MS$3252$ Problem Set

Simple Linear Regression

Fanfare International, Inc., designs, distributes, and markets ceiling fans and light$-$

ing fixtures. The company's product line includes $120$ basic models of ceiling fans

and $138$ compatible fan light kits and table lamps. These products are marketed

to over $1000$ lighting showrooms and electrical wholesales that supply the remod$-$

eling and new construction markets. The product line is distributed by a sales

organization of $58$ independent sales representatives.

In the summer of $2013,$ Fanfare decided it needed to develop forecasts of future sales

to help determine future salesforce needs, capital expenditures, and so on$.$ The file

named $"$PS$1.$xlsx$"($sheet $="$PS$1\_$Q$1")$ contains the year and month the data being

collected, monthly sales data, as well as three additional variables for the period

July $2009$ through April $2013.$ The variables are defined as follows

SALES $=$ total monthly sales in thousands of dollars

ADEx$=$ advertising expense in thousands of dollars

MTGRATE $=$ mortgage rate for $30-$year loans $(\%)$

HSTARTS $=$ housing starts in thousands of units

As a consultant to Fanfare, your job is to try each of the three variables in a simple

linear regression, decide and justify which is the best variable to create a forecasting

model for Fanfare.

Answer: From the t$-$test results of the three simple linear regressions, it is found

that each of the three variables has significant linear relationship with SALES $($p$-$

value close to $0).$ Comparatively, HSTARTS seems to be the best single variable

to forecast Fanfare's future sales, as HSTARTS explains more of the variation in

SALES $(R^2=0\mathrm{.}6843).$

The Health Department would like to use regression analysis to estimate the total

cost of maintenance by the number of patient$-$days. The hospitals' total maintenance

costs and number of patient$-$days for seven months are listed in the file named

$"$PS$1.$xlsx$"($sheet $="$PS$1\_$Q$2").$

$($a$)$ Write down the estimated regression equation if we consider total maintenance

costs as the response and the patient$-$days as the predictor.

hat$(Y)=3430\mathrm{.}943+0\mathrm{.}759x$

where $Y=$ total maintenance costs, $x=$ number of patient$-$days

$($b$)$ Interpret the meaning of the estimated slope parameter.