Question 1083604: You are estimating the cost of engine overhauls. A sample of 99 repairs showed an average overhaul of 285 hours with a standard deviation of 60 hours. Calculate a 95% confidence interval for the average overhaul. (Carry intermediate calculations to three decimal places.)
271.00 ≤ μ ≤ 299.00
273.18 ≤ μ ≤ 296.82
268.20 ≤ μ ≤ 301.80
281.61 ≤ μ ≤ 288.39
Found 2 solutions by Boreal, jim_thompson5910:
Answer by Boreal(15235)   (Show Source):
Answer by jim_thompson5910(35256)  (Show Source):
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Answer: Choice B) 273.18 ≤ μ ≤ 296.82
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Explanation:
At 95% confidence, the critical z value is z = 1.960
See this table.
Look at the bottom where it shows "95%" in the confidence level section. Then look at the value just above the "95%" and you'll see 1.960
So in short, z = 1.960
The sample mean is xbar = 285 which is given
The sample standard deviation is s = 60.
The sample size is given to be n = 99.
We don't know the population standard deviation sigma, but since n > 30 we can use the standard normal Z distribution.
The sample standard deviation s approximates the population standard deviation.
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Let's use the four values mentioned above (z = 1.960, xbar = 285, s = 60, n = 99) to compute the upper and lower endpoints of the confidence interval
L = lower endpoint of confidence interval
L = xbar - z*s/sqrt(n)
L = 285 - 1.96*60/sqrt(99)
L = 285 - 117.6/sqrt(99)
L = 285 - 117.6/9.95
L = 285 - 11.819
L = 273.181
which rounds to 273.18
U = upper endpoint of confidence interval
U = xbar + z*s/sqrt(n)
U = 285 + 1.96*60/sqrt(99)
U = 285 + 117.6/sqrt(99)
U = 285 + 117.6/9.95
U = 285 + 11.819
U = 296.819
which rounds to 296.82
The confidence interval is therefore (L, U) = (273.18, 296.82)
Put another way, we are 95% confident that the population mean mu is between 273.18 and 296.82. We can write it like so
273.18 ≤ μ ≤ 296.82
which is the final answer.
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