1. Texas Weather Example Multiple Linear Regression

2. Data Response (Y) – Average January High Temp Predictors: –Latitude –Elevation –Longitude Units – n=16 County Weather Stations CountyTempLatElevLong Harris5629.7674195.367 Dallas4832.8544096.85 Kennedy6026.9332597.8 Midland4631.952851102.183 Deaf Smith3834.83840102.467 Knox4633.45146199.633 Maverick5328.7815100.483 Nolan4632.452380100.533 El Paso4431.83918106.4 Collington4134.852040100.217 Pecos4730.8673000102.9 Sherman3636.353693102.083 Travis5230.359797.7 Zapata6026.931599.283 Lasalle5628.4545999.217 Cameron6225.91997.433

3. Estimating the Full Model Temp =     LAT   ELEV   LONG  CoefficientsStandard Errort StatP-value Intercept151.297625.133366.0197926.03E-05 Lat-1.993230.13639-14.61425.23E-09 Elev-0.000960.000568-1.683440.118102 Long-0.384710.228584-1.683020.118185

4. Testing the Full Model H 0 :        0 H A : Not all  i = 0 TS: F obs = MSR/MSE = 491.138 P-Value: P(F≥491.138)  0 ANOVA dfSSMSFSignificance F Regression3934.328311.4427491.1388.1236E-13 Residual127.6094940.634125 Total15941.9375

5. Testing Individual Partial Coefficients H 0 :  i = 0 H A :  i ≠ 0 TS: t obs = b i /SE(b i ) Latitude: t obs = -14.61 P-value  0 Elevation: t obs = -1.68 P-value =.1182 Longitude: t obs = -1.68 P-value =.1182 CoefficientsStandard Errort StatP-value Intercept151.297625.133366.0197926.03E-05 Lat-1.993230.13639-14.61425.23E-09 Elev-0.000960.000568-1.683440.118102 Long-0.384710.228584-1.683020.118185

6. Comparing Regression Models Note: Controlling for ELEV and LAT, LONG does not appear significant (at  =.10 level) and same result holds for LONG. Test whether after controlling for LAT, neither ELEV or LONG related to TEMP H 0 :      H A :    and/or   ≠ 0 Complete Model: –Temp =     LAT   ELEV   LONG  Reduced Model –Temp =     LAT 

7. Complete and Reduced Models Complete ANOVA (n=16, k=3) dfSSMS Regression3934.328311.4427 Residual127.6094940.634125 Total15941.9375 Reduced ANOVA (g=1) dfSSMS Regression1881.003 Residual1460.93454.352465 Total15941.9375

8. Test of H 0 :      SSR c = 934.328, SSE c = 7.609 SSR r = 881.003 N=16, k=3, g=1

9. Model with Latitude and Elevation Temp =     LAT   ELEV  Coefficie nts Standard Errort StatP-value Intercept109.25892.97857287436.681621.64E-14 Lat-1.832160.103801087-17.65071.83E-10 Elev-0.001850.000218782-8.439211.24E-06 ANOVA dfSSMSFSignificance F Regression2932.532466.266644.4469.91E-14 Residual139.405680.72351 Total15941.938