Mathematics 376 – Probability and Statistics 2 Lab Project 5 – A Regression Case-Study April 26 and 28, 2006

Background

In class, we have discussed the basic methods for obtaining the least-squares estimators for the coefficients β_{i} in simple linear models

(1)  Y = β_{0} + β_{1}x_{1} + ⋯ + β_{k}x_{k} + ϵ

and techniques for hypothesis testing concerning the β_{i} (see pages 553, 587 in the text for convenient summaries). In this lab project, we will work through a “case-study” of how these methods might be used in a realistic statistical study.

The “Story”

Utility companies, which must plan the operation and expansion of electricity generating facilities, 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 daily mean temperature, in degrees Fahrenheit (x_{1}) and the cost per kilowatt-hour (in dollars) (x_{1}) per household. The company officials expected the demand for electricity to rise in cold weather (due to heating), fall when the weather was moderate, then increase again in hot weather (due to air conditioning). They also expected demand to decrease as the cost increased. Because of the expectation about the dependence on x_{1}, a model just involving x_{1} to the first power was thought to be inappropriate, so they set up a model of the form:

(1)  Y = β_{0} + β_{1}x_{1} + β_{2}x_{1}^{2} + β_{3}x_{2} + β_{4}x_{2}x_{1} + β_{5}x_{2}x_{1}^{2} + ε

The data they had to work with was as follows:

• With x_{2} = .08 (i.e. 8 cents per kilowatt-hour):

\begin{matrix}x_{1} (temp) & 31 & 34 & 39 & 42 & 47 & 56 & 62 & 66 & 68 & 71 & 75 & 78 \\ y (demand) & 55 & 49 & 46 & 47 & 40 & 43 & 41 & 46 & 44 & 51 & 62 & 73 \end{matrix}

• With x_{2} = .1 (i.e. 10 cents per kilowatt-hour):

\begin{matrix}x_{1} (temp) & 32 & 36 & 39 & 42 & 48 & 56 & 62 & 66 & 68 & 72 & 75 & 79 \\ y (demand) & 50 & 44 & 42 & 42 & 38 & 40 & 39 & 44 & 40 & 44 & 50 & 55 \end{matrix}

(Note: The data were from different locations and months so the temperatures don’t match up exactly.)

Lab Questions

A) Start by computing the least squares estimators for the coefficients β_{i} in the model

(1). B) What does the model predict about the demand in a month with average temperature

50 degrees Fahrenheit, if the cost is .12 (12 cents per kilowatt-hour)? Report an estimated value, and a 95% confidence interval. (See Section 11.12 in the text and the notes from 4/19 for the needed formulas.)

C) Now, we want to consider a different question. Namely: Did including the x_{2} terms in (1) really give us any added predictive power? Think of (1) as the complete model as in class on 4/24, and consider the reduced model

(2)  Y = β_{0} + β_{1}x_{1} + β_{2}x_{1}^{2} + ε

where the x_{2} terms have been removed (i.e. β_{3} = β_{4} = β_{5} = 0). Is there sufficient evidence to indicate that the model (1) gives us a significantly better fit to the data? Test using the procedure outlined in class on 4/24 and in section 11.14.

Assignment

One Maple worksheet from each lab group. Due by the end of the day on Friday, April 28 in Swords 335.