Question

Thirty-two small communities in Connecticut (population near 10,000 each) gave an average of x = 138.5 reported cases of larceny per year. Assume that σ is known to be 43.7 cases per year.

(a) Find a 90% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

lower limit
upper limit
margin of error

(b) Find a 95% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

lower limit
upper limit
margin of error

(c) Find a 99% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

lower limit
upper limit
margin of error

Answer 1

Solution:
Given in the question
Number of sample = 32
Sample mean = 138.5
Population standard deviation() = 43.7
Solution(a)
90% confidence interval can be calculated as
Mean +/- Zalpha/2*/sqrt(n)
here alpha = 0.1, alpha/2 = 0.05, from Z table we found Zalpha/2 = 1.645
138.5 +/- 1.645*43.7/sqrt(32)
138.5 +/- 12.7
So 90% confidence interval is 125.8 to 151.2
Lower limit = 125.8
Upper limit = 151.2
Margin of error = Zalpha/2*/sqrt(n) = 1.645*43.7/sqrt(32) = 12.7
Solution(b)
95% confidence interval can be calculated as
Mean +/- Zalpha/2*/sqrt(n)
here alpha = 0.05, alpha/2 = 0.025, from Z table we found Zalpha/2 = 1.96
138.5 +/- 1.96*43.7/sqrt(32)
138.5 +/- 15.1
So 90% confidence interval is 123.4 to 153.6
Lower limit = 123.4
Upper limit = 153.6
Margin of error = Zalpha/2*/sqrt(n) = 1.96*43.7/sqrt(32) = 15.1
Solution(c)
99% confidence interval can be calculated as
Mean +/- Zalpha/2*/sqrt(n)
here alpha = 0.01, alpha/2 = 0.005, from Z table we found Zalpha/2 = 2.575
138.5 +/- 2.575*43.7/sqrt(32)
138.5 +/- 19.9
So 90% confidence interval is 118.6 to 158.4
Lower limit = 118.6
Upper limit = 158.4
Margin of error = Zalpha/2*/sqrt(n) = 2.575*43.7/sqrt(32) = 19.9