Question

The manager of Petco, a nationwide chain of pet supply stores, wants to study characteristics of the customers, particularly the amount of money spent by customers and whether the customers own only one dog, only one car, or more than on dog and/or cat. The results from a sample of 70 customers are as follows: Amount of money spent: LaTeX: X-bar\:=\:\text{21.34}X − b a r = 21.34 LaTeX: S\:=\:\text{9.22}S = 9.22 Thirty-seven customers own only a dog. Twenty-six customers own only a cat. Seven customers own more than one dog and/or cat. Construct a 95% confidence interval estimate for the population mean amount spent in the pet supply store.

Answer 1

The 95% confidence interval would be given by (19.141;23.538)

Explanation:

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=21.34` represent the sample mean for the sample

`\mu` population mean (variable of interest)

s=9.22 represent the sample standard deviation

n=70 represent the sample size

2) Confidence interval

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

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

In order to calculate the critical value
`t_(\alpha/2)` we need to find first the degrees of freedom, given by:

`df=n-1=70-1=69`

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(0.025,69)".And we see that
`t_(\alpha/2)=1.994`

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

`21.74-1.994(9.22)/(√(70))=19.141`

`21.74+1.994(9.22)/(√(70))=23.538`

So on this case the 95% confidence interval would be given by (19.141;23.538)