A random sample of 64 credit sales in a department store showed an average sale of $75.00. From past data, it is known that the standard deviation of the population of sales is $24.00.
1. Construct the 96% confidence interval for the population mean and provide the lower and upper bound of the confidence interval.
2. Assume that the information about average sales was lost. You decided to repeat the study and collect information for another sample. This time you obtain a random sample of 400 sales which accidentally resulted in the same average sale of $75.00. Construct the 96% confidence interval for the population mean and provide the lower and upper bound of the confidence interval.

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= 75[/tex] represent the sample mean

[tex]\mu[/tex] population mean (variable of interest)

[tex]\sigma=24[/tex] represent the population standard deviation

n=64 represent the sample size

Part a

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

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

Since the Confidence is 0.96 or 96%, the value of [tex]\alpha=0.04[/tex] and [tex]\alpha/2 =0.02[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.02,0,1)".And we see that [tex]z_{\alpha/2}=2.054[/tex]

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

[tex]75-2.054\frac{24}{\sqrt{64}}=68.838[/tex]

[tex]75+2.054\frac{24}{\sqrt{64}}=81.162[/tex]

So on this case the 96% confidence interval would be given by (68.838;81.162)

Part b

[tex]75-2.054\frac{24}{\sqrt{400}}=72.535[/tex]

[tex]75+2.054\frac{24}{\sqrt{400}}=77.465[/tex]

So on this case the 96% confidence interval would be given by (72.535;77.465)