Question

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.

Answer 1

The 90% confidence interval would be given by (110.168;117.832)

The 95% confidence interval would be given by (119.393;118.607)

Explanation:

Assuming this info:" A random sample of 45 home theater systems has a mean price of ​$114.00. Assume the population standard deviation is ​$15.30. Construct a​ 90% and 95 confidence interval for the population mean".

1) Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

`\bar X=114` represent the sample mean for the sample

`\mu` population mean (variable of interest)

s=15.30 represent the sample standard deviation

n=45 represent the sample size

2) 90% confidence interval

The confidence interval for the mean is given by the following formula:

`\bar X \pm t_(\alpha/2)(s)/(√(n))` (1)

The degrees of freedom are given by:

`df=n-1=45-1=44`

Since the Confidence is 0.90 or 90%, the value of
`\alpha=0.1` and
`\alpha/2 =0.05`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.05,44)".And we see that
`t_(\alpha/2)=1.68`

Now we have everything in order to replace into formula (1):

`114-1.68(15.30)/(√(45))=110.168`

`114+1.68(15.30)/(√(45))=117.832`

So on this case the 90% confidence interval would be given by (110.168;117.832)

3) 95% confidence interval

Since the Confidence is 0.95 or 95%, the value of
`\alpha=0.05` and
`\alpha/2 =0.025`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,44)".And we see that
`t_(\alpha/2)=2.02`

Now we have everything in order to replace into formula (1):

`114-2.02(15.30)/(√(45))=119.393`

`114+2.02(15.30)/(√(45))=118.607`

So on this case the 95% confidence interval would be given by (119.393;118.607)