# Unlocking the Power of Linear Regression Analysis in Predictive Analytics

Embark on a journey of statistical mastery with our comprehensive guide to linear regression analysis. Dive into the intricacies of predictive analytics, unraveling the secrets behind house sale price predictions. From assessing model assumptions to deciphering the significance of key predictors, this resource equips you with the knowledge to make informed decisions in the realm of statistical modeling. Explore the nuanced world of regression coefficients, understand the implications of heteroscedasticity, and gain valuable insights into autocorrelation. Whether you're a student delving into statistical assignments or a professional navigating real-world data, this material serves as your compass in the fascinating landscape of linear regression.

## Problem Description:

This linear regression assignment involves conducting a thorough analysis of a regression model to predict the logarithm of house sale prices based on various independent variables. The goal is to assess the model's assumptions, identify significant predictors, and provide meaningful interpretations.

### Analysis Overview:

1. Normality and Linearity Assumptions:

• Histogram of Standardized Residuals: Indicates approximately normal distribution with minor deviations.
• Normal P-P plot: Confirms linearity assumption without significant violations.

2. Heteroscedasticity Check:

• Plot of ZPRED and ZRESID: Suggests no heteroscedasticity issue.
• Possible remedies for heteroscedasticity are discussed.

3. Model Coefficients:

• Coefficient summary with unstandardized and standardized coefficients, t-values, and significance levels.

Coefficient

Model

Unstandardized Coefficients

Standardized Coefficients

t

Sig.

Collinearity Statistics

B

Std. Error

Beta

Tolerance

VIF

1

(Constant)

8.780

.110

79.834

.000

log of house size

.360

.011

.311

31.673

.000

.253

3.954

Res: Bedrooms

-.021

.003

-.041

-6.401

.000

.592

1.690

total number of bathrooms (full, half and   quarter combined)

.013

.004

.025

3.048

.002

.352

2.843

yr91

-.420

.011

-.246

-38.705

.000

.605

1.653

yr92

-.400

.010

-.262

-39.686

.000

.561

1.784

yr93

-.374

.010

-.250

-37.982

.000

.566

1.768

yr94

-.347

.010

-.231

-36.131

.000

.596

1.678

yr95

-.337

.011

-.211

-31.686

.000

.552

1.810

yr96

-.284

.010

-.190

-29.729

.000

.599

1.668

yr97

-.198

.009

-.144

-21.885

.000

.564

1.775

yr98

-.097

.009

-.074

-11.215

.000

.560

1.786

.156

.004

.351

34.949

.000

.242

4.130

Res: Bath: Full count

.022

.006

.023

3.951

.000

.700

1.429

total number of fireplace (single story +   multi story)

.036

.005

.043

7.045

.000

.659

1.516

dummy for low quality house

-.012

.008

-.013

-1.490

.136

.340

2.943

Alogna

-.016

.051

-.002

-.320

.749

.933

1.072

Blackdia

.132

.052

.014

2.522

.012

.836

1.196

Bothell

-.020

.022

-.005

-.887

.375

.891

1.123

Burien

-.175

.018

-.051

-9.776

.000

.905

1.105

Carnation

.117

.062

.010

1.900

.057

.833

1.200

Clydehil

.235

.049

.025

4.847

.000

.935

1.070

Covingto

-.084

.032

-.014

-2.606

.009

.827

1.210

Desmoine

-.224

.018

-.072

-12.786

.000

.761

1.315

Duvall

.027

.036

.004

.751

.453

.693

1.442

enumclaw

.178

.028

.044

6.467

.000

.530

1.887

Federal

-.140

.013

-.085

-10.447

.000

.369

2.713

Issaquah

.032

.027

.007

1.184

.236

.778

1.285

Kent

-.135

.013

-.057

-9.982

.000

.746

1.340

Lakefore

-.075

.025

-.016

-3.002

.003

.874

1.144

Medina

.311

.039

.041

7.976

.000

.924

1.083

Mapleval

-.021

.029

-.005

-.737

.461

.635

1.575

Pacific

-.086

.038

-.012

-2.245

.025

.892

1.121

Redmond

-.009

.014

-.004

-.645

.519

.639

1.565

Renton

-.080

.013

-.036

-6.363

.000

.764

1.309

Shoreline

-.161

.020

-.045

-8.210

.000

.800

1.250

sammamis

-.059

.034

-.009

-1.739

.082

.866

1.155

Seatac

-.228

.018

-.066

-12.785

.000

.930

1.076

Tukwila

-.124

.028

-.027

-4.503

.000

.665

1.503

Woodinvi

.017

.029

.003

.606

.545

.891

1.123

Yarrowpo

.087

.093

.005

.944

.345

.981

1.019

Month

.002

.003

.012

.625

.532

.065

15.479

Winter

-.035

.024

-.033

-1.459

.145

.047

21.127

Spring

.001

.017

.002

.087

.931

.077

12.943

Summer

-.013

.010

-.014

-1.308

.191

.229

4.373

yr2000

.086

.009

.062

9.575

.000

.578

1.729

property tax rate (use this instead of   tax rate variable)

-.012

.005

-.020

-2.190

.029

.296

3.381

log of auto non-retail accessibiliy

-.019

.006

-.041

-3.155

.002

.148

6.755

log of lot size

.034

.006

.038

5.770

.000

.571

1.752

Bellevue

-.011

.010

-.008

-1.180

.238

.578

1.730

AM Single-Occupancy Vehicle Travel Time   to CBD

-.010

.000

-.301

-21.828

.000

.128

7.788

Property crime rate

-.001

.000

-.039

-3.946

.000

.255

3.916

lake view

.321

.014

.114

22.316

.000

.941

1.063

Figure 1: A summary of the coefficients

4. Significant Independent Variables:

• Identified at different significance levels (1%, 5%, 10%).
• Interpretations provided for selected variables.

5. Durbin-Watson Statistic:

• DW statistic of 1.091 indicates positive autocorrelation, discussed further.

Significant Predictors:

a) IVs at 1% (1%) level:

• List of variables significant at a 1% significance level.

b) IVs at 5% (5%) level:

• List of variables significant at a 5% significance level.

c) IVs at 10% (10%) level:

• List of variables significant at a 10% significance level.

Interpretations:

d) Selected Interpretations:

• Interpretations for size of the house, quality, and lake view.

Durbin-Watson Statistic:

e) Autocorrelation Check:

• Explanation of the DW statistic indicating positive autocorrelation.

Part 2: LIMDEP Model Summary:

B Std. Err. t P Lower Upper
constant 8.287492 .1048458 79.04 0.000 8.081975 8.493009
lnsqfttotl .3573995 .0113487 31.49 0.000 .335154 .37964510
bedrooms -.021482 .0031982 -6.72 0.000 -.027751 -.015213
bathroom .0142773 .0043052 3.32 0.001 .0058384 .0227162
yr91 -.41144 .0109074 -37.72 0.000 -.4328205 -.3900596
yr92 -.393549 .0099867 -39.41 0.000 -.4131249 -.3739732
yr93 -.3680253 .0097556 -37.72 0.000 -.387148 -.3489025
yr94 -.3413064 .0095704 -35.66 0.000 -.3600662 -.3225465
yr95 -.3269956 .0106567 -30.68 0.000 -.3478846 -.3061067
yr96 -.2799998 .009532 -29.37 0.000 -.2986842 -.2613155
yr97 -.1968071 .0090232 -21.81 0.000 -.2144942 -.17912
yr98 -.0940165 .0086182 -10.91 0.000 -.1109098 -.0771233
bathfull .0228463 .0055808 4.09 0.000 .011907 .0337856
bldggrad .15711 .0044595 35.23 0.000 .1483685 .1658516
fireplac .0371908 .0051326 7.25 0.000 .02713 .0472517
lgrad -.0167043 .0079252 -2.11 0.035 -.0322393 -.0011694
alogna -.019829 .0507218 -0.39 0.696 -.1192531 .079595
blackdia .2381418 .0527658 4.51 0.000 .1347112 .3415724
bothell -.0303834 .0223452 -1.36 0.174 -.074184 .0134172
burien -.1616938 .0176853 -9.14 0.000 -.1963602 -.1270275
carnatio .3190485 .0619847 5.15 0.000 .1975472 .4405498
clydehil .2166883 .0487557 4.44 0.000 .1211181 .3122585
covingto -.0778158 .0330907 -2.35 0.019 -.1426796 -.012952
desmoine -.1946276 .0172268 -11.30 0.000 -.2283953 -.1608599
duvall .1607877 .0364608 4.41 0.000 .0893178 .2322576
enumclaw .2064602 .0275777 7.49 0.000 .1524028 .2605176
federalw -.1443671 .0133985 -10.77 0.000 -.1706306 -.1181035
issaquah .0938296 .0269198 3.49 0.000 .0410618 .1465973
kent -.1256169 .0134561 -9.34 0.000 -.1519932 -.0992406
lakefore -.0661567 .0248777 -2.66 0.008 -.1149217 -.0173918
medina .2963711 .0395984 7.48 0.000 .2187509 .3739913
mapleval .0503367 .0288636 1.74 0.081 -.0062413 .1069147
pacific -.0913396 .0380909 -2.40 0.017 -.1660047 -.0166746
redmond -.0327365 .0135666 -2.41 0.016 -.0593295 -.0061436
renton -.0614642 .0126152 -4.87 0.000 -.0861922 -.0367361
shorelin -.1517455 .0190818 -7.95 0.000 -.1891493 -.1143418
sammamis -.0239609 .0338048 -0.71 0.478 -.0902246 .0423028
seatac -.227214 .0177901 -12.77 0.000 -.2620859 -.1923422
tukwila -.1310121 .0274384 -4.77 0.000 -.1847964 -.0772278
woodinvi -.0069122 .0282333 -0.24 0.807 -.0622546 .0484303
yarrowpo .0622008 .0926457 0.67 0.502 -.1194018 .2438034
month .0017777 .0026589 0.67 0.504 -.0034342 .0069896
winter -.0344254 .0242339 -1.42 0.155 -.0819283 .0130774
spring .002014 .016871 0.12 0.905 -.0310562 .0350843
summer -.0140388 .0101285 -1.39 0.166 -.0338927 .005815
yr2000 .0827017 .0090152 9.17 0.000 .0650303 .1003731
revtaxkc -.0164611 .0062426 -2.64 0.008 -.0286977 -.0042245
lnretac .0369578 .0058725 6.29 0.000 .0254467 .0484689
lnlotsize .0373261 .0059069 6.32 0.000 .0257474 .0489047
bellevue -.0151431 .009563 -1.58 0.113 -.0338883 .003602
cbd_ama -.0078237 .0004304 -18.18 0.000 -.0086674 -.00698
pcrate -.0004554 .0001356 -3.36 0.001 -.0007212 -.0001896
lakeview .3297589 .0143683 22.95 0.000 .3015944 .3579234

Figure 2: LIMDEP Model Summary

• R-square: 0.7365

b) Comparison with SPSS Model:

• Slight difference in adjusted R-square values.

Significant Predictors in LIMDEP Model:

d) IVs at 1%, 5%, and 10% levels:

• Significant variables at different significance levels.