Utility companies, which must plan the operation and

Chapter 11, Problem 92E

(choose chapter or problem)

Utility companies, which must plan the operation and expansion of electricity generation, are vitally interested in predicting customer demand over both short and long periods of time. A short-term study was conducted to investigate the effect of each month’s mean daily temperature x1 and of cost per kilowatt-hour, x2 on the mean daily consumption (in kWh) per household. The company officials expected the demand for electricity to rise in cold weather (due to heating), fall when the weather was moderate, and rise again when the temperature rose and there was a need for air conditioning. They expected demand to decrease as the cost per kilowatt-hour increased, reflecting greater attention to conservation. Data were available for 2 years, a period during which the cost per kilowatt-hour x2 increased due to the increasing costs of fuel. The company officials fitted the model

   \(Y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{1}^{2}+\beta_{3} x_{2}+\beta_{4} x_{1} x_{2}+\beta_{5} x_{1}^{2} x_{2}+\varepsilon\)

to the data in the following table and obtained \(y=325.606-11.383 x_{1}+.113 x_{1}^{2}+.113 x_{1}^{2}-21.699 x_{2}+.873 x_{1} x_{2}-.009 x^{2}\) with SSE = 152.177.

When the model \(Y=\beta_{0}-\beta_{1} x_{1}+\beta_{2} x_{1}^{2}+\varepsilon\) was fit, the prediction equation was \(y=130.009-3.302 x_{1}+.033 x_{1}^{2}\) with SSE = 465.134. Test whether the terms involving \(x_{2}\left(x_{2}, x_{1} x_{2}, x_{1}^{2} x_{2}\right)\) contribute to a significantly better fit of the model to the data. Give bounds for the attained significance level.

Equation transcription:

Text transcription:

Y=beta{0}+beta{1} x{1}+\beta_{2} x{1}^{2}+\beta_{3} x{2}+\beta_{4} x{1} x{2}+\beta{5} x{1}^{2} x{2}+\varepsilon

y=325.606-11.383 x{1}+.113 x{1}^{2}+.113 x{1}^{2}-21.699 x{2}+.873 x{1} x{2}-.009 x^{2}

y=130.009-3.302 x{1}+.033 x{1}^{2}

x{2}(x{2}, x{1} x{2}, x{1}^{2} x{2})