Question

suppose a retailer claims that the average wait time for a customer on its support line is 179 seconds. a random sample of 47 customers had an average wait time of 171 seconds. assume the population standard deviation for wait time is 47 seconds. using a 95% confidence interval, does this sample support the retailers claim?
using a 95% confidence interval, does this sample support the retailers claim? select the correct choice below, and fill in the answer boxes to complete your choice.
(round to two decimal places as needed.)
a. yes, because the retailers claim is between the lower limit of seconds and the upper limit of seconds for the mean wait time.
b. no, because the retailers claim is not between the lower limit of seconds and the upper limit of seconds for the mean wait time.

Explanation:

Step1: Identify the formula for confidence - interval

For a 95% confidence interval when the population standard - deviation $\sigma$ is known, the formula for the confidence interval of the population mean $\mu$ is $\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $z_{\alpha/2}$ is the z - score, $\sigma$ is the population standard deviation, and $n$ is the sample size. For a 95% confidence interval, $\alpha = 1 - 0.95=0.05$, and $\alpha/2 = 0.025$. The $z$ - score $z_{\alpha/2}=z_{0.025}=1.96$.

Step2: Calculate the margin of error $E$

We are given that $\bar{x} = 171$, $\sigma = 47$, and $n = 47$. The margin of error $E=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}=1.96\times\frac{47}{\sqrt{47}}$. First, $\sqrt{47}\approx6.86$. Then $\frac{47}{\sqrt{47}}=\frac{47}{6.86}\approx6.85$. And $E = 1.96\times6.85\approx13.43$.

Step3: Calculate the lower and upper limits of the confidence interval

The lower limit $LL=\bar{x}-E=171 - 13.43 = 157.57$. The upper limit $UL=\bar{x}+E=171 + 13.43 = 184.43$.
The retailer's claim is $\mu = 179$, and $157.57<179<184.43$.

Answer:

A. Yes, because the retailer's claim is between the lower limit of 157.57 seconds and the upper limit of 184.43 seconds for the mean wait time.