Question

We have looked at predicting the price (in s) of New York homes based on the size (in thousands of square feet), using the data in HomesForSaleNY. Two other variables in the dataset are the number of bedrooms and the number of bathrooms. Use technology to create a multiple regression model to predict price based on all three variables: size, number of bedrooms, and number of bathrooms.

Price Size Beds Baths
145 1.3 3 1.5
875 2.9 7 3.75
300 1.5 3 2.5
370 1.1 2 1
268 1.5 2 2
1399 4.8 6 5
1125 3.1 3 2.5
299 1.4 3 2
110 1.2 3 1
2999 6 7 8
170 1 2 1
269 1.5 3 1.5
150 1 2 1.5
288 1.8 3 2.1
350 1.3 3 2
120 0.9 1 1
309 2.4 4 2.5
1500 1.5 2 1.5
635 2.5 4 2.5
350 0.9 2 1
459 1.8 4 2.5
275 2.9 4 1.5
275 1.8 3 2
2500 3.7 3 3
187 1.4 3 1.5
238 1.7 3 1.5
155 0.7 1 1
175 1.6 3 1.5
569 3.2 4 2
105 1.2 2
2.5

a) Which of the variables which are significant at the 5% level?

b) Which variable is the most significant predictor in this model?

c) What price does the model predict for a 1200 square foot (Size = 1.2) New York home with 3 bedrooms and 2 bathrooms? Round your answer to the nearest thousand dollars.

Answer 1

Explanation:

Hello!

The objective is to predict Y: The price of New York homes, based on:

X₁: Size of the house (thousands of square feet)

X₂: Number of bedrooms in the house.

X₃: Number of bathrooms in the house.

The multiple regression model is Yi= α + β₁X₁ + β₂X₂ + β₃X₃ + εi

The estimated multiple regression equation is ^Y= a + b₁X₁ + b₂X₂ + b₃X₃

^Y= -83.49 + 483.89X₁ - 254.86X₂ + 228.92X₃

a)

First I'll test if the multiple regression is significant, if it is, I'll test each variable independently using the F-test and calculated p-values:

1)

H₀: β₁= β₂= β₃= 0

H₁: At least one βi≠0

α: 0.05

F= 28.50

p-value < 0.0001

The p-value is less than α ⇒ The decision is to reject the null hypothesis.

Using a 5% significance level, there is significant evidence to conclude that the price of the house is influenced by its size, the number of bedrooms in the house modifies its price. and the number of bathrooms.

2) Independent hypotheses:

H₀: β₁= 0

H₁: β₁≠0

α: 0.05

F= 74.71

p-value < 0.001

The p-value is less than α ⇒ The decision is to reject the null hypothesis.

Using a 5% significance level, there is significant evidence to conclude that the size of the house modifies its price.

H₀: β₂= 0

H₁: β₂≠0

α: 0.05

F= 5.96

p-value: 0.0217

The p-value is less than α ⇒ The decision is to reject the null hypothesis.

Using a 5% significance level, there is significant evidence to conclude that the number of bedrooms in the house modifies its price.

H₀: β₃= 0

H₁: β₃≠0

α: 0.05

F= 4.82

p-value: 0.0373

The p-value is less than α ⇒ The decision is to reject the null hypothesis.

Using a 5% significance level, there is significant evidence to conclude that the number of bathrooms in the house modifies its price.

b)

To determine which variable has more influence in the variability of the price of the houses I've calculated the coefficient of determination for each combination:

Coefficient of determination of the multiple regression R²= 0.77

Price vs. Size R²= 0.67

Price vs. Nº Bedrooms R²= 0.29

Price vs. Nº Bathrooms R²= 0.61

The variable that explains better the variability in the average price of the houses is the one with the highest coefficient of determination, in this case, the size of the house seems to be the best predictor of the variability of the house price.

c)

You have to calculate the price ^Y for a house with size X₁= 1.2, bedrooms X₂= 3 and bathrooms X₃= 2

Replace the given values in the estimated equation:

^Y= -83.49 + 483.89*1.2 - 254.86*3 + 228.92*2

^Y= 190.438≅ 191 thousands of dollars

I hope this helps!