Question

Suppose the number of equipment sales and service contracts that a store sold during the last six (6) months for treadmills and exercise bikes was as follows:

                       Treadmill          Exercise Bike
Total sold                   185                           123
Service contracts     67                             55

The store can only sell a service contract on a new piece of equipment. Of the 185 treadmills sold, 67 included a service contract and 118 did not.

Complete the following questions in the space provided below:

1. Construct a 95 percent confidence interval for the difference between the proportions of service contracts sold on treadmills versus exercise bikes.
2. Is there a major difference between the two pieces of equipment? Why or why not?

Answer 1

1.

treadmill service contracts sold = 67

bike service contracts sold = 55

n1 = size of treadmills sample = 185

n2 = size of exercise bikes sample = 123

p1 = sample proportion of treadmill service contracts sold = 67/185 = 0.362162

p2 = sample proportion of exercise bike service contracts sold = 55/123 = 0.447154

?1 = population proportion of all treadmill service contracts sold

?2 = population proportion of all exercise bike service contracts sold

The formula to calculate a 95% C.I. for ?1- ?2 is given by Excel spreadsheet:

The difference between the two proportions (treadmill / bike) on service contracts at 95% C. I. is [-0.9687, 0.02689].

2.

When a company is applying CI formulas to get the difference (proportions between service contracts sold by the treadmills and bikes) of -0.197 to 0.027. So, it is likely that the proportion difference between treadmills and bike service contracts over a long period of time is between -0.197 and 0.027. If reviewing the data, note that 0 is included in the interval. Thus, there may not be any proportional difference between service contracts sold on treadmills and bikes at 95% confidence interval (Zc = 1.96). In conclusion, I have found that if you change the confidence interval to 85% (Zc = 1.44), you will find the C.I. is [-0.167, -0.003]. The difference ?1- ?2 is “likely to be negative”.

The proportion service contract sold by treadmills is less than proportion service contract sold by bikes. So it all truly depends on how close to accurate the company chooses to be. Whether they want the 95% C.I., 85% C.I., or 99% C.I. Obviously, the closer you get to 100% the less chance you will have a negative difference.