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\centerline{Mathematics 376 -- Probability and Statistics 2}
\centerline{Lab Project 4/Problem Set 9 -- Regression Case-Studies}
\centerline{April 27 and 29, 2010}
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{\it Background}
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In class, we have discussed the basic methods for obtaining the 
least-squares estimators for the coefficients $\beta_i$ in simple linear 
models
$$Y = \beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k + \epsilon\leqno(1)$$
and techniques for hypothesis testing concerning the $\beta_i$ 
(see page 585 for the case $k = 1$ and models of the form 
$Y = \beta_0 + \beta_1 x + \epsilon$, and page 618 in the text 
for the general case).
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In this lab project, we will work through some ``case-studies'' of 
how these methods might be used.
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{\it From the Text}
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\item{$\bullet$}  Do problems 11.24 and 11.26 (using Maple is recommended!)
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{\it Additional Problem -- The ``Story''}
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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 
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\item{$\bullet$} $x_1 = $ each month's {\it daily mean temperature, in degrees
Fahrenheit}, and 
\item{$\bullet$} $x_2 = $ the {\it cost per kilowatt-hour (in dollars)} per household.
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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:
$$Y = \beta_0 + \beta_1 x_1 + \beta_2 x_1^2 + \beta_3 x_2 + 
\beta_4 x_2 x_1 + \beta_5 x_2 x_1^2 + \varepsilon\leqno(1)$$
The data they had to work with was as follows:
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\item{$\bullet$} With $x_2 = .08$ (i.e.  $8$ cents per kilowatt-hour):
$$\matrix{x_1\ (temp) & 31 & 34 & 39 & 42 & 47 & 56 & 62 & 66 & 68 & 71 & 75 & 78 \cr 
          y\ (demand) & 55 & 49 & 46 & 47 & 40 & 43 & 41 & 46 & 44 & 51 & 62 & 73\cr}$$
\item{$\bullet$} With $x_2 = .1$  (i.e. $10$ cents per kilowatt-hour):
$$\matrix{x_1\ (temp) & 32 & 36 & 39 & 42 & 48 & 56 & 62 & 66 & 68 & 72 & 75 & 79 \cr 
          y\ (demand) & 50 & 44 & 42 & 42 & 38 & 40 & 39 & 44 & 40 & 44 & 50 & 55\cr}$$
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\noindent
(Note:  The data were from different locations and months so the 
temperatures don't match up exactly.)
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\item{A)}  Start by computing the least squares estimators
for the coefficients $\beta_i$ in the model (1).  
\item{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 11.13 in the text
and the notes from 4/26 for the needed formulas.)
\item{C)}  Now, we want to consider a different question.  Namely:
{\it Did including the $x_2$ terms in (1) really give us any added 
predictive power?}  Think of (1) as the {\it complete} model as in 
class on 4/26, and consider the {\it reduced model}
$$Y = \beta_0 + \beta_1 x_1 + \beta_2 x_1^2 + \varepsilon\leqno(2)$$
where the $x_2$ terms have been removed (i.e. $\beta_3 = \beta_4 = \beta_5 = 0$).
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 4/26 and in section 11.14.
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{\it Assignment}
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One Maple worksheet from each person.  Due in class on Monday,
May 3.
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