Question

A restaurant in London is in its first 12 months of business. The year was somewhat successful, with the owner averaging $1,000 net profit per month after all deductions and expenses. However, he is unhappy with the amount of time it has been taking staff to provide customer service (measured from the time the customer enters to the moment they leave with their food). The restaurant design team was convinced that the restaurant’s design would allow for an average delivery time of approximately 7 minutes. As the owner was concerned they might not be hitting the mark, the design team asked a consultant to conduct a small study to examine customer service delivery time. A sample of 20 restaurant customers was selected, and the delivery time was recorded to the nearest minute.

Minutes:

5, 6, 7, 10, 6, 4, 7, 6, 8, 5, 11, 8, 7, 9, 7, 8, 8, 7, 9, 3

(A) Does the empirical evidence suggest that it takes significantly longer on average to service customers than the 7 minutes anticipated by the design team? Construct a 90% confidence interval a estimate of the average customer service delivery time at this restaurant. Interpret the meaning of the interval. Ensure that the interpretation of the results addresses the owner’s concerns. Please use clear, easy-to-understand, non-technical language.

(B) How would the consultant explain what they did in (A) to the restaurant’s design team - if they do not understand technical statistical language, and you have to explain in an easy-to-understand way?

Answer 1

A)

From visual analysis, we can see that majority of the delivery time was between 7 and 8 minutes. There are also delivery times of 10 and 11 minutes.

Hence, empirical evidence does suggest that it takes significantly longer on average to service customers than the 7 minutes,

90% Confidence interval

Let be the population mean delivery time of the restaurant.

From the given data we calculate the sample mean and sample standard deviation using Excel functions Average and stdev.s

We get,

X̅ = 7.05           ....... Sample Mean
n = 20             ....... Sample Size
s = 1.9595           ....... Sample Standard Deviation

Since the population standard deviation is unknown, we use the t-distribution

For 90% Confidence interval

α = 0.1,      α/2 = 0.05
From t tables of Excel function T.INV.2T (α, degrees of freedom) we find the t value
t = T.INV.2T (0.1, 19) = 1.729
We take the positive value of t

Confidence interval is given by


= (6.2924, 7.8076)

90% Confidence interval is (6.29, 7.81) minutes

Interpretation :

If similar samples as mentioned in the problem are taken , then 90% of the times the population mean of delivery time will lie within 6.29 and 7.81 minutes

Since the required average delivery time of 7 minutes is within the 90% confidence interval (6.29, 7.81) minutes, the owner need not be very worried about meeting the delivery time of 7 minutes.

B)

The consultant would explain as follows :

If we take the 'n' samples of 20 delivery times, with all other conditions remaining similar for all the samples, then 90% of the time the average delivery time of the restaurant will be between the 6.29, 7.81 minutes.

Since the concern is about meeting the target of 7 minutes delivery time, the owner should not be highly worried about meeting the target since 7 minutes is part of the confidence interval. Yet the owner can take some corrective actions to meet the remaining 10% of the times when the average delivery time may go beyond 7 minutes.