The help desk for a software company logs the lengths of times of each call that
ID: 3126621 • Letter: T
Question
The help desk for a software company logs the lengths of times of each call that technicians have with customers. Historical records based on the past 3 years indicate that call length (L) is a normally distributed variable with an average () of 8.00 minutes and a standard deviation () of 1.15 min. For each of the problems below, find the appropriate probability.
A) P(L 5.34)
B) P(L > 8.08)
C) P(7.28 L < 9.69)
D) The call times that enclose the central 89% of the distribution are:
Note: Round your answers to two decimal places for this problem.
Low: to High:
Explanation / Answer
a)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 5.34
u = mean = 8
s = standard deviation = 1.15
Thus,
z = (x - u) / s = -2.313043478
Thus, using a table/technology, the left tailed area of this is
P(z < -2.313043478 ) = 0.010360124 [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 = 8.08
u = mean = 8
s = standard deviation = 1.15
Thus,
z = (x - u) / s = 0.069565217
Thus, using a table/technology, the right tailed area of this is
P(z > 0.069565217 ) = 0.472269861 [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 = 7.28
x2 = upper bound = 9.69
u = mean = 8
s = standard deviation = 1.15
Thus, the two z scores are
z1 = lower z score = (x1 - u)/s = -0.626086957
z2 = upper z score = (x2 - u) / s = 1.469565217
Using table/technology, the left tailed areas between these z scores is
P(z < z1) = 0.265628954
P(z < z2) = 0.929160227
Thus, the area between them, by subtracting these areas, is
P(z1 < z < z2) = 0.663531273 [ANSWER]
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d)
MIDDLE VALUES
As the middle area is
Middle Area = P(x1<x<x2) = 0.89
Then the left tailed area of the left endpoint is
P(x<x1) = (1-P(x1<x<x2))/2 = 0.055
Thus, the z score corresponding to the left endpoint, by table/technology, is
z1 = -1.59819314
By symmetry,
z2 = 1.59819314
As
u = mean = 8
s = standard deviation = 1.15
Then
x1 = u + z1*s = 6.162077889 [ANSWER]
x2 = u + z2*s = 9.837922111 [ANSWER]
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