < x < 548.0) = 0.5 - 0.15865525393145707

P(448.0 < x < 548.0) = {0.34}

d) x = 630

P(x < 630.0)=?

The z-score at x = 630.0 is,

z = 0.82

e) x = 395

P(x < 395.0)=?

The z-score at x = 395.0 is,

z = -1.53

PS: you have to refer the z score table to find the final probabilities.

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