Question

Suppose that the national average for the math portion of the College Board's SAT is 535. The College Board periodically rescales the test scores such that the standard deviation is approximately 75. 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 685?______ %

(c) What percentage of students have an SAT math score between 460 and 535?______ %

(d) What is the z-score for a 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

Solution:-

μ = 535, σ = 75

(a) The percentage of students have an SAT math score greater than 610 is 15.87%.

x = 610

By applying normal distribution:-

z = 1

P(z > 1) = 0.1587

P(z > 1) = 15.87%

(b) The percentage of students have an SAT math score greater than 685 is 2.28%.

x = 685

By applying normal distribution:-

z = 2

P(z > 2) = 0.0228

P(z > 2) = 2.28%

(c) The percentage of students have an SAT math score between 460 and 535 is 34.13%.

x₁ = 460

x₂ = 535

By applying normal distribution:-

z(x₁ = 460) = - 1

z(x₂ = 535) = 0

P(-1 < z < 0) = P(z > - 1) - P(z > 0)

P(-1 < z < 0) = 0.8413 - 0.50

P(-1 < z < 0) = 0.3413

P(-1 < z < 0) = 34.13%

(d) The z-score for a student with an SAT math score of 630 is 1.27.

x = 630

By applying normal distribution:-

z = 1.27

(e) The z-score for a student with an SAT math score of 395 is - 1.87.

x = 395

By applying normal distribution:-

z = - 1.87