Question

Problem 2-23

Suppose that the national average for the math portion of the College Board's SAT is 534. 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.

(a)	What percentage of students have an SAT math score greater than 584?
	%
(b)	What percentage of students have an SAT math score greater than 634?
	%
(c)	What percentage of students have an SAT math score between 484 and 534?
	%
(d)	What is the z-score for student with an SAT math score of 635?
(e)	What is the z-score for a student with an SAT math score of 425?

Answer 1

a. What percentage of students has an SAT math score greater than 584?

We have ? = 534 and ? = 50

Percentage of students have an SAT math score greater than 584, we have X ? 584

P (X > 584) = P (X?? > 584?534) = P ((X??)/? > (584?534)/50)

Since Z = x?? /? = (584?534)/50 =1

Now we have,

P (X > 584) =P (Z > 1)

Use the standard normal table for the value of z, now we have

P (Z > 1) = 0.1587

Percentage of students have an SAT math score greater than 584 is 0.1587 or 15.87%

b. What percentage of students has an SAT math score greater than 634?

Percentage of students have an SAT math score greater than 634, we have X ? 634

P (X > 634) = P (X?? > 634?534) = P ((X??)/? > (634?534)/50)

Since Z = x?? /? = (634?534)/50 =2

Now we have,

P (X > 634) =P (Z > 2)

Use the standard normal table for the value of z, now we have

P (Z > 2) = 0.0228

Percentage of students have an SAT math score greater than 634 is 0.0228 or 2.28%

(c)           What percentage of students has an SAT math score between 484 and 534?

P (484< X > 534) = P ((484 -534)/50 <X??/? > (534?534)/50)

= P (-50/50 < Z > 0/50)

= P (-1 < Z > 0)

P (484< X > 534) = 0.3413 or 34.13%

Percentage of students have an SAT math score between 484 and 534 is 0.3413 or 34.13%

(d)          What is the z-score for student with an SAT math score of 635?

The z-score for student with an SAT math score of 635

Z = x?? /? = (635?534)/50

=101/50 = 2.02  

(e)          What is the z-score for a student with an SAT math score of 425?

The z-score for student with an SAT math score of 425

Z = x?? /? = (425?534)/50

= -109/50 = -2.18