Question

The scores of high school seniors on the ACT college entrance examination in a recent year had mean μ = 20.8 and standard deviation σ = 4.8. The distribution of scores is only roughly Normal.

(a) What is the approximate probability that a single student randomly chosen from all those taking the test scores 21 or higher? (Round your answer to four decimal places.)


(b) Now take an SRS of 25 students who took the test. What are the mean and standard deviation of the sample mean score x of these 25 students? (Round your answers to two decimal places.)

μx	=
σx	=

(c) What is the approximate probability that the mean score

x

of these students is 21 or higher? (Round your answer to four decimal places.)


(d) Which of your two Normal probability calculations in (a) and (c) is more accurate? Why?

The answer to (a) is more accurate because an individual score should have a distribution farther from Normal.The answer to (a) is more accurate because an individual score should have a distribution that is exactly Normal.    The answer to (a) is more accurate because an individual score should have a distribution closer to Normal.The answer to (c) is more accurate because x should have a distribution closer to Normal.The answer to (c) is more accurate because x should have a distribution that is farther from Normal.

Answer 1

Solution :

Given that ,

mean = = 20.8

standard deviation = = 4.8

a) P(x 21 ) = 1 - P(x   21)

= 1 - P[(x - ) / (21 -20.8) /4.8 ]

= 1 -  P(z 0.04 )  

= 1 - 0.516 = 0.4840

Probability = 0.4840

b)

n = 25

= = 20.8  

= / n = 4.8 / 25 = 0.96

c) P( ≥ 21 ) = 1 - P( 21)

= 1 - P[( - ) /   (21 - 20.8) /0.96 ]

= 1 - P(z 0.21 )

= 1 - 0.5832 = 0.4168

Probability = 0.4168

d)The answer to (a) is more accurate because an individual score should have a distribution that is exactly Normal.