A real estate association in a suburban community would like to study the relati
ID: 3226069 • Letter: A
Question
A real estate association in a suburban community would like to study the relationship between the size of a single-family house (as measured by the number of rooms) and the selling price of the house (in thousands of dollars). Two different neighborhoods are included in the study, one on the east side of the community (=0) and the other on the west side (=1). A random sample of 20 houses was selected, with the results given below. Complete parts (a) through (g). For parts (a) through (d), do not include an interaction term.
a. State the multiple regression equation that predicts the selling price, based on the number of rooms, X1, and the neighborhood, X2.
Yi = ____ + ( ____)X1i +(____)X2i
(Round to three decimal places as needed. Do not include the $ symbol in your answers.)
b. Interpret the regression coefficients in (a).
Holding constant whether a house is in the east or west side of town, for each increase of 1 room in the house, the predicted price of the home is estimated to increase by ___ thousand dollars. Holding constant the number of rooms in the home, the presence of the home on the ___ side of town is estimated to increase the predicted price of the home by ____ thousand dollars over the price of a home on the ____ side of town.
(Round to three decimal places as needed. Do not include the $ symbol in your answers.)
c. At the 0.05 level of significance, determine whether each independent variable makes a contribution to the regression model.
Test the first independent variable, Rooms. Determine the null and alternative hypotheses.
H0: 1 ___ 0
H1: 1____0
The test statistic for the first independent variable, Rooms, is t Subscript STAT = ____.
(Round to three decimal places as needed.)
The p-value for the first independent variable, Rooms, = ____
(Round to four decimal places as needed.)
Since the p-value is ____ than the value of ___ , ____ the null hypotheses. The first independent variable, Rooms, ____ to make a contribution to the regression model.
Test the second independent variable, Neighborhood. Determine the null and alternative hypotheses.
H0: 2 ___ 0
H1: 2____0
The test statistic for the second independent variable, Neighborhood, is t Subscript STAT = ____.
(Round to three decimal places as needed.)
The p-value for the second independent variable, Neighborhood, = ____
(Round to four decimal places as needed.)
Since the p-value is ____ than the value of ___ , ____ the null hypotheses. The second independent variable, Neighborhood, ____ to make a contribution to the regression model.
d. Construct and interpret a 95% confidence interval estimate of the population slope of the relationship between selling price and number of rooms.
Taking into account the effect of ____ the estimated effect of a 1 room increase is to change the ____ by ____ to _____ thousand dollars.
(Round to three decimal places as needed. Do not include the $ symbol in your answers. Use ascending order.)
e. Construct and interpret a 95% confidence interval estimate of the population slope of the relationship between selling price and neighborhood.
Taking into account the effect of ____ the estimated effect of the home being on the west side of town instead of the east side of town is to change the ____ by ____ to ____thousand dollars. (Round to three decimal places as needed. Do not include the $ symbol in your answers. Use ascending order.)
f. Add an interaction term to the model and, at the 0.05 level of significance, determine whether it makes a significant contribution to the model.
State the multiple regression equation that predicts the selling price, based on the number of rooms, X1, the neighborhood, X2, and the interaction term, X3 =X1 ×X2.
Yi= ____ + (____) X1i +(____) X2i +(____) X3i
(Round to three decimal places as needed. Do not include the $ symbol in your answers.)
Test the interaction term. Determine the null and alternative hypotheses.
H0: ____ 0
H1: ____ 0
The test statistic for the interaction term is tSTAT = ____
(Round to three decimal places as needed.)
The p-value for the interaction term is ____
(Round to four decimal places as needed.)
Since the p-value is ____ than the value of , ____ the null hypothesis. The interaction term ____ to make a contribution to the regression model.
g. On the basis of the results of (c) and (f), which model is most appropriate? Explain.
The model ____ the interaction term appears to be the most appropriate because the interaction term ____ to make a contribution to the regression model.
Price Rooms Neighborhood 305.7 6 0 307.5 8 0 340.2 9 0 346.5 12 0 308.2 8 0 338.8 9 0 334.1 11 0 312.2 8 0 327.8 9 0 335.4 9 0 319.4 7 1 383.8 13 1 339.9 10 1 348.7 10 1 346.1 9 1 327.2 8 1 332.9 8 1 345.8 9 1 363.3 11 1 351.9 9 1Explanation / Answer
Solution:
The regression output for the given regression model is summarised as below:
Regression Statistics
Multiple R
0.915140709
R Square
0.837482517
Adjusted R Square
0.818362813
Standard Error
8.351339721
Observations
20
ANOVA
df
SS
MS
F
Significance F
Regression
2
6109.939123
3054.96956
43.80206511
1.96178E-07
Residual
17
1185.662877
69.7448751
Total
19
7295.602
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
246.4872515
10.70814557
23.0186684
2.97395E-14
223.8950393
269.0794636
Rooms
8.893567251
1.165997094
7.62743518
6.94368E-07
6.433528442
11.35360606
Neighborhood
15.81321637
3.780061154
4.1833229
0.000623675
7.837984538
23.78844821
Part a
The multiple regression equation is given as below:
Y = 246.487 + 8.894*X1 + 15.813*X2
Part b
The y-intercept for the regression equation is given as 246.487 and it indicates the value of dependent variable selling price when both independent variables are zero.
Holding constant whether a house is in the east or west side of town, for each increase of 1 room in the house, the predicted price of the home is estimated to increase by 8.894 thousand dollars. Holding constant the number of rooms in the home, the presence of the home on the East side of town is estimated to increase the predicted price of the home by 15.813 thousand dollars over the price of a home on the west side of town.
Part c
The null and alternative hypothesis for this test is given as below:
H0: 1 = 0
H1: 1 0
The test statistic for this test is given as below:
Test statistic = tSTAT = 7.627
P-value = 0.0000
Alpha = 0.05
Since the p-value is less than the value alpha, we reject the null hypothesis.
Now, we have to test for second independent variable.
The null and alternative hypothesis for this test is given as below:
H0: 1 = 0
H1: 1 0
The test statistic for this test is given as below:
Test statistic = tSTAT = 4.183
P-value = 0.0006
Alpha = 0.05
Since the p-value is less than the value alpha, we reject the null hypothesis.
Part d
The 95% confidence interval for population slope for number of rooms is given as below:
Confidence interval = (6.433528442, 11.35360606)
Regression Statistics
Multiple R
0.915140709
R Square
0.837482517
Adjusted R Square
0.818362813
Standard Error
8.351339721
Observations
20
ANOVA
df
SS
MS
F
Significance F
Regression
2
6109.939123
3054.96956
43.80206511
1.96178E-07
Residual
17
1185.662877
69.7448751
Total
19
7295.602
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
246.4872515
10.70814557
23.0186684
2.97395E-14
223.8950393
269.0794636
Rooms
8.893567251
1.165997094
7.62743518
6.94368E-07
6.433528442
11.35360606
Neighborhood
15.81321637
3.780061154
4.1833229
0.000623675
7.837984538
23.78844821
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