# Question

The Snow Pea Restaurant is a Chinese carryout/delivery restaurant. Most of Snow Pea’s deliveries are within a 10-mile radius, but it occasionally delivers to customers more than 10 miles away. Snow Pea employs a number of delivery people, four of whom are relatively new hires. The restaurant has recently been receiving customer complaints about excessively long delivery times. Therefore, Snow Pea has collected data on a random sample of deliveries by its four new delivery people during the peak dinner time. The data are in the file Delivery Times.xlsx.

The variables are as follows:

• Deliverer: which person made the delivery?

• Prep Time: time from when order was placed until delivery person started driving it to the

Customer

• Travel Time: time to drive from Snow Pea to customer

• Distance: distance (miles) from Snow Pea to customer

Solve the following problems and then, based on your analysis, write a report that makes reasonable recommendations to Snow Pea management.

1. Snow Pea is concerned that one or more of the new delivery people might be slower than others.

a. Let µDi and µTi be the mean delivery time and mean total time for delivery person i, where the total time is the sum of the delivery and prep times. Find 95% confidence intervals for each of these means for each delivery person. Although these might be interesting, give two reasons why they are not really fair measures for comparing the efficiency of the delivery people.

b. Responding to the criticisms in part a, find a 95% confidence interval for the mean speed of delivery for each delivery person, where speed is measured as miles per hour during the trip from Snow Pea to the customer. Then find 95% confidence intervals for the mean difference in speed between each pair of delivery people.

2. Snow Pea would like to advertise that it can achieve a total delivery time of no more than M minutes for all customers within a 10-mile radius. On all orders that take more than M minutes, Snow Pea will give the customers a $10 certificate on their next purchase.

a. Assuming for now that the delivery people in the sample are representative of all of Snow Pea’s delivery people, find a 95% confidence interval for the proportion of deliveries (within the 10-mile limit) that will be on time if M25 minutes; if M30 minutes; if M35 minutes.

b. Suppose Snow Pea makes 1000 deliveries within the 10-mile limit. For each of the values of M in part a, find a 95% confidence interval for the total dollar amount of certificates it will have to pay for being late.

3. The policy in the previous problem is simple to state and simple to administer. However, it is somewhat unfair to customers who live close to Snow Pea—they will never get $10 certificates.

A fairer, but more complex, policy is the following. Snow Pea first analyzes the data and finds that total delivery times can be predicted fairly well with the equation Predicted Delivery Time 14.8 + 2.06*Distance. Also, most of these predictions are within 5 minutes of the actual delivery times. Therefore, whenever Snow Pea receives an order over the phone, it looks up the customer’s address in its computerized geographical database to find distance, calculates the predicted delivery time based on this equation, rounds this to the nearest minute, adds 5 minutes, and guarantees this delivery time or else a $10 certificate. It does this for all customers, even those beyond the 10-mile limit.

a. Assuming again that the delivery people in the sample are representative of all of Snow Pea’s delivery people, find a 95% confidence interval for the proportion of all deliveries that will be within the guaranteed total delivery time.

b. Suppose Snow Pea makes 1000 deliveries. Find a 95% confidence interval for the total dollar amount of certificates it will have to pay for being late.

The variables are as follows:

• Deliverer: which person made the delivery?

• Prep Time: time from when order was placed until delivery person started driving it to the

Customer

• Travel Time: time to drive from Snow Pea to customer

• Distance: distance (miles) from Snow Pea to customer

Solve the following problems and then, based on your analysis, write a report that makes reasonable recommendations to Snow Pea management.

1. Snow Pea is concerned that one or more of the new delivery people might be slower than others.

a. Let µDi and µTi be the mean delivery time and mean total time for delivery person i, where the total time is the sum of the delivery and prep times. Find 95% confidence intervals for each of these means for each delivery person. Although these might be interesting, give two reasons why they are not really fair measures for comparing the efficiency of the delivery people.

b. Responding to the criticisms in part a, find a 95% confidence interval for the mean speed of delivery for each delivery person, where speed is measured as miles per hour during the trip from Snow Pea to the customer. Then find 95% confidence intervals for the mean difference in speed between each pair of delivery people.

2. Snow Pea would like to advertise that it can achieve a total delivery time of no more than M minutes for all customers within a 10-mile radius. On all orders that take more than M minutes, Snow Pea will give the customers a $10 certificate on their next purchase.

a. Assuming for now that the delivery people in the sample are representative of all of Snow Pea’s delivery people, find a 95% confidence interval for the proportion of deliveries (within the 10-mile limit) that will be on time if M25 minutes; if M30 minutes; if M35 minutes.

b. Suppose Snow Pea makes 1000 deliveries within the 10-mile limit. For each of the values of M in part a, find a 95% confidence interval for the total dollar amount of certificates it will have to pay for being late.

3. The policy in the previous problem is simple to state and simple to administer. However, it is somewhat unfair to customers who live close to Snow Pea—they will never get $10 certificates.

A fairer, but more complex, policy is the following. Snow Pea first analyzes the data and finds that total delivery times can be predicted fairly well with the equation Predicted Delivery Time 14.8 + 2.06*Distance. Also, most of these predictions are within 5 minutes of the actual delivery times. Therefore, whenever Snow Pea receives an order over the phone, it looks up the customer’s address in its computerized geographical database to find distance, calculates the predicted delivery time based on this equation, rounds this to the nearest minute, adds 5 minutes, and guarantees this delivery time or else a $10 certificate. It does this for all customers, even those beyond the 10-mile limit.

a. Assuming again that the delivery people in the sample are representative of all of Snow Pea’s delivery people, find a 95% confidence interval for the proportion of all deliveries that will be within the guaranteed total delivery time.

b. Suppose Snow Pea makes 1000 deliveries. Find a 95% confidence interval for the total dollar amount of certificates it will have to pay for being late.

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