You are given the sample mean and the population standard de

You are given the sample mean and the population standard deviation. Use this information to construct the 90% and 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. If convenient, use technology to construct the confidence intervals.

A random sample of 45 home theater systems has a mean price of $138.00. Assume the population standard deviation is $16.60.

Construct a 90% confidence interval for the population mean.

The 90% confidence interval is (___,___)
(Round to two decimal places as needed.)

Construct a 95% confidence interval for the population mean.

The 95% confidence interval is (____,____)
(Round to two decimal places as needed.)

Interpret the results. Choose the correct answer below.

A. With 90% confidence, it can be said that the sample mean price lies in the first interval. With 95% confidence, it can be said that the sample mean price lies in the second interval. The 95% confidence interval is wider than the 90%.

B. With 90% confidence, it can be said that the population mean price lies in the first interval. With 95% confidence, it can be said that the population mean price lies in the second interval. The 95% confidence interval is wider than the 90%.

C. With 90% confidence, it can be said that the population mean price lies in the first interval. With 95% confidence, it can be said that the population mean price lies in the second interval. The 95% confidence interval is narrower than the 90%.

Solution

a)
AT 90% CONFIDENCE INTERVAL
CI = x ± Z a/2 * (sd/ Sqrt(n))
Where,
x = Mean
sd = Standard Deviation
a = 1 - (Confidence Level/100)
Za/2 = Z-table value
CI = Confidence Interval
Mean(x)=138
Standard deviation( sd )=16.6
Sample Size(n)=45
Confidence Interval = [ 138 ± Z a/2 ( 16.6/ Sqrt ( 45) ) ]
= [ 138 - 1.64 * (2.475) , 138 + 1.64 * (2.475) ]
= [ 133.94,142.06 ]

b)
AT 95% CONFIDENCE INTERVAL
Confidence Interval = [ 138 ± Z a/2 ( 16.6/ Sqrt ( 45) ) ]
= [ 138 - 1.96 * (2.475) , 138 + 1.96 * (2.475) ]
= [ 133.15,142.85 ]

c)
B. With 90% confidence, it can be said that the population mean price lies in the first interval. With 95% confidence, it can be said that the population mean price lies in the second interval. The 95% confidence interval is wider than the 90%