Question

 

Number of Customers	Waiting Time (Minutes)
5	47
2	27
4	36
4	44
6	52
3	21
7	82
3	43
8	69
4	28

Many small restaurants in Portland, Oregon, and other cities across the United States do not take reservations. Owners say that with smaller capacity, no-shows are costly, and they would rather have their staff focused on customer service rather than maintaining a reservation system.† However, it is important to be able to give reasonable estimates of waiting time when customers arrive and put their name on the waiting list. The file RestaurantLine contains 10 observations of number of people in line ahead of a customer (independent variable x) and actual waiting time (in minutes) (dependent variable y). The estimated regression equation is:

ŷ = 4.35 + 8.81x

and MSE = 94.42.

(a)

Develop a point estimate (in min) for a customer who arrives with four people on the wait-list. (Round your answer to two decimal places.)

ŷ^* =  min

(b)

Develop a 95% confidence interval for the mean waiting time (in min) for a customer who arrives with four customers already in line. (Round your answers to two decimal places.)

min to  min

(c)

Develop a 95% prediction interval for Roger and Sherry Davy's waiting time (in min) if there are four customers in line when they arrive. (Round your answers to two decimal places.)

min to  min

(d)

Discuss the difference between part (b) and part (c).

The prediction interval is much  ---Select--- wider narrower than the confidence interval. This is because it is  ---Select--- more less difficult to predict the waiting time for an individual customer arriving with four people in line than it is to estimate the mean waiting time for a customer arriving with four people in line.

Answer 1

Given that number of people in line ahead of a customer (independent variable x) and actual waiting time (in minutes) (dependent variable y). The sample data is

Number of Customers (x)	Waiting Time (Minutes) (y)
5	47
2	27
4	36
4	44
6	52
3	21
7	82
3	43
8	69
4	28

We calculate the following

The estimated regression equation is:

MSE=94.42

The estimated standard error of regression is

(a) Develop a point estimate (in min) for a customer who arrives with four people on the wait-list. (Round your answer to two decimal places.)

Using the regression equation, the predicted value of y for is

ans: ŷ* = 39.59 min

(b) Develop a 95% confidence interval for the mean waiting time (in min) for a customer who arrives with four customers already in line. (Round your answers to two decimal places.)

The standard error of expected value of y is

95% confidence level corresponds to a significance level

The right tail critical value is

The degrees of freedom for t are n-2=10-2=8

Using the t tables for df=8 and the area under the curve=0.025, we get

The 95% confidence interval is

ans: 32.12 min to 47.06 min

(c) Develop a 95% prediction interval for Roger and Sherry Davy's waiting time (in min) if there are four customers in line when they arrive. (Round your answers to two decimal places.)

The standard error of a predicted value of y is

The 95% prediction interval is

ans: 15.97  min to 63.21 min

(d) Discuss the difference between part (b) and part (c).

We can see that the width of the prediction interval is 63.21-15.97=47.24 and it s greater than

the width of the confidence interval, which is 47.06-32.12 = 14.94

ans: The prediction interval is much wider than the confidence interval. This is because it is more difficult to predict the waiting time for an individual customer arriving with four people in line than it is to estimate the mean waiting time for a customer arriving with four people in line.