4) The standard deviation of the number of Metropolitan New York Bus Authority b
ID: 3133117 • Letter: 4
Question
4) The standard deviation of the number of Metropolitan New York Bus Authority busses that
are being repaired on a given day is 6. The Chief Operating Officer takes a sample of 50 days
and finds that the mean number of busses being repaired on each day is 82.
a) What is a point estimate of the mean number of busses being repaired on each day?
b) Construct a 90% CI of the mean number of busses being repaired on each day.
c) Construct a 99% CI of the mean number of busses being repaired on each day
Explanation / Answer
a) What is a point estimate of the mean number of busses being repaired on each day?
It is the sample mean,
X = 82 [ANSWER]
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b) Construct a 90% CI of the mean number of busses being repaired on each day.
Note that
Margin of Error E = z(alpha/2) * s / sqrt(n)
Lower Bound = X - z(alpha/2) * s / sqrt(n)
Upper Bound = X + z(alpha/2) * s / sqrt(n)
where
alpha/2 = (1 - confidence level)/2 = 0.05
X = sample mean = 82
z(alpha/2) = critical z for the confidence interval = 1.644853627
s = sample standard deviation = 6
n = sample size = 50
Thus,
Margin of Error E = 1.395704584
Lower bound = 80.60429542
Upper bound = 83.39570458
Thus, the confidence interval is
( 80.60429542 , 83.39570458 ) [ANSWER]
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c) Construct a 99% CI of the mean number of busses being repaired on each day
Note that
Margin of Error E = z(alpha/2) * s / sqrt(n)
Lower Bound = X - z(alpha/2) * s / sqrt(n)
Upper Bound = X + z(alpha/2) * s / sqrt(n)
where
alpha/2 = (1 - confidence level)/2 = 0.005
X = sample mean = 82
z(alpha/2) = critical z for the confidence interval = 2.575829304
s = sample standard deviation = 6
n = sample size = 50
Thus,
Margin of Error E = 2.185663641
Lower bound = 79.81433636
Upper bound = 84.18566364
Thus, the confidence interval is
( 79.81433636 , 84.18566364 ) [ANSWER]
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