Stine and Foster Problem 5-39: Gasoline Sales

A service station near an interstate highway sells three grades of gasoline: regular, plus, and premium. During the last week, the manager counted the number of cars that purchased these types of gasoline. He kept the counts separate for weekdays and the weekend. The data table has two categorical variables. One distinguishes weekdays from weekends, and the other indicates the type of gas (regular, plus, or premium).

(a)Find the contingency table defined by the day of the week and the type of gas. Include the marginal distributions.
The marginal distributions are the borders of the contingency table:
Grade of Gasoline / Premium / 21.38
Plus / 14.93
Regular / 63.69
Type of Day / Weekday / 76.58
Weekend / 23.42
(b)Find the conditional distribution of purchase type for weekday purchases.
Premium / Plus / Regular
Weekday Purchases / 18.61 / 15.21 / 66.17
(c)Find the conditional distribution of premium purchases during the week and weekend.
Premium
Weekday Purchase / 66.67
Weekend Purchases / 33.33
(d)Does the fact that your answers to parts (b) and (c) are different indicate association?"
No. These conditional distributions are not directly comparable. One refers to the type of gas given the purchase happens on a weekday, whereas the second refers to the timing of the purchase given that premium gas was bought. To illustrate association, one can, for example, compare the conditional distribution of purchases of plus and regular gas to the answer in part c.
Premium / Plus / Regular / Marginal
Weekday Purchase / 66.67 / 78.03 / 79.57 / 76.58
Weekend Purchases / 33.33 / 21.97 / 20.43 / 23.42
The columns are comparable. You cannot compare some rows and some columns.
(e)The owner of the station would like to develop better ties to customers who buy premium gas. (The owner expects these customers to be more affluent and likely to purchase other services from the station.) If the owner wants to meet more of these customers, when should the owner be around the station: on weekdays or weekends?"
First we ask is there a significant relationship by looking the Chi-Squared test and the mosaic plot. The low p-value allows us to reject the assertion that there is no relationship. Since the horizontal lines in the mosaic plot are not at the same level (also as discussed in part d), that is also an indication that premium gasoline sales are relatively larger on weekends. Therefore, if the owner wants to meet affluent customers, the weekend is the best time to be at the convenience store.