SOLUTION: The quality control manager at a light bulb factory needs to estimate the mean life of a large shipment of light bulbs. The standard deviation is 75 hours. A random sample of 36 li

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Question 1120034: The quality control manager at a light bulb factory needs to estimate the mean life of a large shipment of light bulbs. The standard deviation is 75 hours. A random sample of 36 light bulbs indicated a sample mean life of 260 hours. Complete parts (a) through (d) below.
a. Construct a 95% confidence interval estimate for the population mean life of light bulbs in this shipment.
The 95% confidence interval estimate is from a lower limit of
nothing hours to an upper limit of
nothing hours.
(Round to one decimal place as needed.)
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
the 95% confidence interval requires a critical z-score of plus or minus 1.959963986.

the z-score formula is:

z = (x - m) / s

z is the z-score.
x is the raw score.
m is the raw mean.
s is the standard error.

your raw mean is 260 hours.
your population standard deviation is 75 hours.
your sample size is 36.

formula for standard error is:

standard error = standard deviation / square root of size.

in your problem, you get:

s = 75 / sqrt(36) = 75 / 6 = 12.5.

use the critical z-scores and the standard error and the mean to find the critical raw scores.

formula is z = (x - m) / s

z = plus or minus 1.959963986.

s = 12.5

on the low side, the z-score formula becomes:

-1.959963986 = (x - 260) / 12.5

solve for x to get x = 235.5004502

on the high side, the z-score formula becomes:

1.959963986 = (x - 260) / 12.5

solve for x to get x = 284.4995498.

that's your 95% confidence interval.

following is a picture of the 95% confidence interval using z-scores.

following is a picture of the 95% confidence interval using raw scores.

in both, the area under the normal distribution curve is shown as .95.

that's the same as 95%.

your confidence interval is the shaded area on the graph.

the critical z-score was found using the inverse norm function of the TI-84 Plus calculator.