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Tony has asked you for some assistance in interpreting the data that he has collected. In particular, he needs to know if the true average delivery time for Pronto Pizza is greater than 25 minutes. Use the data in the file PRONTO.XLSX to answer his question. A description of this data set is given in the Data Description section. Also, examine the data for further information that might help Tony in making his decision about the 29-minute delivery guarantee and in improving his pizza delivery service. The Case Questions will assist you in your analysis of the data. Use important details from your analysis to support your recommendations.

1. Is there sufficient evidence in the data to conclude that the average time to deliver a pizza, once the order is placed, is greater than 25 minutes? Perform an appropriate statistical analysis and attach any supporting output.

2. What percentage of the time will Pronto fail to meets its guaranteed time of 29 minutes or less? Explain how you arrived at your answer and discuss any assumptions that you are making. Will they meet their requirement of failing to meet the guarantee 5% of the time or less?

3. Does the day of the week have an effect on delivery time? Attach any supporting output for your answer.

4. Does the time of day have an effect on delivery time? Attach any supporting output for your answer.

5. Discuss Tony Scapelli's data design and collection. Comment on any potential problems resulting from the way in which the data were collected. Would you recommend that he collect additional data in order to better evaluate the pizza delivery service? Explain.

6. Based on your analysis of the data, what action (or actions) would you recommend to the owners of Pronto Pizza to improve their operation? Attach any supporting output that led to your conclusion.

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Pronto Pizza is a family-owned pizza restaurant in Vinemont, a small town of 20,000 people in upstate New York. Antonio Scapelli started the business 30 years ago as Antonio's Restaurant with just a few thousand dollars. Antonio, his wife, and their children, most of whom are now grown, operate the business. Several years ago, one of Antonio's sons, Tony, Jr., graduated from NYU with an undergraduate degree in business administration. After graduation, he came back to manage the family business. Pronto Pizza was one of the earliest pizza restaurants to offer pizza delivery to homes. Fortunately, Tony had the foresight to make this business decision a few years ago. At the same time, he changed the restaurant's name from Antonio's to Pronto Pizza to emphasize the pizza delivery service. The restaurant has thrived since then, and has become one of the leading businesses in the area. While many of their customers still "dine-in" at the restaurant, nearly 90% of Pronto's current business is derived from the pizza delivery service. Recently, one of the national-chain fast food pizza delivery services found its way to Vinemont, New York. In order to attract business, this new competitor has guaranteed delivery of its pizzas within 30 minutes after the order is placed. If the delivery is not made within 30 minutes, the customer receives the order without charge. Before long, there were signs that this new pizza restaurant was taking business away from Pronto Pizza. Tony realized that Pronto Pizza would have to offer a similar guarantee in order to remain competitive. After a careful cost analysis. Tony determined that to offer a guarantee of 29 minutes or less. Pronto's average delivery time would have to be 25 minutes or less. Tony thought that this would limit the percentage of "free pizzas" under the guarantee to about 5% of all deliveries, which he had figured to be the break-even point for such a promotion. To be sure of Pronto's ability to deliver on a promise of 29 minutes or less, Tony knew that he needed to collect data on Pronto's pizza deliveries.
Pronto Pizza's delivery service operates from 4:00 p.m. to midnight every day of the week. After an order for a pizza is phoned in, one of the two cooks is given the order for preparation. When the crust is prepared and the ingredients have been added, the pizza is placed on the belt of the conveyor oven. The speed of the conveyor is set so that pizzas come out perfectly, time after time. Once the pizza is ready and one of Pronto's drivers is available to make the delivery, the pizza is taken in a heat-insulated bag to the customer. Pronto uses approximately five to six drivers each night for deliveries. Most of the drivers hired by Pronto Pizza are juniors and seniors at the local high school. Given the large number of deliveries made each evening, Tony knew that he could not possibly monitor every single delivery. He had thought of the possibility of having someone else collect the data, but given the importance of accurate data, he decided to make all of the measurements himself. This, of course, meant taking a random sample of, rather than all, deliveries over some time period. Tony decided to monitor deliveries over the course of a full month. During ...

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2.A group of 25 subjects have their diastolic blood pressures measured. The results, in SPSS are:

|N |Valid |25 |
| |Missing|0 |
|Median |85.00 |
|Mode |82.00 |
|Minimum |55.00 |
|Maximum |110.00 |
|Percentiles|25 |71.00 |
| |-------|--------|
| |50 |85.00 |
| |-------|--------|
| |75 |98.00 |

Don't worry about values exactly at the endpoints of these intervals. Do the calculations roughly.
(1 point each)

a. What percentage of subjects were from 55 to 85?
b. What percentage of subjects were < 85?
c. What percentage of subjects were from 71 to 85?
d. What percentage of subjects were > 71?
e. What percentage of subjects were > 98?
f. Is there one value more common than the rest, and if so, what is it?

3. Assume you have already been give a Z value. This saves you a step. Consider and determine the following probabilities (1 point each).

A. Pr (-1 < Z < 1)
B. Pr (0 < Z < 1)
C. Pr (Z > 1)
D. Pr (-1 < Z < 0)
E. Pr (Z < -1)
F. Pr (Z > -2)
G. Pr (-1 < Z < 2)

4. Suppose the mean systolic blood pressure in a group of individuals is 150 mmHg, with a standard deviation of 15. Assuming SBP follows a normal distribution in this population, compute (1 point each):

A. Pr (135 < value < 165)
B. Pr (value > 165)
C. Pr (value < 135)
D. Pr (138.75 < value < 161.25)

5 Compute the 5th, 50th, and 95th percentiles of SBP from the previous question. (3 points: 1 each).

In questions 6 - 8, use the 1 and 2 SD rules, without the table.

6.In general, what percentage of a Gaussian data set is within 1 SD of the mean? What percentage is within 2 SD's of the mean? (2 points: 1 each)

7.If the mean grade on an exam was 80, SD = 6, where did about 68% of the grades fall? How about 95%? Assume the grades are Gaussian. (2 points).

8.Consider the following data: 1, 1, 2, 2, 4, 5, 6, 9, 40, 200

Use the 68% and 95% rules to test the normality of these data. (2 points).

9.A researcher studying a subtype of lymphocytes obtains a sample mean of 100 per mL, and a standard deviation of 20, with 25 subjects. Within what interval can you be roughly 68% sure the population mean number of these cells per mL lies? How about 95% sure? (2 points)

10.A researcher has a sample of 500 subjects. The mean is 40, median is 20, range 10-100. (2 points each)

a.Could this researcher calculate a useful interval with 95% probability of containing the population mean (using the mean and SEM)? Explain

b.Could the researcher use the mean and SD to usefully estimate where 95% of the individual subject values were? Explain

c)If there were 10 subjects, would your answers to a and b change?

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