Question

A sample of 169 cans of coffee showed an average weight of 64 ounces. The standard deviation of the population is known to be 1.8 ounces.

(a) Construct a 68.26% confidence interval for the mean of the population (in ounces). (Round your answers to two decimal places.)

(b) Construct a 97% confidence interval for the mean of the population (in ounces). (Round your answers to two decimal places.)

(c) Discuss why the answers in parts (a) and (b) are different.

Select the correct statement:

A. As the level of confidence increases, the confidence interval becomes wider.

B. As the level of confidence increases, the confidence interval becomes narrower.

C. As the level of confidence increases, the entire confidence interval shifts farther to the left.

D. As the level of confidence increases, the entire confidence interval shifts farther to the right.

E. The answers in parts (a) and (b) are actually the same.

Answer 1

To construct a 68.26% confidence interval, calculate the margin of error and add/subtract it from the sample mean. To construct a 97% confidence interval, use a different critical value for the margin of error. As the level of confidence increases, the confidence interval becomes wider.

Step-by-step explanation:

(a) To construct a 68.26% confidence interval, we need to find the margin of error first. The margin of error can be calculated by multiplying the standard deviation by the critical value, which is 1 for a 68.26% confidence level:

Margin of Error = 1.8 * 1 = 1.8 ounces

To find the confidence interval, we add and subtract the margin of error from the sample mean:

Confidence Interval = 64 +/- 1.8

Confidence Interval = (62.2, 65.8)

(b) To construct a 97% confidence interval, we need to find the margin of error for a 97% confidence level, which is calculated with a critical value of 2.33:

Margin of Error = 1.8 * 2.33 = 4.194 ounces

Confidence Interval = 64 +/- 4.194

Confidence Interval = (59.806, 68.194)

(c) The answers in parts (a) and (b) are different because the level of confidence increases from 68.26% to 97%. As the level of confidence increases, the confidence interval becomes wider. Therefore, the correct statement is A. As the level of confidence increases, the confidence interval becomes wider.