Question

1.             Develop a simple linear regression model to predict the price of a house based upon the living area (square feet) using a 95% level of confidence.

1.             Write the reqression equation
2.             Discuss the statistical significance of the model as a whole using the appropriate regression statistic at a 95% level of confidence.
3.              Discuss the statistical significance of the coefficient for the independent variable using the appropriate regression statistic at a 95% level of confidence.
4.             Interpret the coefficient for the independent variable.
5.             What percentage of the observed variation in housing prices is explained by the model?
6.              Predict the value of a house with 3,000 square feet of living area.

2.             Develop a simple linear regression model to predict the price of a house based upon the number of bedrooms using a 95% level of confidence.

1.             Write the reqression equation
2.             Discuss the statistical significance of the model as a whole using the appropriate regression statistic at a 95% level of confidence.
3.              Discuss the statistical significance of the coefficient for the independent variable using the appropriate regression statistic at a 95% level of confidence.
4.             Interpret the coefficient for the independent variable.
5.             What percentage of the observed variation in housing prices is explained by the model?
6.              Predict the value of a house with 3 bedrooms.

3.             Develop a simple linear regression model to predict the price of a house based upon the number of bathrooms using a 95% level of confidence.

1.             Write the reqression equation
2.             Discuss the statistical significance of the model as a whole using the appropriate regression statistic at a 95% level of confidence.
3.              Discuss the statistical significance of the coefficient for the independent variable using the appropriate regression statistic at a 95% level of confidence.
4.             Interpret the coefficient for the independent variable.
5.             What percentage of the observed variation in housing prices is explained by the model?
6.              Predict the value of a house with 2.5 bathrooms.

4.             Develop a simple linear regression model to predict the price of a house based upon its age using a 95% level of confidence.

1.             Write the reqression equation
2.             Discuss the statistical significance of the model as a whole using the appropriate regression statistic at a 95% level of confidence.
3.              Discuss the statistical significance of the coefficient for the independent variable using the appropriate regression statistic at a 95% level of confidence.
4.             Interpret the coefficient for the independent variable.
5.             What percentage of the observed variation in housing prices is explained by the model?
6.              Predict the value of a house that is 50 years old.

5.             Compare the preceding four simple linear regression models to determine which model is the preferred model. Use the Significance F values, p-values for independent variable coefficients, R-squared or Adjusted R-squared values (as appropriate), and standard errors to explain your selection.
6.             Calculate the predicted sale price of a 50 year old house with 3,000 square feet of living area, 3 bedrooms, and 2.5 bathrooms using your preferred regression model from part 5.

Prepare a single Microsoft Excel file, using a separate worksheet for each regression model, to document your regression analyses. Prepare a single Microsoft Word document that outlines your responses for each portions of the case study.

Selling Price        Age (Years)         Living Area (Sq Feet)       No. Bathrooms No Bedrooms

$92,000                 18           1,527     2              4

$211,002              0              2,195     2.5          4

$115,000              14           1,480     1.5          3

$113,000              53           1,452     2              3

$216,300              0              2,360     2.5          4

$145,000              32           1,440     1              3

$114,000              14           1,480     2.5          2

$139,050              125         1,879     2.5          3

$104,000              14           1,480     1.5          3

$169,900              11           1,792     2.5          3

$177,900              2              1,386     2.5          3

$133,000              14           1,676     2              2

$185,000              0              768         2              4

$115,000              16           1,560     1.5          3

$100,000              91           1,000     1              3

$117,000              15           1,676     1.5          4

$150,000              11           1,656     1.5          3

$187,500              11           2,300     1.5          3

$107,000              25           1,712     1              3

$126,900              26           1,350     1.5          3

$147,000              15           1,676     2.5          3

$62,000                 103         1,317     1.5          3

$101,000              30           1,056     2              3

$143,500              13           912         1              3

$113,400              18           1,232     2              2

$112,000              36           1,280     1              3

$112,500              43           1,232     1              3

$97,000                 45           1,406     1.5          3

$121,000              6              1,164     2              3

$65,720                 123         1,198     1              3

$225,000              10           2,206     2.5          4

Answer 1

Develop a simple linear regression model to predict the price of a house based upon the living area (square feet) using a 95% level of confidence

Dependent variable = price
Independent variable = area
1. Put the values in excel as shown below.

2. We use the regression option under the Data analysis tab.

3. Input the data as shown below.

4. The output will be generated as follows

5. We formulate the regression equation using the output (highlighted in yellow).

Write the reqression equation
Price = 39438.54 + 61.8185 Area

Discuss the statistical significance of the model as a whole using the appropriate regression statistic at a 95% level of confidence.

For this we look that pvalue of the Anova. The Pvalue of anova (highlighted in green)
Since the pvalue is less than 0.05, the model is significant.

Discuss the statistical significance of the coefficient for the independent variable using the appropriate regression statistic at a 95% level of confidence.
For this, we look that pvalue of the coefficient . The Pvalue of the coefficient (highlighted in orange)
Since the pvalue is less than 0.05, hence the variable is significant in predicting the dependent variable.

Interpret the coefficient for the independent variable.
One unit increase in the area, increases the price by 61.81 dollars

What percentage of the observed variation in housing prices is explained by the model?
Variation explained by the model is given by the Rsquare (highlighted in blue) which 0.3444
Note - Higher the number, better the model is.

Predict the value of a house with 3,000 square feet of living area.
Price = 39438.54 + 61.8185 Area
Price = 39438.54 + 61.8185 (3000) =224894.30

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Follow the similar steps for the next variable. The screenshot are provided in case you get stuck.

Develop a simple linear regression model to predict the price of a house based upon the number of bedrooms using a 95% level of confidence.

Dependent variable = price
Independent variable = bedrooms

Write the reqression equation
Price = 30701.86 + 32988.68 Bedroom

Discuss the statistical significance of the model as a whole using the appropriate regression statistic at a 95% level of confidence.

For this we look that pvalue of the Anova. The Pvalue of anova (highlighted in green)
Since the pvalue is less than 0.05, the model is significant.

Discuss the statistical significance of the coefficient for the independent variable using the appropriate regression statistic at a 95% level of confidence.
For this we look that pvalue of the coefficient . The Pvalue of the coefficient (highlighted in orange)
Since the pvalue is less than 0.05, hence the variable is significant in predicting the dependent variable.

Interpret the coefficient for the independent variable.
One unit increase in the bedroom increases the price by 32988.68 dollars

What percentage of the observed variation in housing prices is explained by the model?
Variation explained by the model is given by the Rsquare (highlighted in blue) which 0.1873
Note - Higher the number, better the model is.

Predict the value of a house with 3 bedrooms.
Price = 30701.86 + 32988.68 Bedroom
Price = 30701.86 + 32988.68 (3)=129667.93

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Develop a simple linear regression model to predict the price of a house based upon the number of bathrooms using a 95% level of confidence.

Dependent variable = price
Independent variable = bathroom

Write the reqression equation
Price = 63247.78 + 39596.15 bathroom

Discuss the statistical significance of the model as a whole using the appropriate regression statistic at a 95% level of confidence.

For this we look that pvalue of the Anova. The Pvalue of anova (highlighted in green)
Since the pvalue is less than 0.05, the model is significant.

Discuss the statistical significance of the coefficient for the independent variable using the appropriate regression statistic at a 95% level of confidence.
For this we look that pvalue of the coefficient . The Pvalue of the coefficient (highlighted in orange)
Since the pvalue is less than 0.05, hence the variable is significant in predicting the dependent variable.

Interpret the coefficient for the independent variable.
One unit increase in the bathroom increases the price by 39596.157 dollars

What percentage of the observed variation in housing prices is explained by the model?
Variation explained by the model is given by the Rsquare (highlighted in blue) which 0.2924
Note - Higher the number, better the model is.

Predict the value of a house with 2.5 bathrooms.
Price = 63247.78 + 39596.15 bathroom
Price = 63247.78 + 39596.15 (2.5)=162238.18

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Develop a simple linear regression model to predict the price of a house based upon its age using a 95% level of confidence.

Dependent variable = price
Independent variable = age

Write the reqression equation
Price = 152772.22 -660.88 age

Discuss the statistical significance of the model as a whole using the appropriate regression statistic at a 95% level of confidence.

For this we look that pvalue of the Anova. The Pvalue of anova (highlighted in green)
Since the pvalue is less than 0.05, the model is significant.

Discuss the statistical significance of the coefficient for the independent variable using the appropriate regression statistic at a 95% level of confidence.
For this we look that pvalue of the coefficient . The Pvalue of the coefficient (highlighted in orange)
Since the pvalue is less than 0.05, hence the variable is significant in predicting the dependent variable.

Interpret the coefficient for the independent variable.
One  unit increase in the age decreases the price by 660.8855
dollars

What percentage of the observed variation in housing prices is explained by the model?
Variation explained by the model is given by the Rsquare (highlighted in blue) which 0.3066
Note - Higher the number, better the model is.

Predict the value of a house that is 50 years old.
Price = 152772.22 -660.88 age
Price = 152772.22 -660.88 (50)= 119727.95

Compare the preceding four simple linear regression models to determine which model is the preferred model. Use the Significance F values, p-values for independent variable coefficients, R-squared or Adjusted R-squared values (as appropriate), and standard errors to explain your selection.
Calculate the predicted sale price of a 50 year old house with 3,000 square feet of living area, 3 bedrooms, and 2.5 bathrooms using your preferred regression model from part 5.

This is a summary of all the model. We see that all the model are significant and the variable are significant predictors of the dependent variable.

However we see that the Rsquare for model with Area is higher compared to other. This indicates that the model is able to explain a higher percentage of the variability of y.

Hence I would prefer the model with Area.

Price = 39438.54 + 61.8185 Area
Price = 39438.54 + 61.8185 (3000) =224894.30