Question

One can calculate the 95% confidence interval for the mean with the population standard deviation known. This will give us an upper and a lower confidence limit. What happens if we decide to calculate the 99% confidence interval? Describe how the increase in the confidence level has changed the width of the confidence interval. Do the same for the confidence interval set at 80%. Include an example with actual numerical values for the intervals in your post to help with your explanations.

Answer 1

50 people have average balance in their bank account $16000 with population standard deviation of $3500.

make 95 % confidence interval.

x?= 16000

sigma= 3500

n= 50

alpha=0.05 then Z(alpha/2)= 1.96

Margin of error E= Z(alpha/2)*sigma/sqrt(n)=970.1505

95% Confidence interval for population mean =(x?-E,x?+E)

lower bound= 15029.8495

upper bound= 16970.1505

now construct 99% CI for the above problem

x?= 16000

sigma= 3500

n= 50

alpha=0.01 then Z(alpha/2)=2.576

Margin of error E=Z(alpha/2)*sigma/sqrt(n)=1275.054948

99% Confidence interval for population mean =(x?-E,x?+E)

lower bound= 14724.94505

upper bound= 17275.05495

now contruct 80 % CI for the above problem

x?= 16000

sigma= 3500

n= 50

alpha=0.2 then Z(alpha/2)= 1.282

Margin of error E= Z(alpha/2)*sigma/sqrt(n)=634.5576254

80% Confidence interval for population mean =(x?-E,x?+E)

rounded lower bound= 15365.4424

rounded upper bound= 16634.5576

we noticed from the above three case that increasing confidence level make interval wider. and decreasing confidence level make intervel narrow.

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