Question

In: Statistics and Probability

A real estate agent wants to develop a model to predict the selling price of a...

A real estate agent wants to develop a model to predict the selling price of a home. The agent believes that the most important variables in determining the price of a house are its:

  1. Size of the house (square footage)
  2. Number of bedrooms
  3. Lot size where the house is built (also in square feet).

He collects from the market the data on a number of houses, which include for every house the information listed above as well as the house price.

- Construct and report a regression model that allows predicting the selling price of the house if one knows house size, number of bedrooms, and a lot size.

- Characterize the model goodness of fit.

-Provide a plain interpretation for the coefficients (the slopes) of the model built in (a) above.  For each predictor variable, write your interpretation clearly and fully.

-Assuming that no required conditions are violated, test the validity of the regression model overall.  Specifically, write the hypothesis being tested, p-value for this hypothesis, and your conclusions.  Use 5% significance level.

-Are there any “useless” (not significant) independent variables in the regression model?  If yes, list them, and for each variable explain why it is insignificant (i.e. write down the hypotheses being tested, and p-value for each of them).  Use 5% significance level.

-Check all appropriate required conditions and for every required condition comment if it is violated or not.

Price Bedrooms House Size Lot Size
124100 3 1290 3900
218300 4 2080 6600
117800 3 1250 3750
168300 3 1550 4650
120400 3 1360 4050
159200 3 1450 4200
158000 4 2110 6600
73800 2 1270 4200
142500 4 1940 6300
160100 3 1290 4050
199200 4 2190 6900
179200 4 2030 6300
153800 3 1310 4350
150900 4 2300 7200
180100 4 1870 5700
132600 4 1920 6000
147200 4 1530 4500
149800 3 1350 4200
151500 3 1590 5100
132800 4 1680 5100
115300 3 1370 4200
196600 4 2130 6450
217400 4 1840 5700
106100 3 1600 4950
220900 4 2330 7200
162000 4 2290 6900
179000 4 2270 6900
107700 4 1910 5550
136900 4 2150 6450
115400 3 1230 3600
118500 3 1410 4500
208600 5 2360 7200
186700 4 2320 7050
131800 4 1530 4950
149400 3 1280 3900
155600 4 1690 5250
160300 3 1560 4800
131200 4 1810 5550
107300 3 1240 4050
109700 3 1320 4200
203100 4 1870 5700
144800 4 1920 6000
150400 3 1520 4800
96400 2 1070 3450
153500 3 1570 4800
139900 4 2260 7050
146900 4 1970 6000
136800 3 1360 4200
96400 3 1290 4050
148400 3 1550 5100
143100 2 1220 3750
191800 5 2330 7350
102000 3 1460 4500
147500 3 1410 4350
184300 4 2300 7050
178100 4 2220 6750
267800 5 2980 9150
245700 5 2950 9000
107000 3 1550 4800
137700 4 2010 6150
88900 3 1570 4800
98700 4 1660 5100
181200 4 2310 7350
199500 4 2200 6750
162400 4 1590 4950
125500 3 1360 4350
165400 4 2310 7350
209400 5 2790 8400
129800 4 1540 4950
192000 4 1780 5400
124700 3 1320 4350
147300 4 1780 5250
154700 4 1980 6000
122200 4 1590 5100
125000 4 1830 5850
253200 5 2340 7500
157800 3 1540 4800
123700 3 1200 3750
125500 4 1560 4650
130000 4 1520 4650
179800 4 2070 6150
150200 4 1840 5700
160900 4 1950 5850
153200 3 1280 4050
204200 4 2310 7050
215800 4 2380 7200
159700 3 1580 4800
180800 4 2140 6600
178800 5 2300 7050
120200 3 1370 4500
134200 4 1590 5100
134800 3 1480 4650
161500 4 1870 5700
155400 3 1520 4500
113200 3 1250 3750
180500 3 1320 3900
218100 5 2980 9000
117500 3 1570 4950
157400 3 1560 5100
155900 4 1620 4800

Solutions

Expert Solution

a. The regression model is

Price= +Bedrooms+HouseSize+LotSize

b. Coefficient of determination= 55.999% meaning that 55.999% of Dependent vaiable(Price) is determined by Independent coefficients (Bedrooms,HouseSize,LotSize).

c. If all other independent variables are fixed at some value then and increase in the bedroom by 1 unit will result in a 2306.080821 units increase in price.

If all other independent variables are fixed at some value then and increase in the HouseSize by 1 unit will result in a 74.29680602 units increase in price.

If all other independent variables are fixed at some value then and increase in the LotSize by 1 unit will result in a 4.36378288 units decrease in price.

37717.59451 units of price in price are independent of the variables.

d.

For Null Hypothesis

p value for intercept<0.05 and p value of bedroom, HouseSize and LotSize>0.05. Therefore the coefficients of Bedroom, HouseSize and Lotsize are in the confidence interval but the overall validity of the regression model is still questionable.

Do comment for queries and leave a positive remark

SUMMARY OUTPUT Regression Statistics Multiple R 0.748329963 R Square 0.559997733 Adjusted R 0.546247662 Standard E 25022.70761 Observatic 100 ANOVA df F ignificance F 40.7269 4.57E-17 Regressior Residual Total 3 96 99 SS M S 76501718347 2.55E+10 60109046053 6.26E+08 1.36611E+11 Coefficients Standard Error Stat P-value Lower 95% Upper 95% Lower 0.05% Upper 0.05% Intercept 37717.5945114176.74195 2.660526 0.009145 9576.984 65858.2 37708.6874 37726.5016 Bedrooms 2306.080821 6994.19244 0.329714 0.742335 -11577.3 16189.44 2301.68643 2310.47521 House Size 74.29680602 52.97857934 1.402393 0.164023 -30.8648 179.4585 74.2635201 74.330092 Lot Size -4.36378288 17.0240013 -0.25633 0.798244 -38.1562 29.42859 -4.3744789 -4.35308685 RESIDUAL OUTPUT Observatior Predicted PriceResiduals 1 123459.9635 640.0364917 2 172678.3073 45621.69269 3 121142.6587 -3342.658699 4 139504.2959 28795.70409 5 128006.1725 -7606.172498 6 134038.3176 25161.68239 7 174907.2115 -16907.21149 8 118358.8117 -44558.8117 9 163585.8893 -21085.88933 10 122805.3961 37294.60392 11 179541.8211 19658.17889 12 170272.6019 8927.398125 13 122982.1973 30817.80267 14 186405.3349 -35505.33491 15 161003.3826 19096.61736 16 163409.0881 -30809.08808 17 140979.008 6220.991953 18 126608.637 23191.36299 19 140512.4659 10987.53414 20 149505.2592 -16705.25922 21 128094.5731 -12794.57313 22 177047.715 19552.28495 23 158774.4785 58625.52154 24 141910.0014 -35810.00135 25 188634.2391 32265.76091 26 186971.5017 -24971.50171 27 185485.5656 -6485.565592 28 164629.8223 -56929.82231 29 178533.6512 -41633.65117 30 120311.29 -4911.290011 31 129757.3105 -11257.3105 32 193169.2241 15430.77591 33 188545.8385 -1845.838462 34 139015.3058 -7215.305751 35 122716.9954 26683.00455 36 149593.6599 6006.340149 37 139592.6965 20707.30346 38 157200.1417 -26000.14171 39 119090.5558 -11790.55578 40 124379.7328 -14679.73282 41 161003.3826 42096.61736 42 163409.0881 -18609.08808 43 136620.8243 13779.1757 44 106772.2877 -10372.28766 45 140335.6646 13164.3354 46 184088.0301 -44188.0301 47 167123.9284 -20223.92838 48 127351.6051 9448.394934 49 122805.3961 -26405.39608 50 137540.5936 10859.40638 51 116607.6737 26492.3263 52 190285.7525 1514.247521 53 133472.1508 -31472.1508 54 130411.8779 17088.12206 55 187059.9023 -2759.902341 56 182425.2927 -4325.292723 57 230723.8672 37076.13279 58 229149.5305 16550.46954 59 138849.7285 -31849.72848 60 169441.2332 -31741.23319 61 140335.6646 -51435.6646 62 148019.3231 -49319.3231 63 186493.7355 -5293.735537 64 180939.3566 18560.6434 65 143473.1141 18926.88589 66 126697.0376 -1197.037634 67 186493.7355 -21093.73554 68 219880.3112 -10480.31123 69 139758.2738 -9958.273811 70 155625.805 36374.19504 71 123725.1654 974.8346071 72 156280.3724 -8980.372393 73 167866.8964 -13166.89644 74 142818.5467 -20618.54668 75 157376.943 -32376.94297 76 190374.1531 62825.84689 77 138106.7604 19693.23958 78 117427.8184 6272.181602 79 142553.3448 -17053.3448 80 139581.4726 -9581.472555 81 173899.0415 5900.958452 82 158774.4785 -8574.478458 83 166292.5597 -5392.559689 84 122062.428 31137.57198 85 187802.8704 16397.1296 86 192349.0794 23450.92061 87 141078.6327 18621.36734 88 177136.1157 3663.884327 89 189365.9832 -10565.98316 90 126785.4383 -6585.438262 91 142818.5467 -8618.54668 92 134303.5195 496.4805072 93 161003.3826 496.6173611 94 137929.9592 17470.04083 95 121142.6587 -7942.658699 96 125688.8677 54811.13231 97 231378.4346 -13278.43464 98 139681.0972 -22181.09717 99 138283.5617 19116.43832 100 146356.5857 9543.414275

| 37717.59451

| 2306.080821

| 74.29680602

-4.36378288

Intercept Bedrooms House Size Lot Size Coefficients 37717.59451 2306.080821 74.29680602 -4.36378288 Standard Errort Stat P-value 14176.74195 2.660526 0.009144764 6994.19244 0.329714 0.742334676 52.97857934 1.402393 0.164023325 17.0240013 -0.25633 0.798243601

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SUMMARY OUTPUT Regression Statistics Multiple R 0.748329963 R Square 0.559997733 Adjusted R 0.546247662 Standard E 25022.70761 Observatic 100 ANOVA df F ignificance F 40.7269 4.57E-17 Regressior Residual Total 3 96 99 SS M S 76501718347 2.55E+10 60109046053 6.26E+08 1.36611E+11 Coefficients Standard Error Stat P-value Lower 95% Upper 95% Lower 0.05% Upper 0.05% Intercept 37717.5945114176.74195 2.660526 0.009145 9576.984 65858.2 37708.6874 37726.5016 Bedrooms 2306.080821 6994.19244 0.329714 0.742335 -11577.3 16189.44 2301.68643 2310.47521 House Size 74.29680602 52.97857934 1.402393 0.164023 -30.8648 179.4585 74.2635201 74.330092 Lot Size -4.36378288 17.0240013 -0.25633 0.798244 -38.1562 29.42859 -4.3744789 -4.35308685 RESIDUAL OUTPUT Observatior Predicted PriceResiduals 1 123459.9635 640.0364917 2 172678.3073 45621.69269 3 121142.6587 -3342.658699 4 139504.2959 28795.70409 5 128006.1725 -7606.172498 6 134038.3176 25161.68239 7 174907.2115 -16907.21149 8 118358.8117 -44558.8117 9 163585.8893 -21085.88933 10 122805.3961 37294.60392 11 179541.8211 19658.17889 12 170272.6019 8927.398125 13 122982.1973 30817.80267 14 186405.3349 -35505.33491 15 161003.3826 19096.61736 16 163409.0881 -30809.08808 17 140979.008 6220.991953 18 126608.637 23191.36299 19 140512.4659 10987.53414 20 149505.2592 -16705.25922 21 128094.5731 -12794.57313 22 177047.715 19552.28495 23 158774.4785 58625.52154 24 141910.0014 -35810.00135 25 188634.2391 32265.76091 26 186971.5017 -24971.50171 27 185485.5656 -6485.565592 28 164629.8223 -56929.82231 29 178533.6512 -41633.65117 30 120311.29 -4911.290011 31 129757.3105 -11257.3105 32 193169.2241 15430.77591 33 188545.8385 -1845.838462 34 139015.3058 -7215.305751 35 122716.9954 26683.00455 36 149593.6599 6006.340149 37 139592.6965 20707.30346 38 157200.1417 -26000.14171 39 119090.5558 -11790.55578 40 124379.7328 -14679.73282 41 161003.3826 42096.61736 42 163409.0881 -18609.08808 43 136620.8243 13779.1757 44 106772.2877 -10372.28766 45 140335.6646 13164.3354 46 184088.0301 -44188.0301 47 167123.9284 -20223.92838 48 127351.6051 9448.394934 49 122805.3961 -26405.39608 50 137540.5936 10859.40638 51 116607.6737 26492.3263 52 190285.7525 1514.247521 53 133472.1508 -31472.1508 54 130411.8779 17088.12206 55 187059.9023 -2759.902341 56 182425.2927 -4325.292723 57 230723.8672 37076.13279 58 229149.5305 16550.46954 59 138849.7285 -31849.72848 60 169441.2332 -31741.23319 61 140335.6646 -51435.6646 62 148019.3231 -49319.3231 63 186493.7355 -5293.735537 64 180939.3566 18560.6434 65 143473.1141 18926.88589 66 126697.0376 -1197.037634 67 186493.7355 -21093.73554 68 219880.3112 -10480.31123 69 139758.2738 -9958.273811 70 155625.805 36374.19504 71 123725.1654 974.8346071 72 156280.3724 -8980.372393 73 167866.8964 -13166.89644 74 142818.5467 -20618.54668 75 157376.943 -32376.94297 76 190374.1531 62825.84689 77 138106.7604 19693.23958 78 117427.8184 6272.181602 79 142553.3448 -17053.3448 80 139581.4726 -9581.472555 81 173899.0415 5900.958452 82 158774.4785 -8574.478458 83 166292.5597 -5392.559689 84 122062.428 31137.57198 85 187802.8704 16397.1296 86 192349.0794 23450.92061 87 141078.6327 18621.36734 88 177136.1157 3663.884327 89 189365.9832 -10565.98316 90 126785.4383 -6585.438262 91 142818.5467 -8618.54668 92 134303.5195 496.4805072 93 161003.3826 496.6173611 94 137929.9592 17470.04083 95 121142.6587 -7942.658699 96 125688.8677 54811.13231 97 231378.4346 -13278.43464 98 139681.0972 -22181.09717 99 138283.5617 19116.43832 100 146356.5857 9543.414275

| 37717.59451

| 2306.080821

| 74.29680602

-4.36378288

Intercept Bedrooms House Size Lot Size Coefficients 37717.59451 2306.080821 74.29680602 -4.36378288 Standard Errort Stat P-value 14176.74195 2.660526 0.009144764 6994.19244 0.329714 0.742334676 52.97857934 1.402393 0.164023325 17.0240013 -0.25633 0.798243601

Ho: H = Coefficients H:p Coefficients

orchestra answered 1 year ago
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