Chad Dobson is considering real estate investment near a community college. The accompanying data file includes monthly rents (in $) for 27 houses, along with three characteristics of the home: number of bedrooms (Beds), number of bathrooms (Baths), and square footage (Sqft). Rent Beds Baths Sqft 2950 4 4 1453 2400 4 2 1476 2375 3 3 1132 2375 3 3 1132 2350 4 2.5 1589 2000 3 2.5 1459 1935 3 2 1200 1825 3 2 1248 1810 2 2 898 1735 3 2.5 1060 1695 3 2 1100 1405 3 1 1030 1375 2 1 924 1365 2 1 974 1325 2 2 988 1275 2 2 880 1200 1 1 712 1180 2 1.5 890 1180 2 2 960 1115 2 1 1020 1100 2 1 903 1060 1 1 724 1007 3 2 1260 850 2 1.5 890 810 1 1 570 785 1 1 475 744 2 1 930 1. Estimate the linear model that uses Rent as the response variable. Note: Negative values should be indicated by a minus sign. Round your answers to 2 decimal places. 2. Estimate the exponential model that uses log of Rent as the response variable. Note: Negative values should be indicated by a minus sign. Round your answers to 2 decimal places. 3. Compute the predicted rent for a 1,500-square-foot house with three bedrooms and two bathrooms for the linear and the exponential models (ignore the significance tests). Note: Do not round intermediate calculations and round your final answers to 2 decimal places
Chad Dobson is considering real estate investment near a community college. The accompanying data file includes monthly rents (in $) for 27 houses, along with three characteristics of the home: number of bedrooms (Beds), number of bathrooms (Baths), and square footage (Sqft).
| Rent | Beds | Baths | Sqft |
| 2950 | 4 | 4 | 1453 |
| 2400 | 4 | 2 | 1476 |
| 2375 | 3 | 3 | 1132 |
| 2375 | 3 | 3 | 1132 |
| 2350 | 4 | 2.5 | 1589 |
| 2000 | 3 | 2.5 | 1459 |
| 1935 | 3 | 2 | 1200 |
| 1825 | 3 | 2 | 1248 |
| 1810 | 2 | 2 | 898 |
| 1735 | 3 | 2.5 | 1060 |
| 1695 | 3 | 2 | 1100 |
| 1405 | 3 | 1 | 1030 |
| 1375 | 2 | 1 | 924 |
| 1365 | 2 | 1 | 974 |
| 1325 | 2 | 2 | 988 |
| 1275 | 2 | 2 | 880 |
| 1200 | 1 | 1 | 712 |
| 1180 | 2 | 1.5 | 890 |
| 1180 | 2 | 2 | 960 |
| 1115 | 2 | 1 | 1020 |
| 1100 | 2 | 1 | 903 |
| 1060 | 1 | 1 | 724 |
| 1007 | 3 | 2 | 1260 |
| 850 | 2 | 1.5 | 890 |
| 810 | 1 | 1 | 570 |
| 785 | 1 | 1 | 475 |
| 744 | 2 | 1 | 930 |
1. Estimate the linear model that uses Rent as the response variable.
Note: Negative values should be indicated by a minus sign. Round your answers to 2 decimal places.
2. Estimate the exponential model that uses log of Rent as the response variable.
Note: Negative values should be indicated by a minus sign. Round your answers to 2 decimal places.
3. Compute the predicted rent for a 1,500-square-foot house with three bedrooms and two bathrooms for the linear and the exponential models (ignore the significance tests).
Note: Do not round intermediate calculations and round your final answers to 2 decimal places
The model is given by:
Rent = β0 + β1 × Beds + β2 × Baths + β3 × Sqft + ε
where β0 is the intercept, β1, β2, and β3 are the regression coefficients for Beds, Baths, and Sqft respectively, and ε is the error term.
The formulas for the parameter estimates are:
β̂0 = y̅ - β̂1 x̅1 - β̂2 x̅2 - β̂3 x̅3
β̂1 = ∑(xi1 - x̅1)(yi - y̅) / ∑(xi1 - x̅1)^2
β̂2 = ∑(xi2 - x̅2)(yi - y̅) / ∑(xi2 - x̅2)^2
β̂3 = ∑(xi3 - x̅3)(yi - y̅) / ∑(xi3 - x̅3)^2
where x̅1, x̅2, and x̅3 are the means of Beds, Baths, and Sqft, respectively, and y̅ is the mean of Rent.
Using the formulas and the data, we can compute the parameter estimates:
y̅ = 1535.56
x̅1 = 2.444
x̅2 = 1.944
x̅3 = 995.07
Sum of squares:
SSxx1 = ∑(xi1 - x̅1)^2 = 16.938
SSxx2 = ∑(xi2 - x̅2)^2 = 4.055
SSxx3 = ∑(xi3 - x̅3)^2 = 1208295.937
SSxy1 = ∑(xi1 - x̅1)(yi - y̅) = -1285.19
SSxy2 = ∑(xi2 - x̅2)(yi - y̅) = -838.06
SSxy3 = ∑(xi3 - x̅3)(yi - y̅) = 659328.94
Using the above values, we can compute the parameter estimates:
Next, we calculate the regression coefficients:
beta1 = SSxy1 / SSxx1 = -1285.19 / 16.938 = -75.85
beta2 = SSxy2 / SSxx2 = -838.06 / 4.055 = -206.68
beta3 = SSxy3 / SSxx3 = 659328.94 / 1208295.937 = 0.546
Finally, we can use the regression coefficients to calculate the intercept:
beta0 = mean(Rent) - beta1*mean(Beds) - beta2*mean(Baths) - beta3*mean(Sqft)
= 1535.56 - (-75.85)*2.444 - (-206.68)*1.944 - 0.546*995.07
= 1096.56
Thus, the multiple linear regression model is:
Rent = 1096.56 - 75.85Beds - 206.68Baths + 0.546*Sqft + ε
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