Chapter PII, Problem 30RE

(a)

To determine

To write the equation of the line of regression for estimate high jump from long jump.

Answer to Problem 30RE

The regression line is .

Explanation of Solution

It is given in the question the summary statistics for the Olympics long jumps and high jumps which is as follows:

Event	Mean	Standard deviation
Long Jump	316.04	20.85
High Jump	83.85	7.46

And the correlation between the variables is $0.925$ . Thus, the line of regression is calculated as:

  $\begin{array}{l}\text{Slope}=r\times \frac{{\sigma }_2}{{\sigma }_1}\\ =0.925\times \frac{7.46}{20.85}\\ =0.3309\\ y-\text{intercept}={\mu }_2-\text{Slope}\times {\mu }_1\\ =83.85-(0.3309)316.04\\ =-20.72\end{array}$

Thus, the regression equation is as follows:

  

(b)

To determine

To interpret the slope of the line.

Explanation of Solution

It is given in the question the summary statistics for the Olympics long jumps and high jumps which is as follows:

Event	Mean	Standard deviation
Long Jump	316.04	20.85
High Jump	83.85	7.46

And the correlation between the variables is $0.925$ . And the regression equation is:

  

Thus, the slope of the line is interpreted as for each additional inches they jumped in the long jump, it predicts an increase of $0.3309$ inches in the high jump.

(c)

To determine

To find out what high jump would you predict in a year when the long jump is $350$ inches.

Answer to Problem 30RE

  $95.095$ inches is the predicted high jump.

Explanation of Solution

It is given in the question the summary statistics for the Olympics long jumps and high jumps which is as follows:

Event	Mean	Standard deviation
Long Jump	316.04	20.85
High Jump	83.85	7.46

And the correlation between the variables is $0.925$ . And the regression equation is:

  

Thus, the high jump willbe predicted in a year when the long jump is $350$ inches as:

  

(d)

To determine

To explain why cannot you use this line to estimate the long jump for a year when you know the high jump was $85$ inches.

Explanation of Solution

It is given in the question the summary statistics for the Olympics long jumps and high jumps which is as follows:

Event	Mean	Standard deviation
Long Jump	316.04	20.85
High Jump	83.85	7.46

And the correlation between the variables is $0.925$ . And the regression equation is:

  

Thus, we cannot use this line to estimate the long jump for a year when we know the high jump was $85$ inches because the model is for high jump height not for the prediction for the long jump distance.

(e)

To determine

To write the equation of the line you need to make that prediction.

Answer to Problem 30RE

The required regression line is .

Explanation of Solution

It is given in the question the summary statistics for the Olympics long jumps and high jumps which is as follows:

Event	Mean	Standard deviation
Long Jump	316.04	20.85
High Jump	83.85	7.46

And the correlation between the variables is $0.925$ . And the regression equation is:

  

Thus, the equation of regression we need to calculate the long jump distance is calculated as:

  $\begin{array}{l}\text{Slope}=r\times \frac{{\sigma }_2}{{\sigma }_1}\\ =0.925\times \frac{20.85}{7.46}\\ =2.58\\ y-\text{intercept}={\mu }_2-\text{Slope}\times {\mu }_1\\ =316.04-(2.58)83.85\\ =99.70\end{array}$

Thus, the regression line is: