Question

the trade magazine qsr routinely checks the drive - through service times of fast - food restaurants. a 90% confidence interval that results from examining 781 customers in taco bells drive - through has a lower bound of 153.6 seconds and an upper bound of 156.8 seconds. complete parts (a) through (c)
(a) what is the mean service time from the 781 customers?
the mean service time from the 781 customers is $square$ seconds
(type an integer or a decimal. do not round.)
(b) what is the margin of error for the confidence interval?
the margin of error is $square$ seconds.
(type an integer or a decimal. do not round.)
(c) interpret the confidence interval.
select the correct choice below and fill in the answer boxes to complete your choice
(type integers or decimals. do not round.)
oa. there is a $square\\%$ probability that the mean drive - through service time of taco bell is between $square$ seconds and $square$ seconds
ob. one can be $square\\%$ confident that the mean drive - through service time of taco bell is $square$ seconds.
oc. the mean drive - through service time of taco bell is $square$ seconds $square\\%$ of the time
od. one can be $square\\%$ confident that the mean drive - through service time of taco bell is between $square$ seconds and $square$ seconds

Explanation:

(a)

Step1: Recall the formula for the confidence interval

The confidence interval is given by \(\bar{x}-E\) (lower bound) and \(\bar{x} + E\) (upper bound), where \(\bar{x}\) is the sample mean and \(E\) is the margin of error. Also, \(\bar{x}=\frac{(\text{lower bound}+\text{upper bound})}{2}\)

Step2: Calculate the sample mean

Substitute the lower bound \(L = 153.6\) and upper bound \(U=156.8\) into the formula \(\bar{x}=\frac{153.6 + 156.8}{2}=\frac{310.4}{2}=155.2\)

(b)

Step1: Recall the formula for the margin of error

\(E=\bar{x}-L\) (or \(E = U-\bar{x}\)). Since \(\bar{x}=155.2\) and \(L = 153.6\)

Step2: Calculate the margin of error

\(E=155.2-153.6 = 1.6\)

(c)

Step1: Recall the interpretation of a confidence interval

A \(C\%\) confidence interval means that if we were to take many samples and construct confidence intervals in the same way, approximately \(C\%\) of those intervals would contain the True population parameter. In the case of a \(90\%\) confidence interval for the mean \(\mu\) (service time), we can say that one can be \(90\%\) confident that the True population mean \(\mu\) lies within the calculated interval \((153.6,156.8)\)

Answer:

(a)

155.2

(b)

1.6

(c)

D. One can be 90% confident that the mean drive - through service time of Taco Bell is between 153.6 seconds and 156.8 seconds