The scores of 12th-grade students on the National Assessment of Educational Prog
ID: 3126317 • Letter: T
Question
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 = 308 and standard deviation = 38.
Choose one 12th-grader at random. What is the probability (±0.1) that his or her score is higher than 308? _____
Higher than 346 (±0.001)? _____
Now choose an SRS of 4 twelfth-graders and calculate their mean score x. If you did this many times, what would be the mean of all the x-values? _____
What would be the standard deviation (±0.1) of all the x-values? _____
What is the probability that the mean score for your SRS is higher than 308? (±0.1) _____ Higher than 346? (±0.0001) _____
Explanation / Answer
A)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 308
u = mean = 308
s = standard deviation = 38
Thus,
z = (x - u) / s = 0
Thus, using a table/technology, the right tailed area of this is
P(z > 0 ) = 0.5 [ANSWER]
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b)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 346
u = mean = 308
s = standard deviation = 38
Thus,
z = (x - u) / s = 1
Thus, using a table/technology, the right tailed area of this is
P(z > 1 ) = 0.158655254 [answer]
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c)
It will have the same mean,
u(X) = 308 [ANSWER]
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d)
The standard deviation will decrease by a factor of sqrt(n),
sigma(X) = sigma/sqrt(n) = 38/sqrt(4) = 19 [ANSWER]
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e)
We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as
x = critical value = 308
u = mean = 308
n = sample size = 4
s = standard deviation = 38
Thus,
z = (x - u) * sqrt(n) / s = 0
Thus, using a table/technology, the right tailed area of this is
P(z > 0 ) = 0.5 [ANSWER]
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f)
We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as
x = critical value = 346
u = mean = 308
n = sample size = 4
s = standard deviation = 38
Thus,
z = (x - u) * sqrt(n) / s = 2
Thus, using a table/technology, the right tailed area of this is
P(z > 2 ) = 0.022750132 [ANSWER]