Chapter PII, Problem 11RE

(a)

To determine

To find out what is the value of the correlation coefficient between speed and density.

Answer to Problem 11RE

  $r=-0.9844$ .

Explanation of Solution

It is given in the question that highway planners investigated the relationship between traffic density and the average speed of the traffic. The mean traffic density is $68.6$ cpm ad standard deviation is $27.07$ cpm. And the mean of the speed is $26.38$ mph with standard deviation of $9.68$ mph. the regression line is as:

  

Now, to find the correlation r we need to substitute the values into the linear regression slope and then solve r . thus, we have,

  $\begin{array}{l}b=r\frac{s_x}{s_y}\\ \Rightarrow -0.352=r\frac{9.68}{27.07}\\ \Rightarrow r=-0.9844\end{array}$

(b)

To determine

To find out what percent of the variation in average speed is explained by traffic density.

Answer to Problem 11RE

  $96.9\%$ of the variation in average speed is explained by traffic density.

Explanation of Solution

It is given in the question that highway planners investigated the relationship between traffic density and the average speed of the traffic. The mean traffic density is $68.6$ cpm ad standard deviation is $27.07$ cpm. And the mean of the speed is $26.38$ mph with standard deviation of $9.68$ mph. the regression line is as:

  

Now, to find the percent of the variation in average speed accounted for by traffic density, we must find $r^2$ , thus, we have,

  $\begin{array}{l}$ r^2={(-0.9844)}^2$\Rightarrow r^2=0.969\\ \Rightarrow r^2=96.9\%\end{array}$

  $96.9\%$ of the variation in average speed is explained by traffic density.

(c)

To determine

To predict the average speed to traffic on the thoroughfare when the traffic density is $50$ cpm.

Answer to Problem 11RE

The average speed to traffic on the thoroughfareis $32.95$ .

Explanation of Solution

It is given in the question that highway planners investigated the relationship between traffic density and the average speed of the traffic. The mean traffic density is $68.6$ cpm ad standard deviation is $27.07$ cpm. And the mean of the speed is $26.38$ mph with standard deviation of $9.68$ mph. the regression line is as:

  

Thus, to predict the average speed to traffic on the thoroughfare when the traffic density is $50$ cpm is calculated as:

  

(d)

To determine

To find out what is the value of the residual for a traffic density of $56$ cpm with an observed speed of $32.5$ mph.

Answer to Problem 11RE

The residual is $1.662$ .

Explanation of Solution

It is given in the question that highway planners investigated the relationship between traffic density and the average speed of the traffic. The mean traffic density is $68.6$ cpm ad standard deviation is $27.07$ cpm. And the mean of the speed is $26.38$ mph with standard deviation of $9.68$ mph. the regression line is as:

  

Now, we have to predict the speed for a traffic density of $56$ cpm as:

  

Thus, the residual is calculated as with an observed speed of $32.5$ mph,

  $\begin{array}{l}=\text{Observed}-\text{Predicted}\\ =32.5-30.383\\ =1.662\end{array}$

(e)

To determine

To explain will the slope increase, decrease or remain the same if we redo the analysis and include this point.

Answer to Problem 11RE

The slope would increase.

Explanation of Solution

It is given in the question that highway planners investigated the relationship between traffic density and the average speed of the traffic. The mean traffic density is $68.6$ cpm ad standard deviation is $27.07$ cpm. And the mean of the speed is $26.38$ mph with standard deviation of $9.68$ mph. the regression line is as:

  

Thus, the slope would increase because the new point has enough leverage and is a relatively large number of standard deviations of y away from the average $(\overline{x},\overline{y})$ to influence it to do so.

(f)

To determine

To explain will the correlation become stronger, weaker or remain the same if we redo the analysis and include this point $(125,55)$ .

Answer to Problem 11RE

The strength of the correlation would weaken.

Explanation of Solution

It is given in the question that highway planners investigated the relationship between traffic density and the average speed of the traffic. The mean traffic density is $68.6$ cpm ad standard deviation is $27.07$ cpm. And the mean of the speed is $26.38$ mph with standard deviation of $9.68$ mph. the regression line is as:

  

Thus, the strength of the correlation would weaken upon including the point $(125,55)$ because the point does not follow the negative direction of the general trend of the rest of the data points.

(g)

To determine

To find the value of his calculated correlation between the speed and density.

Answer to Problem 11RE

The correlation would remain the same.

Explanation of Solution

It is given in the question that highway planners investigated the relationship between traffic density and the average speed of the traffic. The mean traffic density is $68.6$ cpm ad standard deviation is $27.07$ cpm. And the mean of the speed is $26.38$ mph with standard deviation of $9.68$ mph. the regression line is as:

  

Thus, the value of his calculated correlation between the speed and density should remain the same as the change in the units does not change the correlation of the variables.