Consider the situation in which a gas company wishes to predict weekly natural gas consumption for its city. In the exercises of Chapter we used the single predictor variable x, average hourly temperature, to predict y, weekly natural gas 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 natural gas 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 natural gas 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 natural gas consumption. The natural gas company has collected data concerning weekly natural gas consumption (y, in MMcF), average hourly temperature (x1, in degrees Fahrenheit), and the chill index (x2) for the last eight weeks. The data are given in Table, and scatter plots of y versus x1 and y versus x2 are given below the data. Moreover, Figure 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 b1 and b2, and report their values. Then interpret b1 and b2.
b. Calculate a point estimate of the mean natural gas 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 natural gas 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.