On the WeissStats site are data on home size (in square feet) and assessed value (in thousands of dollars) for the same homes as in Exercise \(4.73\).

a. Obtain a scatterplot for the data.

b. Decide whether finding a regression line for the data is reasonable. If so, then also do parts (c)-(f)

c. Determine the interpret the regression equation for the data.

d. Identify potential outliers and influential observations.

e. In case a potential outlier is present, remove it and discuss the effect.

f. In case a potential influential observation is present, remove it and discuss the effect.

The below table gives the size of home (in square feet) and the assessed value (in thousands of dollars) for a sample of \(44\) homes.

The below graph represents the given points and home size is on the horizontal axis and the value is on the vertical axis.

It is reasonable to find the regression line for the data if there is no strong curvature present in the scatterplot.

It can be noted that, there is no strong curvature present in the scatterplot, therefore it is reasonable to find the regression line to the data

The homes are valued by their sizes. Therefore the response variable is value and the predictor variable is the home size.

The sample size \(n=44\).

Below are the necessary sums.

\(\sum x_{i}=132455\)

\(\sum y_{i}=19689\)

\(\sum x_{i}^{2}=430481773\)

\(\sum x_{i}y_{i}=62813175\)

To find \(s_{xy}\) and \(s_{xx}\):

\(s_{xx}=430481773-\frac{132455^{2}}{44}=31747067.8864\)

\(s_{xy}=62813175-\frac{(132455)(19689)}{63}=3542572.8409\)

To find the averages:

\(\bar{x}=\frac{132455}{44}=3010.3409\)

\(\bar{y}=\frac{19689}{44}=447.4773\)

Hence the parameters are:

\(b_{1}=\frac{3542572.8409}{31747067.8864}\)

\(=0.1116\)

\(b_{0}=447.4773-(0.1116)\times 3010.3409\)

\(=111.5233\)

The regression equation to predict the value \((y)\) from the home size \((x)\) is,

\(\hat{y}=111.5233+0.1116x\)

From the regression equation, the value of a house is increase on average by \(0.1116\) thousand dollars if the home size is increase by \(1\) square feet.

If a data point lies far from the regression line then it is an outlier.

If the removal of a point causes a considerable change in the regression equation then the point is called an influential observation. That is, the removal of a point causes a considerable change in the direction of the regression line.

The predicted values for the given data are summarized in the below table.

The below graph represents the given points and the fitted regression line.

From the plotted graph,

• The removal of a point does not cause any considerable change in the direction of the regression line therefore there are no potential influential observations.

Not applicable, because it was concluded that there are no outliers in part (d).

Not applicable, because it was concluded that there are no potential influential observation in part (d).