(15.48 S-AQ) The scores of 12th-grade students on the National Assessment of Edu

Question

(15.48 S-AQ) The scores of 12th-grade students on the National Assessment of Educational Progress year 2000 mathematics test have a distribution that is approximately Normal with mean µ = 281 and standard deviation s = 39.

Choose one 12th-grader at random. What is the probability (±±0.1) that his or her score is higher than 281? ______ Higher than 359 (±±0.001)? ______

Now choose an SRS of 16 twelfth-graders and calculate their mean score xx¯. If you did this many times, what would be the mean of all the xx¯-values? ______

What would be the standard deviation (±±0.1) of all the xx¯-values? ______

What is the probability that the mean score for your SRS is higher than 281? (±±0.1) ______ Higher than 359? (±±0.0001) ______

Explanation / Answer

the PDF of normal distribution is = 1/ * 2 * e ^ -(x-u)^2/ 2^2
standard normal distribution is a normal distribution with a,
mean of 0,
standard deviation of 1
equation of the normal curve is ( Z )= x - u / sd ~ N(0,1)
mean ( u ) = 281
standard Deviation ( sd )= 39
PART A.
GREATER THAN
P(X > 281) = (281-281)/39
= 0/39 = 0
= P ( Z >0) From Standard Normal Table
= 0.5
P(X > 359) = (359-281)/39
= 78/39 = 2
= P ( Z >2) From Standard Normal Table
= 0.0228
PART B.
When size = 16
mean of the sampling distribution ( x ) = 281
standard Deviation ( sd )= 39/ Sqrt ( 16 ) =9.75
sample size (n) = 16

PART C.
a.
GREATER THAN
P(X > 281) = (281-281)/39/ Sqrt ( 16 )
= 0/9.75= 0
= P ( Z >0) From Standard Normal Table
= 0.5
b.
GREATER THAN
P(X > 359) = (359-281)/39/ Sqrt ( 16 )
= 78/9.75= 8
= P ( Z >8) From Standard Normal Table
= 0

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