# Question: A highway department is studying the relationship between traffic flow

A highway department is studying the relationship between traffic flow and speed during rush hour on Highway 193. The data in the file TrafficFlow were collected on Highway 193 during 100 recent rush hours.

a. Develop a scatter chart for these data. What does the scatter chart indicate about the relationship between vehicle speed and traffic flow?

b. Develop an estimated simple linear regression equation for the data. How much variation in the sample values of traffic flow is explained by this regression model? Use a 0.05 level of significance to test the relationship between vehicle speed and traffic flow. What is the interpretation of this relationship?

c. Develop an estimated quadratic regression equation for the data. How much variation in the sample values of traffic flow does this regression model explain? Is the overall regression relationship significant at a 0.05 level of significance? If so, then test the relationship between each of the independent variables and the dependent variable at a

0.05 level of significance. How would you interpret this model? Is this model superior to the model you developed in part b?

d. As an alternative to fitting a second-order model, fit a model using a piecewise linear regression with a single knot. What value of vehicle speed appears to be a good point for the placement of the knot? Does the estimated piecewise linear regression provide a better fit than the estimated quadratic regression developed in part c? Explain.

e. Separate the data into two sets such that one data set contains the observations of vehicle speed less than the value of the knot from part d and the other data set contains the observations of vehicle speed greater than or equal to the value of the knot from part d. Then fit a simple linear regression equation to each data set. How does this pair of regression equations compare to the single piecewise linear regression with the single knot from part d? In particular, compare predicted values of traffic flow for values of the speed slightly above and slightly below the knot value from part d.

f. What other independent variables could you include in your regression model to explain more variation in traffic flow?

a. Develop a scatter chart for these data. What does the scatter chart indicate about the relationship between vehicle speed and traffic flow?

b. Develop an estimated simple linear regression equation for the data. How much variation in the sample values of traffic flow is explained by this regression model? Use a 0.05 level of significance to test the relationship between vehicle speed and traffic flow. What is the interpretation of this relationship?

c. Develop an estimated quadratic regression equation for the data. How much variation in the sample values of traffic flow does this regression model explain? Is the overall regression relationship significant at a 0.05 level of significance? If so, then test the relationship between each of the independent variables and the dependent variable at a

0.05 level of significance. How would you interpret this model? Is this model superior to the model you developed in part b?

d. As an alternative to fitting a second-order model, fit a model using a piecewise linear regression with a single knot. What value of vehicle speed appears to be a good point for the placement of the knot? Does the estimated piecewise linear regression provide a better fit than the estimated quadratic regression developed in part c? Explain.

e. Separate the data into two sets such that one data set contains the observations of vehicle speed less than the value of the knot from part d and the other data set contains the observations of vehicle speed greater than or equal to the value of the knot from part d. Then fit a simple linear regression equation to each data set. How does this pair of regression equations compare to the single piecewise linear regression with the single knot from part d? In particular, compare predicted values of traffic flow for values of the speed slightly above and slightly below the knot value from part d.

f. What other independent variables could you include in your regression model to explain more variation in traffic flow?

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