Step 4: The scores of high school seniors on the ACT college entrance examinatio
ID: 3132173 • Letter: S
Question
Step 4:
The scores of high school seniors on the ACT college entrance examination in a recent year had mean ? = 22.3 and standard deviation ? = 5.2.
The distribution of scores is only roughly Normal.
Which of your two Normal probability calculations in (a) and (c) is more accurate? Why?
The probability in Step 1 is more accurate since the distribution of the population is roughly Normal whereas the sampling distribution is only assumed to be Normal.
can someone help answer this question?
0.0384 0.0344 0.0002 0.2877Explanation / Answer
A)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 27
u = mean = 22.3
s = standard deviation = 5.2
Thus,
z = (x - u) / s = 0.903846154
Thus, using a table/technology, the right tailed area of this is
P(z > 0.903846154 ) = 0.183038492 [answer]
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STEP 2:
It has the same mean,
mean = 22.3 [ANSWER]
But a reduced standard deviation,
standard deviation = sigma/sqrt(n) = 5.2/sqrt(16) = 1.3 [ANSWER]
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STEP 3:
We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as
x = critical value = 27
u = mean = 22.3
n = sample size = 16
s = standard deviation = 5.2
Thus,
z = (x - u) * sqrt(n) / s = 3.615384615
Thus, using a table/technology, the right tailed area of this is
P(z > 3.615384615 ) = 0.000149951 [answer]
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STEP 4:
OPTION C: The probability in Step 3 is more accurate because the distribution is only roughly Normal. [ANSWER]
By central limit theorem, the distirbution of means is always more "normal" than the original distribution.
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