Question

constructing a confidence interval to test a claim about the population...
a chain of hamburger restaurants claims that the population mean of the wait times in their drive - thru for all customers is 4.74 minutes. you work for a competitor and you want to test that claim. to do so, you select a random sample of 40 of the chain’s drive - thru customers and record the wait time in the drive - thru for each. assume it is known that the population standard deviation of the wait times in the drive - thru for the hamburger chain’s restaurants is 2.91 minutes.
based on your sample, follow the steps below to construct a 90% confidence interval for the population mean of the wait times in the drive - thru for all customers. then state whether the confidence interval you construct contradicts the restaurant chain’s claim. (if necessary, consult a list of formulas.)
(a) click on “take sample” to see the results from your random sample of 40 customers.
there is a “take sample” button. then a table with columns “number of customers”, “sample mean”, “sample standard deviation”, “population standard deviation” and rows with values 40, 4.47, 2.63, 2.91 respectively.
enter the values of the sample size, the point estimate for the population mean, the population standard deviation, and the critical value you need for your 90% confidence interval. (choose the correct critical value from the table of critical values provided.) when you are done, select “compute”.
there is a form with fields “sample size:”, “point estimate:”, “population standard deviation:”, “critical value:” and a “compute” button. also, there are fields “standard error:”, “margin of error:”, “90% confidence interval:”. on the right, there is a table of critical values: $z_{0.005}=2.576$, $z_{0.010}=2.326$, $z_{0.025}=1.960$, $z_{0.050}=1.645$, $z_{0.100}=1.282$
(b) based on your sample, graph the 90% confidence interval for the population mean of the wait times in the drive - thru for all customers.

• enter the lower and upper limits on the graph to show your confidence interval.
• for the point ($\bullet$), enter the restaurant chain’s claim of 4.74 minutes.

there is a graph with “90% confidence interval:” at the top. the x - axis ranges from 0.00 to 10.00 with marks at 0.00, 2.00, 4.00, 5.00, 6.00, 8.00, 10.00.

Explanation:

Step1: Identify key values

Sample size $n=40$, point estimate $\bar{x}=4.47$, $\sigma=2.91$, critical value $z_{0.05}=1.645$ (for 90% CI)

Step2: Calculate standard error

Standard error $SE=\frac{\sigma}{\sqrt{n}}=\frac{2.91}{\sqrt{40}}\approx0.461$

Step3: Calculate margin of error

Margin of error $E=z_{0.05}\times SE=1.645\times0.461\approx0.758$

Step4: Compute confidence interval

Lower limit: $4.47 - 0.758\approx3.71$; Upper limit: $4.47 + 0.758\approx5.23$

Step5: Check claim consistency

Claim value 4.74 is within (3.71,5.23), so no contradiction

Answer:

(a) Sample size: 40; Point estimate: 4.47; Population standard deviation: 2.91; Critical value: 1.645; Standard error≈0.461; Margin of error≈0.758; 90% confidence interval≈(3.71,5.23)
(b) Lower limit≈3.71, Upper limit≈5.23, Point (•) at 4.74
The confidence interval does not contradict the claim.