91Ó°ÊÓ

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 \(x_{1}\) and of cost per kilowatt-hour, \(x_{2}\) on the mean daily consumption (in \(\mathrm{kWh}\) ) 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 \(x_{2}\) 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 \(\hat{y}=325.606-11.383 x_{1}+.113 x_{1}^{2}-21.699 x_{2}+.873 x_{1} x_{2}-.009 x_{1}^{2} x_{2}\) with \(\mathrm{SSE}=152.177\) When the model \(Y=\beta_{0}-\beta_{1} x_{1}+\beta_{2} x_{1}^{2}+\varepsilon\) was fit, the prediction equation was \(\hat{y}=130.009-3.302 x_{1}+.033 x_{1}^{2}\) with \(\mathrm{SSE}=465.134 .\) 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.

Step by step solution

01

State Hypotheses

We want to test whether the terms involving \(x_2\) significantly improve the model. The null hypothesis \(H_0\) is that the coefficients of \(x_2, x_1 x_2, x_1^2 x_2\) are zero. The alternative hypothesis \(H_a\) is that at least one of these coefficients is not zero.

02

Determine Test Statistic

We use the F-test to compare the two models. The test statistic is given by \[ F = \frac{(SSER - SSEF) / (df_R - df_F)}{SSEF / df_F} \] where \(SSER\) is the sum of squared errors for the reduced model, \(SSEF\) is the sum for the full model, \(df_R\) is the degrees of freedom for the reduced model, and \(df_F\) for the full model.

03

Calculate SSE and Degrees of Freedom

From the problem, \(SSER = 465.134\) and \(SSEF = 152.177\). The full model has 6 parameters (including the intercept), and the reduced model has 3 parameters. Assuming the use of the same number of observations \(n\), the degrees of freedom are \(df_R = n - 3\) and \(df_F = n - 6\).

04

Compute F-statistic

Substitute the given values into the F-statistic formula:\[ F = \frac{(465.134 - 152.177) / (3)}{152.177 / (n - 6)} \]This can be simplified further once \(n\) (the number of observations) is known.

05

Determine Significance

With the calculated \(F\)-value, consult the \(F\)-distribution table to find the \(p\)-value using the degrees of freedom \((3, n-6)\). This tells us the significance level.

06

Make a Conclusion

If the \(p\)-value is less than the significance level (often 0.05), we reject the null hypothesis. If it is greater, we fail to reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiple Regression Analysis

Multiple regression analysis is a statistical method used to understand relationships among multiple variables. In this exercise, the utility company wants to predict electricity consumption based on various factors. Here, the dependent variable (the outcome we want to predict) is the mean daily consumption of electricity per household. On the other hand, the independent variables are the mean daily temperature, both linear and quadratic terms

• Linear term: Temperature (\[x_1\])
• Quadratic term: Temperature squared (\[x_1^2\])
• Cost per kilowatt-hour (\[x_2\])
• Interaction terms (\[x_1 \times x_2\], and \[x_1^2 \times x_2\])

These components construct a model that tries to predict how the electricity demand changes under different temperatures and costs. This model is essential for companies to plan their operations efficiently and make informed decisions about future electricity generation needs.

F-Test

The F-test is a statistical test used in this context to compare two competing models to see if the more complex model (with more variables) provides a significantly better prediction of the dependent variable.
The null hypothesis (\[H_0\]) in this scenario assumes that the additional variables \[x_2, x_1 x_2, x_1^2 x_2\] do not improve the model fit. The alternative hypothesis (\[H_a\]) suggests that at least one of these terms does improve the model.
The test statistic in the F-test is calculated using the formula: \[ F = \frac{(SSER - SSEF) / (df_R - df_F)}{SSEF / df_F} \]where

• \[SSER\] is the sum of squared errors for the reduced model
• \[SSEF\] is the sum for the full model
• \[df_R\] represents the degrees of freedom for the reduced model
• \[df_F\] for the full model

This statistical test is crucial in evaluating whether incorporating extra terms in the model significantly improves its predictive capability.

Model Comparison

Model comparison is vital to determine which statistical model fits the data better. In this exercise, the objective is to evaluate if including terms related to the cost per kilowatt-hour \[x_2\] and its interactions with temperature leads to a more accurate prediction of electricity consumption.
The first model only considers temperature (\[x_1\] and \[x_1^2\]), while the second, more comprehensive model, incorporates \[x_2\] and the interaction terms (\[x_1 \times x_2\], and \[x_1^2 \times x_2\]).
The balance between model complexity and accuracy is essential. A model with too many parameters may fit the current data very well but perform poorly on new data—a phenomenon known as overfitting. Conversely, a simpler model may not capture all underlying patterns.
Using the F-test, students compare these models by calculating the F-statistic and checking its significance against established criteria (like the p-value threshold of 0.05). This process helps determine if the additional variables genuinely contribute valuable information to the model's predictions.