In: Math
Thirty-three 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 41.5 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 |
Solution :
Given that,
= 138.5
= 41.5
n = 33
(a)
At 90% confidence level the z is ,
= 1 - 90% = 1 - 0.90 = 0.10
/ 2 = 0.10 / 2 = 0.05
Z/2
= Z0.05 = 1.645
Margin of error = E = Z/2* (
/
n)
= 1.645 * (41.5 / 33)
= 11.9
Margin of error = E = 11.9
At 90% confidence interval estimate of the population mean is,
- E <
<
+ E
138.5 - 11.9 < < 138.5 + 11.9
126.6 < < 150.4
Lower limit = 126.6
Upper limit = 150.4
(b)
At 95% confidence level the z is ,
= 1 - 95% = 1 - 0.95 = 0.05
/ 2 = 0.05 / 2 = 0.025
Z/2
= Z0.025 = 1.96
Margin of error = E = Z/2* (
/
n)
= 1.96 * (41.5 / 33)
= 14.2
Margin of error = E = 14.2
At 90% confidence interval estimate of the population mean is,
- E <
<
+ E
138.5 - 14.2 < < 138.5 + 14.2
124.3 < < 152.7
Lower limit = 124.3
Upper limit = 152.7
(C)
At 99% confidence level the z is ,
= 1 - 99% = 1 - 0.99 = 0.01
/ 2 = 0.01 / 2 = 0.005
Z/2
= Z0.005 = 2.576
Margin of error = E = Z/2* (
/
n)
= 2.576 * (41.5 / 33)
= 18.6
Margin of error = E = 18.6
At 99% confidence interval estimate of the population mean is,
- E <
<
+ E
138.5 - 18.6 < < 138.5 + 18.6
119.9 < < 157.1
Lower limit = 119.9
Upper limit = 157.1