3. In the workbook on page 14, we solved the followingproblem:
A new type of light bulb has been developed which is believed tolast longer than
ordinary light bulbs. To determine the average life of the newlight bulb, a random sample of
100 light bulbs was tested. The sample had a mean life of 1960hours. Estimate the true mean
life of the new light bulb using a 97% confidence interval. Stateyour final conclusion using a
clear, complete sentence. Assume the population standard deviationis 142 hours.
For the confidence interval we calculated in this workbook problem,the confidence level
is 97%.
Using ZInterval on the calculator we found the 97% confidenceinterval for the mean life
was 1929.2 to 1990.8 hours. Based on this sample, we are 97%confident that the mean life
expectancy of all the new lightbulbs is between 1929.2 and 1990.8hours.
The length of a confidence interval is (upper bound- lower bound).For this
confidence interval, the length is (1990.8-1929.2) = 61.6hours.
To find the Margin of Error, using the length of a confidenceinterval, we use
EE = mmmmmmmmmmmm oooo eeeeeeeeee = llllnnnnnnnn oooo CCCC
22 . For this CI, EE = 61.6
2 = 30.8 hoooooooo.
To find the center of the CI, CCCCCCCCCCCC = (uuuuuuuuuubbbbbbbbbb+llllllllll bbbbbbbbbb)
2
. The center of this CI is
CCCCCCCCCCCC = uuuuuuuuuu bbbbbbbbbb+llllllllll bbbbbbbbbb
2 = 1990.8+1929.2
2 = 3920
2 = 1960 hoooooooo, which is the sample
mean. Remember every confidence interval for the population mean iscentered at the sample
mean.
(a) Now calculate confidence intervals for the mean expectancy ofall the new lightbulbs using
confidence levels of 90%, 92%, 94%, 98%, and 99%. Determine thelength, the margin of error
and center for each CI. I have filled in the information for the97% confidence interval we
formed in the workbook.
Confidence level   Confidence Interval Length of CIMargin of Error Center of CI
90%
92%
94%
97% 1929.2 to 1990.8 hours   61.6 hours   30.8hours 1960 hours
98%
99%
(b) As the confidence level increases, what happens to the width ofthe confidence interval? Use
clear, complete sentences to state and justify your answer.
(c) As the confidence level increases, what happens to the marginof error? Use clear, complete
sentences to state and justify your answer.
(d) As the confidence level increases, what happens to the centerof the confidence interval? Use
clear, complete sentences to state and justify youranswer.
