Question

9. Exercise 4.9 In a study of housing demand, the county assessor develops the following regression model to estimate the market value (i.e., selling price) of residential property within her jurisdiction. The assessor suspects that important variables affecting selling price (YY , measured in thousands of dollars) are the size of a house (X1X1 , measured in hundreds of square feet), the total number of rooms (X2X2 ), age (X3X3 ), and whether or not the house has an attached garage (X4X4 , No=0, Yes=1No=0, Yes=1 ). Y=α+β1X1+β2X2+β3X3+β4X4+εY=α+β1X1+β2X2+β3X3+β4X4+ε Now suppose that the estimate of the model produces following results: a=166.048a=166.048 , b1=3.459b1=3.459 , b2=8.015b2=8.015 , b3=−0.319b3=−0.319 , b4=1.186b4=1.186 , sb1=1.079sb1=1.079 , sb2=5.288sb2=5.288 , sb3=0.789sb3=0.789 , sb4=12.252sb4=12.252 , R2=0.838R2=0.838 , F-statistic=12.919F-statistic=12.919 , and se=13.702se=13.702 . Note that the sample consists of 15 randomly selected observations. According to the estimated model, holding all else constant, an additional 100 square feet of area means the market value selector 1    increases decreases by approximately selector 2    $3,459 $8,015 $319 . Which of the independent variables (if any) appears to be statistically significant (at the 0.05 level) in explaining the market value of residential property? Check all that apply. Size of the house (X1X1 ) Total number of rooms (X2X2 ) Age (X3X3 ) Having an attached garage (X4X4 ) What proportion of the total variation in sales is explained by the regression equation? 0.838 0.789 0.129 The given F-value shows that the assessor selector 1    cannot can reject the null hypothesis that neither of the independent variables explains a significant (at the 0.05 level) proportion of the variation in income. Which of the following is an approximate 95 percent prediction interval for the selling price of a 15-year-old house having 18 hundred sq. ft., 5 rooms, and an attached garage? (237.382, 292.190) (157.232, 212.040) (170.934, 198.338)

Answer 1

The slope coefficient of X1 (measured in 100 feets) is 3.459. So, holding all else constant, an additional 100 square feet of area means the market value increases by 3.459 * 1000 = $3,459.

Degree of freedom = n - k - 1= 15 - 4 - 1 = 10
where k is number of predictors.
Critical value of t at significance level 0.05 and df = 10 is, 2.23
t = Coefficient / Standard error
t for X1 = b1 / sb1 = 3.459 / 1.079 = 3.21
t for X2 = b2 / sb2 = 8.015 / 5.288 = 1.52
t for X3 = b3 / sb3 = −0.319 / 0.789 = -0.40
t for X4 = b4 / sb4 = 1.186 / 12.252 = 0.10
Since, t value for X1 is only greater than the critical value, X1 is significant.
The answer is Size of the house (X1)

The proportion of the total variation in sales is explained by the regression equation is R2=0.838
0.838

Numerator df = k = 4
Denominator df = n - k - 1 = 15 - 4 - 1 = 10
Critical value of F at df = 4, 10 and 0.05 significance level is 3.48
Since the observed F is greater than the critical value, we can reject the null hypothesis that neither of the independent variables explains a significant (at the 0.05 level) proportion of the variation in income.

Predicted value of selling price of a 15-year-old house having 18 hundred sq. ft., 5 rooms, and an attached garage

= 166.048 + 3.459 * 18 + 8.015 * 5 - 0.319 * 15 + 1.186 * 1 = 264.786

The predicted value lies in the interval (237.382, 292.190). So, the correct answer is,

(237.382, 292.190)