a. Construct a correlation matrix showing the correlations between the following

Question

a.     Construct a correlation matrix showing the correlations between the following variables: (Note, leave out Surface Condition Code)

Average Speed –All:             

Wind Speed:  

                                    Air temp:       

                                    Rel Humid;                

Visibility:      

                                    Surface Temperature:                        

Total Vehicles:                      

        Which variable has the highest correlation with Average Speed?

b.Develop a multiple regression model using Average Speed as the dependent variable and the following variables as independent variables:

           

Wind Speed:  

                                    Air temp:       

                                    Rel Humid;                

Visibility:      

                                    Surface Temperature:                        

Total Vehicles:

                   Show the regression equation.

c. After examining the results from parts a. and b., can you identify any potential multicollinearity issues? Explain.

d. Determine whether the overall regression model is statistically significant at an alpha level = 0.05. State the appropriate null and alternative hypothesis and use the p-value approach to conduct the hypothesis test.

e.Identify which of the independent variables are not statistically significant at the 0.05 alpha level. Use the p-value approach to make this determination.

f.Calculate the predicted average speed for a five-minute time segment under the following conditions:

Wind Speed = 44 mph

                                    Air temp = 36 degrees

                                    Rel Humid = 59                     

Visibility = 7.15 miles           

                                    Surface Temperature = 51.98            

Total Vehicles = 48

Average Speed- All Wind Spd Air Temp Rel Hum Visibility Surface Condition Code Surface Temp Total Vehicles All 68.62 21 42 64 15.07 Wet 44.312 29.00 70.04 26 40 82 6.14 Wet 43.1276 24.00 67.95 21 39 88 1.45 Wet 42.818 20.00 68.86 19 37 87 1.9 Wet 42.6272 22.00 69.88 18 37 88 2.49 Wet 42.3815 25.00 68.45 17 38 89 2.84 Wet 42.08 22.00 65.52 16 36 91 5.41 Wet 41.774 21.00 65.14 18 35 92 4.97 Wet 41.468 22.00 67.04 17 35 95 5.64 Wet 41.162 25.00 68.33 21 36 96 11.31 Wet 40.862 21.00 70.23 22 34 98 6.36 Wet 40.406 22.00 66.80 25 35 98 6.29 Wet 39.794 25.00 66.29 24 33 99 6.07 Wet 39.407 24.00 69.69 25 33 100 5.17 Wet 38.957 13.00 64.83 19 33 100 6.87 Wet 39.038 12.00 62.44 19 33 100 7.14 Wet 39.038 16.00 65.54 17 33 100 6.96 Wet 39.002 24.00 65.00 14 33 98 13.35 Wet 38.732 13.00 64.53 12 33 98 19.85 Wet 39.038 17.00 67.06 10 33 98 25.02 Wet 39.11 17.00 66.69 11 35 97 21.62 Wet 39.038 13.00 63.53 12 33 98 21.58 Wet 39.038 19.00 67.68 13 35 94 25.12 Wet 38.732 22.00 66.30 14 35 94 23.56 Wet 38.504 23.00 65.08 11 33 94 24.96 Wet 37.814 12.00 63.75 10 33 95 19.49 Wet 37.814 16.00 69.09 7 33 96 19.74 Wet 37.814 11.00 65.76 8 33 96 12.41 Wet 38.096 17.00 66.29 8 33 96 15.7 Wet 38.318 14.00 67.36 9 33 94 17.9 Wet 38.372 11.00 67.25 10 35 92 12.77 Wet 38.264 16.00 69.58 10 34 90 16.09 Wet 38.12 12.00 65.50 10 35 91 5.9 Wet 37.733 12.00 64.12 11 33 92 5.2 Wet 37.844 17.00 66.35 10 33 93 12.28 Wet 37.958 17.00 64.00 12 35 92 20.54 Wet 37.808 11.00 66.80 14 33 93 6.51 Wet 37.427 15.00 67.94 18 34 94 3.75 Wet 37.166 17.00 70.17 15 32 97 2.17 Wet 36.776 6.00 64.17 16 34 91 2.74 Wet 36.635 6.00 67.00 17 32 93 1.57 Wet 35.978 13.00 67.57 19 32 93 1.17 Wet 34.7135 7.00 68.17 18 32 94 0.95 Wet 33.854 12.00 63.27 20 33 95 0.73 Wet 33.827 11.00 67.31 18 33 96 0.66 Wet 33.701 13.00 62.83 20 33 97 0.57 Wet 33.494 12.00 66.86 18 33 98 0.54 Wet 33.242 7.00 64.57 16 33 99 0.5 Wet 33.143 14.00 66.50 17 33 100 0.37 Wet 33.143 8.00 58.64 21 33 100 0.47 Wet 32.936 14.00 55.00 19 32 100 0.3 Wet 32.936 15.00 50.88 21 32 100 0.21 Wet 33.044 8.00 62.20 21 32 100 0.14 Wet 32.936 5.00 55.07 22 31 100 0.22 Wet 32.936 14.00 47.75 24 31 100 0.48 Wet 32.99 8.00 46.67 26 31 100 0.32 Wet 33.044 6.00 44.00 25 31 100 0.32 Wet 33.044 5.00 30.50 24 31 100 0.25 Wet 33.017 2.00 38.29 30 31 100 0.11 Wet 33.188 7.00 43.75 29 31 100 0.13 Wet 32.792 8.00 34.57 28 31 100 0.18 Wet 32.8145 7.00 35.00 28 31 100 0.2 Wet 32.936 6.00 34.50 26 31 100 0.23 Wet 32.882 6.00 35.86 26 31 100 0.24 Wet 32.936 7.00 34.25 26 31 100 0.23 Wet 32.936 4.00 37.00 30 30 100 0.18 Wet 32.882 3.00 26.33 32 30 100 0.17 Wet 32.792 6.00 36.14 32 30 100 0.21 Wet 32.792 7.00 31.78 30 30 100 0.28 Wet 32.882 9.00 40.50 28 30 100 0.46 Wet 32.792 6.00 35.60 28 30 100 0.47 Wet 32.8145 5.00 30.50 29 30 100 0.45 Wet 32.792 6.00 37.50 29 29 100 0.3 Wet 32.792 8.00 32.00 31 29 99 0.15 Wet 32.837 4.00 38.00 32 28 100 0.1 Wet 32.792 9.00 37.71 34 28 99 0.09 Wet 32.792 14.00 42.00 32 28 99 0.1 Wet 32.837 2.00 43.50 32 28 98 0.12 Wet 32.936 6.00 25.00 34 27 98 0.09 Wet 32.738 15.00 8.25 47 26 91 0.34 Wet 32.63 4.00 5.10 47 26 91 0.4 Wet 32.63 9.00 23.00 46 33 65 7.95 Wet 43.529 10.00 18.25 45 33 67 6.96 Wet 44.294 5.00 17.41 44 33 63 6.69 Wet 41.924 12.00 26.33 42 33 66 7.21 Wet 36.896 3.00 66.3 17 29 98 2.28 Wet 32.486 10.0 61.9 14 29 97 1.47 Wet 32.630 20.0 68.3 13 29 97 1.23 Wet 32.630 23.0 63.3 14 29 97 1.07 Wet 32.630 23.0 61.3 14 29 95 1 Wet 32.630 18.0 65.3 17 29 96 1.04 Wet 32.630 20.0 61.6 16 29 96 0.8 Wet 32.783 16.0 64.4 13 30 96 0.81 Wet 32.936 23.0 67.2 14 30 96 0.71 Wet 32.936 9.0 65.6 13 30 97 0.7 Wet 32.909 14.0 65.2 13 30 97 0.73 Wet 33.650 18.0 67.4 14 30 96 0.72 Wet 34.304 19.0 65.0 14 30 96 0.79 Wet 34.871 21.0 61.3 13 30 97 0.8 Wet 35.006 13.0 67.6 14 30 97 0.84 Wet 35.078 36.0 64.4 12 30 97 0.72 Wet 35.780 20.0 67.9 12 30 97 0.64 Wet 35.528 20.0 67.1 12 30 97 0.37 Wet 34.079 31.0 61.8 10 30 98 0.32 Wet 33.386 32.0 62.6 11 31 97 0.57 Wet 33.242 32.0 65.0 11 30 97 0.37 Wet 33.998 30.0 60.6 7 31 99 0.44 Wet 33.092 31.0 65.3 10 31 98 0.48 Wet 33.044 30.0 64.6 15 31 98 0.5 Wet 33.044 26.0 66.1 14 31 95 0.56 Wet 34.730 28.0 63.6 13 31 96 0.48 Wet 36.806 26.0 64.9 12 30 95 0.53 Wet 36.797 29.0 64.3 15 31 94 0.65 Wet 37.352 34.0 67.7 14 31 95 0.61 Wet 39.481 39.0 67.3 13 31 95 0.55 Wet 40.856 32.0 67.0 14 31 95 0.85 Wet 40.334 31.0 65.7 13 31 95 0.58 Wet 41.612 50.0 65.8 13 31 96 0.5 Wet 42.503 37.0 64.2 17 31 94 0.63 Wet 41.774 26.0 65.9 17 31 95 0.55 Wet 40.712 36.0 68.4 16 31 94 0.68 Wet 41.252 36.0 68.9 15 31 94 0.62 Wet 42.476 34.0 65.3 13 32 96 0.6 Wet 43.538 35.0 67.5 15 31 94 0.64 Wet 43.142 46.0 66.9 14 32 96 0.78 Wet 43.898 37.0 70.1 14 32 94 0.74 Wet 43.988 27.0 67.8 16 31 93 0.81 Wet 45.266 23.0 66.7 16 31 92 0.63 Wet 44.510 38.0 67.4 15 31 94 0.58 Wet 43.592 39.0 64.6 17 31 94 0.59 Wet 39.794 37.0 68.3 14 31 95 0.53 Wet 41.306 36.0 68.0 14 31 97 0.54 Wet 38.876 20.0 69.2 15 31 97 0.51 Wet 41.306 36.0 66.6 13 31 95 0.52 Wet 40.406 24.0 68.1 14 31 96 0.4 Wet 41.306 28.0 65.7 11 31 99 0.29 Wet 38.876 41.0 68.4 13 31 99 0.29 Wet 36.797 39.0 66.2 11 31 99 0.31 Wet 35.270 29.0 67.0 14 31 99 0.5 Wet 34.466 41.0 66.7 14 31 99 0.54 Wet 36.140 39.0 66.5 12 31 99 0.58 Wet 37.292 34.0 65.8 14 31 99 0.7 Wet 38.192 30.0 64.4 14 31 99 0.55 Wet 38.570 33.0 66.4 14 31 98 0.54 Wet 38.660 34.0 64.8 12 31 100 0.47 Wet 37.346 31.0 67.3 9 32 100 0.44 Wet 38.066 32.0 67.8 8 32 100 0.76 Wet 38.552 43.0 66.1 11 32 100 0.62 Wet 39.947 63.0 65.2 11 32 100 0.76 Wet 40.100 39.0 64.6 15 32 99 0.52 Wet 40.406 54.0 68.5 13 32 99 0.53 Wet 39.794 51.0 68.7 12 32 99 0.52 Wet 40.100 43.0 64.5 13 32 99 0.58 Wet 40.406 36.0 67.8 11 32 100 0.55 Wet 39.434 37.0 68.1 12 32 100 0.55 Wet 38.876 44.0 69.1 11 32 100 0.77 Wet 38.570 48.0 67.6 13 32 100 0.63 Wet 39.794 34.0 69.1 12 31 100 0.59 Wet 38.723 56.0 65.0 12 31 100 0.51 Wet 37.652 41.0 68.3 13 31 100 0.68 Wet 37.346 33.0 70.1 12 31 100 0.5 Wet 37.652 48.0 69.6 10 32 100 0.51 Wet 35.753 18.0 67.1 9 31 100 0.33 Wet 35.978 41.0 66.4 9 31 100 0.48 Wet 35.141 35.0 65.3 9 31 100 0.4 Wet 33.773 29.0 64.7 7 31 100 0.43 Wet 33.548 44.0 66.5 7 31 100 0.63 Wet 33.314 38.0 65.9 8 31 100 0.56 Wet 33.188 35.0 61.7 9 31 100 0.45 Wet 33.332 28.0 61.5 11 31 100 0.48 Wet 33.044 34.0 63.4 12 31 100 0.55 Wet 33.044 31.0 63.1 9 31 100 0.67 Wet 33.044 34.0 62.1 10 31 100 1.1 Wet 32.882 35.0 61.8 9 31 100 0.8 Wet 32.837 44.0 62.1 11 31 100 0.54 Wet 32.882 38.0 59.9 9 31 100 0.53 Wet 32.792 31.0 62.5 7 31 100 0.57 Wet 32.792 42.0 61.8 7 31 100 0.65 Wet 32.792 28.0 55.9 7 31 100 0.59 Wet 32.882 36.0 53.8 12 30 100 0.65 Wet 32.792 43.0 52.0 9 30 100 0.81 Wet 32.630 20.0 50.5 7 30 100 0.87 Wet 32.630 26.0 51.2 6 30 100 1.2 Wet 32.630 27.0 50.7 6 30 100 1.29 Wet 32.738 23.0 51.3 6 30 100 1.83 Wet 32.630 23.0 50.6 6 30 100 3.35 Wet 32.630 21.0 47.8 5 30 100 2.55 Wet 32.630 14.0 51.9 5 30 100 4.78 Wet 32.630 32.0 31.00 35 27 98 0.09 Snow 32.171 4.00 33.50 36 27 98 0.06 Snow 32.018 2.00 18.00 36 27 98 0.08 Snow 31.406 4.00 25.14 36 27 97 0.1 Snow 31.388 7.00 27.20 36 27 96 0.1 Snow 31.388 5.00 20.73 35 27 97 0.1 Snow 31.136 11.00 18.50 38 27 96 0.09 Snow 30.974 10.00 22.00 36 27 96 0.09 Snow 31.028 5.00 26.44 38 27 96 0.09 Snow 31.01 9.00 30.64 33 27 95 0.18 Snow 30.974 14.00 24.64 36 27 95 0.18 Snow 30.83 11.00 22.44 36 27 94 0.09 Snow 31.226 9.00 28.60 37 27 94 0.11 Snow 31.136 5.00 30.89 38 27 93 0.12 Snow 31.334 19.00 23.57 39 27 93 0.09 Snow 31.352 7.00 27.20 37 26 93 0.09 Snow 31.496 5.00 22.82 40 26 93 0.08 Snow 31.406 11.00 25.40 40 26 93 0.08 Snow 31.424 5.00 26.56 43 26 94 0.09 Snow 31.28 9.00 29.30 41 26 94 0.15 Snow 31.226 20.00 14.71 41 26 93 0.24 Snow 31.226 14.00 15.18 42 26 93 0.51 Snow 31.937 11.00 10.75 45 26 94 0.42 Snow 32.126 12.00 15.25 43 26 93 0.47 Snow 32.18 8.00 16.25 41 25 93 0.49 Snow 32.33 4.00 59.7 18 29 97 1.13 Snow 31.424 19.0 64.9 19 29 97 1.89 Snow 31.478 17.0 65.1 19 29 97 2.1 Snow 31.586 8.0 63.3 17 29 98 2.54 Snow 31.730 15.0 60.9 18 29 98 2.51 Snow 32.018 11.0 67.4 19 29 98 2.99 Snow 32.072 17.0 68.6 16 29 98 2.26 Snow 32.099 11.0 64.3 17 29 98 1.98 Snow 32.324 11.0

Explanation / Answer

a) Correlation matrix

Wind Spd has the highest correlation with Average Speed- All

b)

Average Speed=-13.79-1.18*Wind Spd+1.37*Air Temp+0.17*Rel Hum-0.28*Visibility+0.97*Surface Temp-0.05*Total Vehicles

c)

No big multicollinearity issues amongst independent variables.

d)

H0: 0=1=2=3=4=5=6=0 (no linear relationship)

H1: at least one i 0   (at least one independent variable affects Y)

F=195.39 and F Crit is 0 and since F>F Crit, reject the null hypothesis. Thus, we conclude that at least one independent variable affects Y

e)

Based on the p-value, Rel Hum and Total Vehicles All are not significant

f)

Average Speed=-13.79-1.18*44+1.37*36+0.17*59-0.28*7.15+0.97*51.98-0.05*48

=39.70

SUMMARY OUTPUT Regression Statistics Multiple R 0.92 R Square 0.85 Adjusted R Square 0.84 Standard Error 6.75 Observations 221 ANOVA df SS MS F Significance F Regression 6 53342.17 8890.36 195.39 0.00 Residual 214 9737.16 45.50 Total 220 63079.33 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept -13.79 17.37 -0.79 0.43 -48.04 20.46 Wind Spd -1.18 0.08 -15.17 0.00 -1.33 -1.02 Air Temp 1.37 0.30 4.54 0.00 0.77 1.96 Rel Hum 0.17 0.11 1.56 0.12 -0.05 0.39 Visibility -0.28 0.11 -2.54 0.01 -0.49 -0.06 Surface Temp 0.97 0.20 4.75 0.00 0.56 1.37 Total Vehicles All -0.05 0.06 -0.89 0.37 -0.16 0.06

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