The national average for the math portion of the College Board\'s SAT test is 54
ID: 3134572 • Letter: T
Question
The national average for the math portion of the College Board's SAT test is 540. The College Board periodically rescales the test scores such that the standard deviation is approximately 75. Answer the following questions using a bell-shaped distribution and the empirical rule for the math test scores.
If required, round your answers to two decimal places.
(a) What percentage of students have an SAT math score greater than 615? % (b) What percentage of students have an SAT math score greater than 690? % (c) What percentage of students have an SAT math score between 465 and 540? % (d) What is the z-score for student with an SAT math score of 625? (e) What is the z-score for a student with an SAT math score of 415Explanation / Answer
A)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 615
u = mean = 540
s = standard deviation = 75
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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b)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 690
u = mean = 540
s = standard deviation = 75
Thus,
z = (x - u) / s = 2
Thus, using a table/technology, the right tailed area of this is
P(z > 2 ) = 0.022750132 [ANSWER]
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c)
We first get the z score for the two values. As z = (x - u) / s, then as
x1 = lower bound = 465
x2 = upper bound = 540
u = mean = 540
s = standard deviation = 75
Thus, the two z scores are
z1 = lower z score = (x1 - u)/s = -1
z2 = upper z score = (x2 - u) / s = 0
Using table/technology, the left tailed areas between these z scores is
P(z < z1) = 0.158655254
P(z < z2) = 0.5
Thus, the area between them, by subtracting these areas, is
P(z1 < z < z2) = 0.341344746 [ANSWER]
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d)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 625
u = mean = 540
s = standard deviation = 75
Thus,
z = (x - u) / s = 1.133333333 [ANSWER]
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e)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 415
u = mean = 540
s = standard deviation = 75
Thus,
z = (x - u) / s = -1.666666667 [ANSWER]