Question

A realtor collects the data set given in DS 12.2.4 concerning the sizes of a random selection of newly constructed houses in a certain area together with their appraised values for tax purposes.

1. Fit a linear regression model with appraised value as the explanatory variable and size as the dependent variable.
2. What is the estimate of the error variance?
A B
Area (square feet) Appraised Value ($1000)
1380 76
3120 216
3520 238
1130 69
1030 50
1720 119
3920 282
1490 81
1860 132
3430 228
2000 145
3660 251
2500 170
1220 71
1390 29

Answer 1

The table is missing in the question. The table is attached below.

Solution :

Let X = appraised value

Y = area (square feet)

The regression line is given by :

`$\hat y = b_0+b_1X$`

`$b_1=(n\sum XY-\sum X \sum Y)/(n \sum X^2-(\sum X)^2)$`

`$=(15(5964990)-(2157)(33370))/(15(404799)-(2157)^2)$`

`$=12.3267$`

`$b_0=(\sum Y)/(n)-b_1(\sum X)/(n)$`

`$=(33370)/(15)-\left(12.3267 * (2157)/(15)\right)$`

`$b_0=452.0841$`

The regression line is :

`$\hat Y = 452.0841+12.3267 X$`

To estimate the error variance, we have:

Error variance,
`$\sigma =\sqrt{\frac{\sum (Y-\hat Y)^2}n-2{}}$`

`$=\sqrt{(587682.3)/(15-2)}$`

`$=212.6178$`