Question: Donna Nickles manages a gasoline station on the corner of Bristol Avenue and Harpst Street in Arcata, California. Her station is a franchise, and the parent company calls her station every day at midnight to give her the prices for various grades of gasoline for the upcoming day. Over the past eight weeks Donna has recorded the price and gallon sales of regular grade gasoline at her station as well as the price of regular grade gasoline charged by her competitor across the street. She is curious about the sensitivity of her sales to the price of regular grade gasoline she charges and the price of regular gasoline charged by her competitor across the street. She also wonders whether her sales differ systematically by day of the week and whether her station has experienced a trend in sales over the past eight weeks. The data collected by Donna for each day of the past eight weeks are provided in the WEB file Gas Station.
a. Construct a time series plot of daily sales, a scatter plot of the price Donna charges for a gallon of regular grade gasoline and daily sales at Donna's station, and a scatter plot of the price Donna's competitor charges for a gallon of regular grade gasoline and daily sales at Donna's station. What types of relationships exist in the data?
b. Use a multiple regression model with the price Donna charges for a gallon of regular grade gasoline and the price Donna's competitor charges for a gallon of regular grade gasoline as causal variables to develop an equation to account for the relationships between these prices and Donna's daily sales in the data. Based on this model, compute an estimate of sales for a day on which Donna is charging $3.50 for a gallon for regular grade gasoline and her competitor is charging $3.45 for a gallon of regular grade gasoline.
c. Use a multiple linear regression model with the trend and dummy variables as follows to develop an equation to account for both trend and seasonal effects in the data: Monday 5 1 if the sales were recorded on a Monday, 0 otherwise; Tuesday 5 1 if the sales were recorded on a Tuesday, 0 otherwise; Saturday 5 1 if the sales were recorded on a Saturday, 0 otherwise; Note that when the values of the six dummy variables are equal to 0, the observation corresponds to Sunday. Based on this model, compute an estimate of sales for Tuesday of the first week after Donna collected her data.
d. Use a multiple regression model with the price Donna charges for a gallon of regular grade gasoline and the price Donna's competitor charges for a gallon of regular grade gasoline as causal variables and the trend and dummy variables from part c to create an equation to account for the relationships between these prices and daily sales as well as the trend and seasonal effects in the data. Based on this model, compute an estimate of sales for Tuesday of the first week after Donna collected her data a day if Donna is charging $3.50 for a gallon for regular grade gasoline and her competitor is charging $3.45 for a gallon of regular grade gasoline.
e. Which of the three models you developed in parts b, c, and d is most effective? Justify your answer.