Question

Instructions: Have students apply critical thinking to the below problems. Each problem is worth two points. A partial point can be given if the student only answered one part of the question correctly. Maximum Points: 8 points (2 points each) Playbill magazine reported that the mean annual household income of its readers is $119,155 (Playbill, January 2006). Assume this estimate of the mean annual household income is based on a sample of 80 households, and, based on past studies the population standard deviation is known to be o = $30,000.00. a. Developed a 90% confidence interval estimate of the population mean b. Developed a 95% confidence interval estimate of the population mean. c. Developed a 99% confidence interval estimate of the population mean d. Discuss what happens to the width of the confidence interval as the confidence level is increased. Does this result seem reasonable? Explain.

Answer 1

a)

We know that the 90% confidence interval for population mean is given by:

(+ Zo 10/2 - AR

30000 (119155 20.05 V 80

30000 (119155 = (1.645) V 80

(119155 5517.4977)

(113637.5023.124672.4977)

b)

We know that the 95% confidence interval for population mean is given by:

(T+Z0,05/ 26 vn

30000 (119155 – 20.025 V

(30000、 (119155 + (1.96) 。 v80

(119155 +6574.0399)

(112580.9601, 125729.0399)

c)

We know that the 99% confidence interval for population mean is given by:

c/T02 王)

30000 (119155 – 20.005 V 80

30000 (119155 + (2.5758) V 80

(119155 + 8639.4958)

​​​​​​​

(110515.5042.127794.4958)

d)

We can see that as the confidence level increases, the width of the confidence interval also increases.