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In: Statistics and Probability

There are numerous variables that are believed to be predictors of housing prices, including living area...

There are numerous variables that are believed to be predictors of housing prices, including living area (square feet), number of bedrooms, and number of bathrooms. The data in the Case Study No. 2.xlsx file pertains to a random sample of houses located in a particular geographic area.

  1. Develop the following simple linear regression models to predict the sale price of a house based upon a 90% level of confidence. Write the regression equation for each model.
    1. Sale price based upon square feet of living area.
    2. Sale price based upon number of bedrooms.
    3. Sale price based upon number of bathrooms.
  2. Develop the following multiple linear regression models to predict the sale price of a house based upon a 90% level of confidence. Write the regression equation for each model.
    1. Sale price based upon square feet of living area and number of bedrooms.
    2. Sale price based upon square feet of living area and number of bathrooms.
    3. Sale price based upon number of bedrooms and number of bathrooms.
    4. Sale price based upon square feet of living area, number of bedrooms, and number of bathrooms.
  3. Discuss the joint statistical significance of each of the preceding simple and multiple linear regression models at a 90% level of confidence and 95% level of confidence.
  4. Discuss the individual statistical significance of the coefficient for each independent variable for each of the preceding simple and multiple linear regression models at a 90% level of confidence and 95% level of confidence.
  5. Compare any of the preceding simple and multiple linear regression models that were found to be jointly and individually statistically significant at a 90% level of confidence and select the preferred regression model. Explain your selection using the appropriate regression statistics.
  6. Interpret the coefficient for each independent variable (or variables) associated with your selected preferred regression model.
  7. Using the preferred regression model, predict the sale price of a house with the following values for the independent variables: 3,000 square feet of living area, 3 bedrooms, and 2.5 bathrooms.

Prepare a single Microsoft Excel file using a separate worksheet for each question and upload your Excel file for grading via the Blackboard submission link.

Selling Price Living Area (Sq Feet) No. Bathrooms No Bedrooms
$145,000 1,152 1 2
$103,000 1,290 1.5 3
$210,000 2,396 1.5 4
$559,000 3,090 4 4
$218,000 1,428 1 3
$262,138 1,631 2.5 3
$125,000 1,368 1 3
$130,000 1,134 1 3
$157,500 1,697 1.5 3
$193,000 1,666 2.5 3
$275,000 1,738 2.5 4
$240,000 1,457 1.5 2
$200,136 1,632 2.5 3
$395,000 2,186 2.5 3
$366,703 2,117 2.5 3
$103,150 936 1 3
$310,000 3,347 2.5 6
$142,900 1,824 2.5 4
$359,770 2,592 3 3

Solutions

Expert Solution

a.Sale price based upon square feet of living area.

The regression equation is given by

Y = 142.2543959X - 23064.61598

Y = Sales price in dollars

X = Square feet of living area.

b.Sale price based upon number of bedrooms.

The regression equation is given by

Y = 40113.90385X + 105696.5769

Y = Sales price in dollars

X = No of Bedrooms

c.Sale price based upon the number of bathrooms

The regression equation is given by

Y = 113369.8846X + 9854.809717

Y = Sales price in dollars

X = No of Bathrooms

a.Sale price based upon square feet of living area and number of bedrooms.

Regression equation is given by

Y = 213.9101105X1 - 74833.98496X2 + 90336.18553

Y = Sales price in dollars

X1 = Square feet of living area.

X2 = No of Bedrooms

b.Sale price based upon square feet of living area and number of bathrooms.

Regression equation is given by

Y = 73.07796607X1 + 71746.36155X2 - 40288.5095

Y = Sales price in dollars

X1 = Square feet of living area.

X2 = No of Bathrooms

c.Sale price based upon the number of bedrooms and number of bathrooms.

Regression equation is given by

Y = -6569.109081X1 + 116149.1231X2 + 25732.37296

Y = Sales price in dollars

X1 = No of Bedrooms

X2 = No of Bathrooms

d.Sale price based upon square feet of living area, number of bedrooms, and number of bathrooms

The regression equation is given by

Y = 148.2555647X1 - 61665.83682X2 + 55016.32877X3 + 57174.06061

Y = Sales price in dollars

X1 = Square feet of living area.

X2 = No of Bedrooms

X3 = No of Bathrooms

Question 3:

At 90% Significance

a. Sale price based upon square feet of living area, number of bedrooms, and number of bathrooms

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.904252688
R Square 0.817672923
Adjusted R Square 0.781207508
Standard Error 55765.64154
Observations 19
ANOVA
df SS MS F Significance F
Regression 3 2.09196E+11 69731957514 22.42324 8.49E-06
Residual 15 46647101649 3109806777
Total 18 2.55843E+11
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 90.0% Upper 90.0%
Intercept 57174.06061 52858.40362 1.081645617 0.296493 -55491 169839.1 -35489.4 149837.5
X Variable 1 148.2555647 40.52520877 3.658354124 0.00233 61.87813 234.633 77.21283 219.2983
X Variable 2 -61665.83682 22374.88195 -2.756029594 0.014707 -109357 -13974.9 -100890 -22441.5
X Variable 3 55016.32877 23821.56476 2.309517839 0.035564 4241.865 105790.8 13255.93 96776.73

At the significance of 95%

b.Sale price based upon square feet of living area, number of bedrooms, and number of bathrooms

Regression Statistics
Multiple R 0.904252688
R Square 0.817672923
Adjusted R Square 0.781207508
Standard Error 55765.64154
Observations 19
ANOVA
df SS MS F Significance F
Regression 3 2.09196E+11 69731957514 22.42324 8.49E-06
Residual 15 46647101649 3109806777
Total 18 2.55843E+11
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept 57174.06061 52858.40362 1.081645617 0.296493 -55491 169839.1 -55491 169839.1
X Variable 1 148.2555647 40.52520877 3.658354124 0.00233 61.87813 234.633 61.87813 234.633
X Variable 2 -61665.83682 22374.88195 -2.756029594 0.014707 -109357 -13974.9 -109357 -13974.9
X Variable 3 55016.32877 23821.56476 2.309517839 0.035564 4241.865 105790.8 4241.865 105790.8

5.Preferred model is based on the regression equation is defined based on R2 value. Higher the value of R2 the better the model suitability explaining the variation in the dependent variable (i.e. Sales price). So the most preferred model is given by

Y = 148.2555647X1 - 61665.83682X2 + 55016.32877X3 + 57174.06061

6. Preferred regression equation is given by

Y = 148.2555647X1 - 61665.83682X2 + 55016.32877X3 + 57174.06061

Y = Sales price in dollars

X1 = Square feet of living area.

X2 = No of Bedrooms

X3 = No of Bathrooms

This means

Variation in Sales price is effected by 148.2 times in Square feet of living area.

Variation in Sales price is effected by -61665.8 times No of Bedrooms.

Variation in Sales price is effected by 55016.3 times No of Bathrooms

7. Regression equation is given by

Y = 148.2555647X1 - 61665.83682X2 + 55016.32877X3 + 57174.06061

Here X1 = 3000sqft

         X2 = 3 Bedrooms

        X3 = 2.5 Bathrooms

So Y = $454484.1

orchestra answered 2 years ago
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