Consider the fuel consumption problem in which a natural gas company wishes to predict weekly fuel consumption for its city. In the exercises of Chapter 13, we used the single predictor variable x, average hourly temperature, to predict y, weekly fuel consumption. We now consider predicting y on the basis of average hourly temperature and a second predictor variable-the chill index. The chill index for a given average hourly temperature expresses the combined effects of all other major weather-related factors that influence fuel consumption, such as wind velocity, sunlight, cloud cover, and the passage of weather fronts. The chill index is expressed as a whole number between 0 and 30. A weekly chill index near 0 indicates that, given the average hourly temperature during the week, all other major weather-related factors will only slightly increase weekly fuel consumption. A weekly chill index near 30 indicates that, given the average hourly temperature during the week, other weather-related factors will greatly increase weekly fuel consumption. The natural gas company has collected data concerning weekly fuel consumption (y, in MMcF of natural gas), average hourly temperature (x1 in degrees Fahrenheit), and the chill index (x2) for the last eight weeks. The data are given in Table 14.3. and scatter plots of y versus A, and y versus .v, are given below the data. Moreover, Figure 14.5 on the next page gives Excel and MINITAB outputs of a regression analysis of these data using the model
y = β0+ β1x1 + β2x2 + ε
a. Using the Excel or MINITAB output (depending on the package used in your class), find (on the output) b1, and b2, the least squares point estimates of β1, and β2, and report their values. Then interpret b1 and b2.
b. Calculate a point estimate of the mean fuel consumption for all weeks that have an average hourly temperature of 40 and a chill index of 10, and a point prediction of the amount of fuel consumed in a single week that has an average hourly temperature of 40 and a chill index of 10. Find this point estimate (prediction), which is given at the bottom of the MINITAB output, and verify that it equals (within rounding) your calculated value.