Recall that the owner of a local health food store recently started a new ad campaign to attract more business and wants to know if average daily sales have increased. Historically average daily sales were approximately $2,700. The upper bound of the 95% range of likely sample means for this one-sided test is approximately $2,843.44. If the owner took a random sample of forty-five days and found that daily average sales were now $2,984, what can she conclude at the 95% confidence level?A. Average daily sales have increasedB. Average daily sales have decreasedC. Average daily sales have remained the sameD. Average daily sales have not increasedE. The answer cannot be determined without further information

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".

Data given and notation

[tex]\bar X=2700[/tex] represent the sample mean

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

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

n=45 represent the sample size

95% confidence interval

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 0.95 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: "=-NORM.INV(0.025,0,1)".And we see that [tex]z_{\alpha/2}=1.96[/tex]

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

[tex]2700-1.96\frac{\sigma}{\sqrt{n}}=LL[/tex]

[tex]2700+1.96\frac{\sigma}{\sqrt{n}}=2843.44[/tex]

So on this case the 95% confidence interval would be given by (LL;2843.44)

And we want to conclude about the value of 2984 from the random sample of 45. Since this value is higher than the upper limit of the confidence interval we can conclude that the average daile sales have increased at 5% of significance.

A. Average daily sales have increased

Correct the value tested is higher than the 95% confidence interval already founded.

B. Average daily sales have decreased

False, we don't have enough evidence to conclude this since the average is higher than the upper limit for the confidence interval.

C. Average daily sales have remained the same

False, we have a value outside of the confidence interval.

D. Average daily sales have not increased

False, we have enough evidence to conclude that the average increase since is above the upper limit from the confidence interval

E. The answer cannot be determined without further information

False, we have enough info to conclude something