Question

The College Board reported the following mean scores for the three parts of the Scholastic Aptitude Test (SAT).†

Critical Reading	502
Mathematics	515
Writing	494

Assume that the population standard deviation on each part of the test is

σ = 100.

(Round your answers to four decimal places.)

(a)

What is the probability a sample of 85 test takers will provide a sample mean test score within 10 points of the population mean of 502 on the Critical Reading part of the test?

(b)

What is the probability a sample of 85 test takers will provide a sample mean test score within 10 points of the population mean of 515 on the Mathematics part of the test?

How does this probability compare to the value computed in part (a)?

It is  ---Select--- greater than less than equal to the value from part (a).

(c)

What is the probability a sample of 100 test takers will provide a sample mean test score within 10 of the population mean of 494 on the writing part of the test?

Answer 1

a)

Here, μ = 502, σ = 10.8465, x1 = 492 and x2 = 512. We need to compute P(492<= X <= 512). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (492 - 502)/10.8465 = -0.92
z2 = (512 - 502)/10.8465 = 0.92

Therefore, we get
P(492 <= X <= 512) = P((512 - 502)/10.8465) <= z <= (512 - 502)/10.8465)
= P(-0.92 <= z <= 0.92) = P(z <= 0.92) - P(z <= -0.92)
= 0.8212 - 0.1788
= 0.6424

b)

Here, μ = 515, σ = 10.8465, x1 = 505 and x2 = 525. We need to compute P(505<= X <= 525). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (505 - 515)/10.8465 = -0.92
z2 = (525 - 515)/10.8465 = 0.92

Therefore, we get
P(505 <= X <= 525) = P((525 - 515)/10.8465) <= z <= (525 - 515)/10.8465)
= P(-0.92 <= z <= 0.92) = P(z <= 0.92) - P(z <= -0.92)
= 0.8212 - 0.1788
= 0.6424

It is equal to the value from part (a)

c)

Here, μ = 494, σ = 10, x1 = 484 and x2 = 504. We need to compute P(484<= X <= 504). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (484 - 494)/10 = -1
z2 = (504 - 494)/10 = 1

Therefore, we get
P(484 <= X <= 504) = P((504 - 494)/10) <= z <= (504 - 494)/10)
= P(-1 <= z <= 1) = P(z <= 1) - P(z <= -1)
= 0.8413 - 0.1587
= 0.6826