Question

Question 3: The Table below summarises the results of traffic monitoring observations at a motorway, in terms of average spacing between vehicles versus vehicle speed. cinema9.58 14.04 19,36 22.68 27 31.32 35.64 39.96 4.28 46 52.9257.2461.5665.88 68.270 Space (m) 11.5 15.3 18.8 19.7 22.5 25.4 27.7 32.2 33.8 42 54.5 60.575.1 77.3 81 100.5 (a) The Figure below shows part of the Minitab output for a linear regression analysis for average spacing against speed. Explain the analysis in the table, calculate the values blocked in the table, and write up the prediction equation for average spacing as a function of vehicle speed. (30%) Regression Analysis: Space versus Speed Model Summary SR-3 R-sq(ad)) R-99 (pred) 8.61557 90.921 90.274 87,378 1 Coefficients Term Coef SE Coef I-Value P-value VIE Constant 5.14 -2.27 0.040 Speed 1.328 11.84 0.000 1.00 (b) Discuss the quality of the linear model for average spacing based on the diagnostics in the Minitab output above and the residuals plot displayed below. (20%)

Answer 1

(a) We have given below data regarding vehicles average Speed (miles/hr) and Average Space (m). Speed (miles/hr) 9.58 14.04 Average Space (m) 11.5 15.3 18.8 19.36 19.7 22.5 22.68 27 31.32 35.64 39.96 44.28 25.4 27.7 32.2 33.8 46 42 54.5 60.5 52.92 57.24 61.56 65.88 75.1 77.3 81 68.2 70 100.5 Taking Speed (miles/hr) as response variable and Average Space (m) as independent predictor, the linear regression output is given below: The regression equation is Average Space (m) = - 11.7 + 1.33 Speed (miles/hr) Predictor Constant Speed (miles/hr) Coef SE Coef -11.656 5.141 1.3285 0.1122 т -2.27 0.040 11.84 0.000 S = 8.61557 R-Sq - 90.98 R-Sq(adj) - 90.38 Analysis of Variance F 140.17 Source Regression Residual Error Total MS 10405 74 P 0.000 DE SS 1 10405 14 1039 15 11444 Unusual Observations Obs 16 Speed (miles/hr) 70.0 Average Space (m) 100.50 Fit SE Fit Residual St Resid 81.34 3.85 19.16 2.49R R denotes an observation with a large standardized residual.

The model is statistically significant as p-value for coefficient of speed (miles/hr) is <0.05. We have R square 90.9% which implies that 90.9% variation is explained by independent variable Speed (miles/hr) in dependent variable Average Space (m). We have fitted regression equation as Average Space (m) = - 11.7 + 1.33 Speed (miles/hr) Unit increase in Speed (miles/hr) will change Average Space (m) by 1.33 times. We have constant term=-11.7 and SE Coef of speed= 0.1122. (b) Residuals versus fitted values have not the random pattern. Hence the assumption of random pattern (independently distributed) assumption is violated. Versus Fits (response is Average Space (m) 20 15 10 - in Residual 0 -5 -10 -15 0 10 20 30 60 70 80 90 40 50 Fitted Value (c) The quadratic model is better than the fitted linear model “Average Space (m) = - 11.7 + 1.33 Speed (miles/hr)" because in case of quadratic model R square is better compared to linear model. In quadratic model, we have R square 98.2% which means 98.2% variation is explained by independent variable Speed (miles/hr) in dependent variable Average Space (m). The fitted quadratic model equation is Space= 18.23-0.5365*Speed+0.02272*Speed^2

Polynomial Regression Analysis: Space versus Speed The regression equation is Space - 18.23 - 0.5365 Speed 0.02272 Speed"2 5 - 3.98536 R-sq - 98.21 R-Sq (adj) - 97.91 Analysis of variance Source SS MS Regression 2 11237.6 5618.01 353.7 Error 13 206.5 15.88 Total 15 11444.1 0.000 (d) Now using quadratic model fit, we predict the average spacing for given below Speed (miles/hr). Speed (miles/hr) 16 42 55 Average Space (m) 16.1 35.2 57.2 We have fitted quadratic model as "Space= 18.23-0.5365* Speed+0.02272*Speed"2" Average spacing for given Speed (miles/hr)=16 is Space= 18.23-0.5365* 16+0.02272* 16^2=15.462 Average spacing for given Speed (miles/hr)=42 is Space= 18.23-0.5365* 42+0.02272*42^2=35.775 Average spacing for given Speed (miles/hr)=55 is Space = 18.23-0.5365*55+0.02272*55^2=57.451 Model is indeed providing suitable predictions as we have negligible difference between observed values of Average Space and expected values of Average space.