Question

In an effort to estimate the mean dollars spent per visit by customers of a food store, the manager has selected a random sample of 100 cash register receipts. The mean of these was $45.67 with a sample standard deviation equal to $12.30. Assuming that he wants to develop a 90 percent confidence interval estimate, the upper limit of the confidence interval estimate is: Question 47 options: about $47.69 about $2.02 approximately $65.90 None of the above

Answer 1

Answer:

$45.67-1.66\frac{12.30}{\sqrt{100}}=43.63$    

$45.67+1.66\frac{12.30}{\sqrt{100}}=47.69$    

And the best answer for this case would be:

About $47.69

Step-by-step explanation:

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".

$\bar X=45.67$ represent the sample mean

$\mu$ population mean (variable of interest)

s=12.3 represent the sample standard deviation

n=100 represent the sample size  

Solution to the problem

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

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

In order to calculate the critical value $t_{\alpha/2}$ we need to find first the degrees of freedom, given by:

$df=n-1=100-1=99$

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(0.05,99)".And we see that $t_{\alpha/2}=1.66$

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

$45.67-1.66\frac{12.30}{\sqrt{100}}=43.63$    

$45.67+1.66\frac{12.30}{\sqrt{100}}=47.69$    

And the best answer for this case would be:

About $47.69