The director for a large consumer products company

Question
1. The director for a large consumer products company wishes to study the effect of a new packaging design on sales of a laundry stain remover. To compare sales with the new and current package, the product with new packaging was made available for one month in a random sample of 11 stores, while a random sample of 9 stores was selected to monitor monthly sales of the product with the current packaging. Mean sales for the new package was 5.51 thousand units with a standard deviation of 1.24 thousand units, while the current package produced mean sales of 5.19 thousand units with a standard deviation of 1.17 thousand units. Let Populations1 and 2 represent the monthly sales of the laundry stain remover with the new and current packages, respectively, and assume that both populations are normally distributed.
(a) Calculate the ratio of the maximum sample variance to the minimum sample variance. Does it appear that the population variances are equal or unequal? Explain.
(b) Construct the appropriate 95% confidence interval for the difference in the mean sales with the new and current packaging, and interpret the result. Also give the point estimate for this difference.
(c) Based on your interval in (b), does there appear to be a difference in the mean sales with the two different packages? Why?

2. Refer to Question 1. An analyst working for the company questioned the study that was conducted, stating that it would have been better to conduct a paired design. The study was therefore repeated with 10 pairs of stores, where the stores in each pair were matched on overall sales volume and other demographic characteristics. The results obtained are given below:

Pair
New Package Current Package

1 2 3 4 5 6 7 8 9 10 4.58 5.19 3.94 6.32 7.68 3.48 5.72 7.04 5.27 5.84 3.97 4.88 4.09 5.87 6.93 4.00 5.08 6.95 4.96 5.13

(a) Construct the appropriate 95% confidence interval for the difference in the mean sales with the new and current packaging, and interpret the result. Also give the point estimate for this difference, and compare it to the point estimate obtained in Question 2(b).
(b) Based on your interval in (a), does there appear to be a difference in the mean sales with the two different packages? Why?
(c) Explain why the two confidence intervals in Questions 2 and 3 suggest different findings when comparing the new and current packaging.

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