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.

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".

[tex]\bar X=114[/tex] represent the sample mean for the sample

[tex]\mu[/tex] 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:

[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (1)

The degrees of freedom are given by:

[tex]df=n-1=45-1=44[/tex]

Since the Confidence is 0.90 or 90%, the value of [tex]\alpha=0.1[/tex] and [tex]\alpha/2 =0.05[/tex], 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 [tex]t_{\alpha/2}=1.68[/tex]

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

[tex]114-1.68\frac{15.30}{\sqrt{45}}=110.168[/tex]

[tex]114+1.68\frac{15.30}{\sqrt{45}}=117.832[/tex]

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 [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], 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 [tex]t_{\alpha/2}=2.02[/tex]

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

[tex]114-2.02\frac{15.30}{\sqrt{45}}=119.393[/tex]

[tex]114+2.02\frac{15.30}{\sqrt{45}}=118.607[/tex]

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