Question

3. In the workbook on page 14, we solved the following problem:
A new type of light bulb has been developed which is believed to last longer than
ordinary light bulbs. To determine the average life of the new light bulb, a random sample of
100 light bulbs was tested. The sample had a mean life of 1960 hours. Estimate the true mean
life of the new light bulb using a 97% confidence interval. State your final conclusion using a
clear, complete sentence. Assume the population standard deviation is 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% confidence interval 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 1990.8 hours.
The length of a confidence interval is (upper bound- lower bound). For this
confidence interval, the length is (1990.8-1929.2) = 61.6 hours.
To find the Margin of Error, using the length of a confidence interval, 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 = (uuuuuuuuuu bbbbbbbbbb+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 is centered at the sample
mean.
(a) Now calculate confidence intervals for the mean expectancy of all the new lightbulbs using
confidence levels of 90%, 92%, 94%, 98%, and 99%. Determine the length, the margin of error
and center for each CI. I have filled in the information for the 97% confidence interval we
formed in the workbook.

Confidence level    Confidence Interval Length of CI Margin of Error Center of CI

90%
92%
94%
97% 1929.2 to 1990.8 hours   61.6 hours    30.8 hours 1960 hours
98%
99%

(b) As the confidence level increases, what happens to the width of the confidence interval? Use
clear, complete sentences to state and justify your answer.

(c) As the confidence level increases, what happens to the margin of error? Use clear, complete
sentences to state and justify your answer.

(d) As the confidence level increases, what happens to the center of the confidence interval? Use
clear, complete sentences to state and justify your answer.

Answer 1

(a)

Standard error of mean = = 142 / = 14.2

Center of CI is always the sample mean

90% confidence interval

Z value for 90% confidence interval is 1.645

Margin of error = 1.645 * 14.2 = 23.359

90% confidence interval of average life of the new light bulb is,

(1960 - 23.4 ,  1960 + 23.4)

(1936.6 , 1983.4)

Length of CI = 2 * Margin of error = 2 * 23.4 = 46.8

92% confidence interval

Z value for 92% confidence interval is 1.75

Margin of error = 1.75 * 14.2 = 24.9

92% confidence interval of average life of the new light bulb is,

(1960 - 24.9 ,  1960 + 24.9)

(1935.1 , 1984.9)

Length of CI = 2 * Margin of error = 2 * 24.9 = 46.8

94% confidence interval

Z value for 94% confidence interval is 1.88

Margin of error = 1.88 * 14.2 = 26.7

94% confidence interval of average life of the new light bulb is,

(1960 - 26.7 ,  1960 + 26.7)

(1933.3 , 1986.7 )

Length of CI = 2 * Margin of error = 2 * 26.7 = 46.8

98% confidence interval

Z value for 98% confidence interval is 2.326

Margin of error = 2.326 * 14.2 = 33.0

98% confidence interval of average life of the new light bulb is,

(1960 - 33.0 ,  1960 + 33.0)

(1927 , 1993)

Length of CI = 2 * Margin of error = 2 * 33.0 = 66.0

99% confidence interval

Z value for 99% confidence interval is 2.576

Margin of error = 2.576 * 14.2 = 36.6

99% confidence interval of average life of the new light bulb is,

(1960 - 36.6 ,  1960 + 36.6)

(1923.4 , 1996.6)

Length of CI = 2 * Margin of error = 2 * 36.6 = 73.2

Confidence level	Confidence Interval	Length of CI	Margin of Error	Center of CI
90%	1936.6 to 1983.4 hours	46.8 hours	23.4 hours	1960 hours
92%	1935.1 to 1984.9 hours	49.8 hours	24.9 hours	1960 hours
94%	1933.3 to 1986.7 hours	53.4 hours	26.7 hours	1960 hours
97%	1929.2 to 1990.8 hours	61.6 hours	30.8 hours	1960 hours
98%	1927 to 1993 hours	66.0 hours	33.0 hours	1960 hours
99%	1923.4 to 1996.6 hours	73.2 hours	36.6 hours	1960 hours

(b)

From the table,

As the confidence level increases, the z score increases and hence the width of the confidence interval increases.

(c)

From the table,

As the confidence level increases, the z score increases and hence the margin of error increases.

(d)

From the table,

As the confidence level increases, the the center of the confidence interval is always same and is equal to the simple mean. This is because the average of lower and upper limit of confidence interval is sample mean.