1 answer

2. [20| Consider the car stopping distance example we studied in Section 2.2, with the model...

Question:

2. [20| Consider the car stopping distance example we studied in Section 2.2, with the model 2 where d is the stopping distance, v is the velocity of the car before braking, tr is the response time, and k is a coefficient related to the ratio of the braking force and the mass fo the car. (a) [5] Fit the model (i.e, determine the parameters t and k) to the data in the first column and the last column of Table 2.4 (on Page 75 in the text, for your convenience, a CSV file containing the data has been posted on CourseSpaces), using the least aquare method. (You will need to use software such as MATLAB or R). Make a plot to compare your predicted stopping distance with the observed distance. (b) 5] Fit the same data to the model T=tr + ku where d/v. Make a plot to compare your predicted stopping dis- tance with the observed distance. (c) [5| Table 2.4 also shows that the variance of the stopping distance increases with the velocity v, which violates the implicit assumption of least square method. Propose a method to consider different vari ances with each v, and fit model 1. Make a plot to compare your predicted stopping distance with the observed distance. Note that your result does not improve the sum of squared residue (otherwise your answer would be identical to Paart (a). In what aspect do you consider this method as an in mprovement to the result in Part (a)?(d) 5 Suggest an improvement to the model, and fit your model to data, and show that it is an improvement. Make a plot to compare your predicted stopping distance with both the observed distance and your predictions in Part (a).

Speed Distance Standard Deviation 20 25 30 35 40 45 142.5 50 55209.5 60 65 292.5 70 75 80 56 73.5 91.5 116 4.5 5.5 10.5 13 16.5 20 24.5 29 35 42 173 248 343 401 464

2. [20| Consider the car stopping distance example we studied in Section 2.2, with the model 2 where d is the stopping distance, v is the velocity of the car before braking, tr is the response time, and k is a coefficient related to the ratio of the braking force and the mass fo the car. (a) [5] Fit the model (i.e, determine the parameters t and k) to the data in the first column and the last column of Table 2.4 (on Page 75 in the text, for your convenience, a CSV file containing the data has been posted on CourseSpaces), using the least aquare method. (You will need to use software such as MATLAB or R). Make a plot to compare your predicted stopping distance with the observed distance. (b) 5] Fit the same data to the model T=tr + ku where d/v. Make a plot to compare your predicted stopping dis- tance with the observed distance. (c) [5| Table 2.4 also shows that the variance of the stopping distance increases with the velocity v, which violates the implicit assumption of least square method. Propose a method to consider different vari ances with each v, and fit model 1. Make a plot to compare your predicted stopping distance with the observed distance. Note that your result does not improve the sum of squared residue (otherwise your answer would be identical to Paart (a). In what aspect do you consider this method as an in mprovement to the result in Part (a)?
(d) 5 Suggest an improvement to the model, and fit your model to data, and show that it is an improvement. Make a plot to compare your predicted stopping distance with both the observed distance and your predictions in Part (a).
Speed Distance Standard Deviation 20 25 30 35 40 45 142.5 50 55209.5 60 65 292.5 70 75 80 56 73.5 91.5 116 4.5 5.5 10.5 13 16.5 20 24.5 29 35 42 173 248 343 401 464

Answers

Using the software Solution of first two parts is provided as below:

clc
clear all
v=[20:5:80];
d=[42 56 73.5 91.5 116 142.5 173 209.5 248 292.5 343 401 464];
X=v;
Y=d./v;
C=polyfit(X,Y,1);
k=C(1)
tr=C(2)

plot(v,d,'*',v,k*v.*v+tr*v,'r');
figure;
k=C(1)
tr=C(2)
plot(X,Y,'*',X,k*X+tr,'r')

.

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