ASIAN BOX & CONTAINER COMPANY
The Asian Box and Container (ABC) Company manufactures containers used in the food industry. Its major product are containers that transport Fortune Cookies from the manufacturer to (primarily) Chinese restaurants in various parts of the United States. In response to requests from these restaurants, ABC has developed a new container that protects large shipments of Fortune Cookies to be delivered to its client restaurants.
The new container costs more per unit than the conventional container that has been in use for many years; thus, use of the new container can be justified only by there being lower product breakage when the new container is used. ABC's management, therefore, will be willing to use the new container on a regular basis only if convincing evidence can be provided that indeed the product breakage is lower with the use of the new container.
ABC hired a consulting firm, Fan & Du, to examine the issue. It told the consulting firm that it wanted an accurate estimate of the product breakage using the new container. Specifically, ABC stated that it wanted an estimate of the number of broken Fortune Cookies per container within one Fortune Cookie, with a probability/confidence of .99.
An initial report from the consulting company indicated that an estimate satisfying the conditions/requirements stated will cost $100,000. The manner of collecting data involved shipping the Fortune Cookies through the distribution system in a way that mirrors, to the degree possible, the actual distribution pattern of ABC. Preliminary studies suggested that the cost of conducting the study would be $200 per container.
In forming its initial report the consulting firm examined the product breakage in 20 new containers. Results are summarized in the Excel output table below:
|
Column1
|
|
Mean
|
25
|
|
Standard Error
|
1.956097081
|
|
Median
|
24.5
|
|
Mode
|
23
|
|
Standard Deviation
|
8.747932086
|
|
Sample Variance
|
76.52631579
|
|
Kurtosis
|
-0.856097105
|
|
Skewness
|
0.131556604
|
|
Range
|
29
|
|
Minimum
|
11
|
|
Maximum
|
40
|
|
Sum
|
500
|
|
Count
|
20
|
|
Confidence Level(95.0%)
|
4.094159515
|
|
|
|
(Q1) Based on these 20 data points, what is a 95% confidence interval for the true average number of broken cookies per box?
Q2) Using these preliminary data, with the approximation [for the rest of the problem] that s = s, would you say that the cost estimate of $100,000 provided by the consulting firm appears to be reasonable?
(Q3) Suppose that ABC establishes a limit of $15,000 (i.e., refusing to spend $100,000) for the estimation process. How might ABC compromise its requirements of accuracy and confidence to accommodate this $15,000 budget? Specifically,
i) if ABC wishes to retain 99% confidence in its interval, what will be the precision that replaces "within one Fortune Cookie" that ABC must settle for?
ii) if ABC wishes, instead, to retain the precision of "within one Fortune Cookie," what will be the lower confidence level that ABC must settle for?
iii)] if instead of either of the above choices, ABC decides to compromise on both aspects, the confidence level and the precision, and decides to have 95% confidence, what will be the precision of the interval?