Chapter 2.4, Problem 72E

(a)

To determine

To find: The comparison of two predicted mean strengths using their difference.

Answer to Problem 72E

Solution: The predicted dominant arm strength for a baseball player is $5.138$ units more than the non-baseball player.

Explanation of Solution

Calculation: It is noted from the referred Exercises 2.70 and 2.71 that the predicted mean strength of the dominant arm $\left({\widehat{y}}_1\right)$ is $17.716{\text{cm}}^{\text{4}}\text{/1000}$ for a non-baseball player and the predicted mean strength of the dominant arm $\left({\widehat{y}}_1\right)$ is $22.854{\text{cm}}^{\text{4}}\text{/1000}$ for a baseball player who uses exercise strength. In both the cases, the non-dominant arm strength is $16{\text{cm}}^{\text{4}}\text{/1000}$.

The difference in the two predicted mean strengths is calculated as:

$\begin{array}{c}{\widehat{y}}_2-{\widehat{y}}_1=\left\{\begin{array}{l}\text{Mean arm strength for}\\ \text{ a baseball player }\end{array}\right\}-\left\{\begin{array}{l}\text{Mean arm strength for }\\ \text{ a non-baseball player }\end{array}\right\}\\ =22.854-17.716\\ =5.138\end{array}$

Therefore, the difference in the mean strengths of a baseball and a non-baseball player is positive. This implies that the predicted dominant arm strength for a baseball player, who uses exercise strength, is greater than the predicted dominant arm strength for a baseball player, who uses control more than the exercise strength. Thus, it can be said that ${\widehat{y}}_2>{\widehat{y}}_1$.

(b)

To determine

To explain: The inference for the difference in the two predicted mean strengths.

Answer to Problem 72E

Solution: There is a positive impact of the baseball throwing exercise over control as the difference in the mean arm strengths of the baseball player and a non-baseball player is positive.

Explanation of Solution

From the referred exercise, it is ascertained that the difference in the predicted arm strengths for a baseball and a non-baseball player is $5.138{\text{cm}}^{\text{4}}\text{/1000}$. The difference is positive for the same non-dominant arm strength as $16{\text{cm}}^{\text{4}}\text{/1000}$ in both the cases, implying that the mean arm strength for a baseball player is relatively more than the mean arm strength for a non-baseball player. It is also provided that the baseball player exercises baseball throwing practice while using the arm strength and a non-baseball player uses control more than the exercise. Hence, it can be said that the baseball throwing exercise has a positive impact and improves the strength of arm bones for a player.

(c)

Section 1

To determine

To find: The predicted dominant arm strengths for non-dominant strengths $12{\text{cm}}^{\text{4}}\text{/1000}$ and $20{\text{cm}}^{\text{4}}\text{/1000}$.

Answer to Problem 72E

Solution: The results obtained are represented in the following table:

		Dominant arm Strength
		Non-Baseball Player $\left({\text{cm}}^{\text{4}}\text{/1000}\right)$	Baseball Player $\left({\text{cm}}^{\text{4}}\text{/1000}\right)$
Non-Dominant Arm Strength	$12{\text{cm}}^{\text{4}}\text{/1000}$	${\widehat{y}}_3=\text{13}\text{.972}$	${\widehat{y}}_4\text{=17}\text{.362}$
	$20{\text{cm}}^{\text{4}}\text{/1000}$	${\widehat{y}}_5=21.46$	${\widehat{y}}_6\text{=28}\text{.346}$

Explanation of Solution

Calculation: The linear regression equations for a non-baseball player is:

$\text{Dominant}=2.74+(0.936\times \text{nondominant)}$

and for a baseball player is:

$\text{Dominant}=0.886+(1.373\times \text{nondominant)}$

From the above part (a), the dominant arm strengths when the non-dominant strength is $16{\text{cm}}^{\text{4}}\text{/1000}$ are obtained as $17.716{\text{cm}}^{\text{4}}\text{/1000}$ for a non-baseball player and $22.854{\text{cm}}^{\text{4}}\text{/1000}$ for a baseball player.

The dominant arm strengths provided the non-dominant arm strength as $12{\text{cm}}^{\text{4}}\text{/1000}$ are calculated as follows:

For a non-baseball player, it is represented as ${\widehat{y}}_3$ and computed as:

$\begin{array}{c}\text{Dominant}=2.74+(0.936\times \text{nondominant)}\\ {\widehat{y}}_3=2.74+(0.936\times \text{12)}\\ {\widehat{y}}_3=\text{13}\text{.972}\end{array}$

For a baseball player, it is represented as ${\widehat{y}}_4$ and computed as:

$\begin{array}{c}\text{Dominant}=0.886+(1.373\times \text{nondominant)}\\ {\widehat{y}}_4=0.886+(1.373\times 12\text{)}\\ {\widehat{y}}_4\text{=17}\text{.362}\end{array}$

The dominant arm strengths provided the non-dominant arm strength as $20{\text{cm}}^{\text{4}}\text{/1000}$ are calculated as follows:

For a non-baseball player, it is represented as ${\widehat{y}}_5$ and computed as:

$\begin{array}{c}\text{Dominant}=2.74+(0.936\times \text{nondominant)}\\ {\widehat{y}}_5=2.74+(0.936\times 20\text{)}\\ {\widehat{y}}_5=21.46\end{array}$

For a baseball player, it is represented as ${\widehat{y}}_6$ and computed as:

$\begin{array}{c}\text{Dominant}=0.886+(1.373\times \text{nondominant)}\\ {\widehat{y}}_6=0.886+(1.373\times 20\text{)}\\ {\widehat{y}}_6\text{=28}\text{.346}\end{array}$

The above results obtained can be represented in the form of a table as follows:

		Dominant arm Strength
		Non-Baseball Player $\left({\text{cm}}^{\text{4}}\text{/1000}\right)$	Baseball Player $\left({\text{cm}}^{\text{4}}\text{/1000}\right)$
Non-Dominant Arm Strength	$12{\text{cm}}^{\text{4}}\text{/1000}$	${\widehat{y}}_3=\text{13}\text{.972}$	${\widehat{y}}_4\text{=17}\text{.362}$
	$20{\text{cm}}^{\text{4}}\text{/1000}$	${\widehat{y}}_5=21.46$	${\widehat{y}}_6\text{=28}\text{.346}$

Section 2:

To determine

To find: The differences in the respective arm strengths.

Answer to Problem 72E

Solution: The differences are $3.39{\text{cm}}^{\text{4}}\text{/1000}$ for $12{\text{cm}}^{\text{4}}\text{/1000}$ and $6.886{\text{cm}}^{\text{4}}\text{/1000}$ for $20{\text{cm}}^{\text{4}}\text{/1000}$.

Explanation of Solution

Calculation: The arm strengths for the non-dominant arm strengths $12{\text{cm}}^{\text{4}}\text{/1000}$ and $20{\text{cm}}^{\text{4}}\text{/1000}$ are obtained from the above part as follows:

		Dominant arm Strength
		Non-Baseball Player $\left({\text{cm}}^{\text{4}}\text{/1000}\right)$	Baseball Player $\left({\text{cm}}^{\text{4}}\text{/1000}\right)$
Non-Dominant Arm Strength	$12{\text{cm}}^{\text{4}}\text{/1000}$	${\widehat{y}}_3=\text{13}\text{.972}$	${\widehat{y}}_4\text{=17}\text{.362}$
	$20{\text{cm}}^{\text{4}}\text{/1000}$	${\widehat{y}}_5=21.46$	${\widehat{y}}_6\text{=28}\text{.346}$

The difference in the arm strengths of baseball and the non-baseball players are calculated as follows:

For non-dominant arm strength $12{\text{cm}}^{\text{4}}\text{/1000}$.

$\begin{array}{c}{\widehat{y}}_4-{\widehat{y}}_3=\left\{\begin{array}{l}\text{Mean arm strength for}\\ \text{ a baseball player }\end{array}\right\}-\left\{\begin{array}{l}\text{Mean arm strength for }\\ \text{ a non-baseball player }\end{array}\right\}\\ =\text{17}\text{.362}-\text{13}\text{.972}\\ =3.39\end{array}$

For non-dominant arm strength $20{\text{cm}}^{\text{4}}\text{/1000}$.

$\begin{array}{c}{\widehat{y}}_6-{\widehat{y}}_5=\left\{\begin{array}{l}\text{Mean arm strength for}\\ \text{ a baseball player }\end{array}\right\}-\left\{\begin{array}{l}\text{Mean arm strength for }\\ \text{ a non-baseball player }\end{array}\right\}\\ =\text{28}\text{.346}-21.46\\ =6.886{\text{cm}}^{\text{4}}\text{/1000}\end{array}$

Hence, the differences are $3.39{\text{cm}}^{\text{4}}\text{/1000}$ for $12{\text{cm}}^{\text{4}}\text{/1000}$ and $6.886{\text{cm}}^{\text{4}}\text{/1000}$ for $20{\text{cm}}^{\text{4}}\text{/1000}$.

Section 3:

To determine

To find: A table for the results of the three calculations.

Answer to Problem 72E

Solution: The resultant table is obtained as follows:

		Dominant arm Strength
		Non-Baseball Player $\left({\text{cm}}^{\text{4}}\text{/1000}\right)$	Baseball Player $\left({\text{cm}}^{\text{4}}\text{/1000}\right)$	Differences
Non-Dominant Arm Strength	$12{\text{cm}}^{\text{4}}\text{/1000}$	${\widehat{y}}_3=\text{13}\text{.972}$	${\widehat{y}}_4\text{=17}\text{.362}$	${\widehat{y}}_4-{\widehat{y}}_3=3.39$
	$16{\text{cm}}^{\text{4}}\text{/1000}$	${\widehat{y}}_1=17.716$	${\widehat{y}}_2=22.854$	${\widehat{y}}_2-{\widehat{y}}_1=5.138$
	$20{\text{cm}}^{\text{4}}\text{/1000}$	${\widehat{y}}_5=21.46$	${\widehat{y}}_6\text{=28}\text{.346}$	${\widehat{y}}_6-{\widehat{y}}_5=6.886$

Explanation of Solution

From sections 1 and 2 of part (c) of exercise 2.72, the following values are obtained:

${\widehat{y}}_1=17.716{\text{cm}}^{\text{4}}\text{/1000}$

${\widehat{y}}_2=22.854{\text{cm}}^{\text{4}}\text{/1000}$

${\widehat{y}}_3=\text{13}\text{.972}$

${\widehat{y}}_4\text{=17}\text{.362}$

Also,

${\widehat{y}}_5=21.46$

${\widehat{y}}_6\text{=28}\text{.346}$

The differences in the estimated strengths have been calculated as:

${\widehat{y}}_2-{\widehat{y}}_1=5.138{\text{cm}}^{\text{4}}\text{/1000}$

${\widehat{y}}_4-{\widehat{y}}_3=3.39{\text{cm}}^{\text{4}}\text{/1000}$

${\widehat{y}}_6-{\widehat{y}}_5=6.886{\text{cm}}^{\text{4}}\text{/1000}$

where, ${\widehat{y}}_1$, ${\widehat{y}}_3,$ and ${\widehat{y}}_5$ represent the dominant arm strengths of the non-baseball players with non-dominant arm strengths as $16{\text{cm}}^{\text{4}}\text{/1000}$, $12{\text{cm}}^{\text{4}}\text{/1000,}$ and $20{\text{cm}}^{\text{4}}\text{/1000,}$ respectively, and ${\widehat{y}}_2$, ${\widehat{y}}_4,$ and ${\widehat{y}}_6$ represent the dominant arm strengths of the baseball players with non-dominant arm strengths as $16{\text{cm}}^{\text{4}}\text{/1000}$, $12{\text{cm}}^{\text{4}}\text{/1000,}$ and $20{\text{cm}}^{\text{4}}\text{/1000,}$ respectively. The $\left({\widehat{y}}_2-{\widehat{y}}_1\right)$, $\left({\widehat{y}}_4-{\widehat{y}}_3\right),$ and $\left({\widehat{y}}_6-{\widehat{y}}_5\right)$ represent the differences in the mean arm strengths of both the players for all the three cases.

The above information can be represented in the form of a table as follows:

		Dominant arm Strength
		Non-Baseball Player $\left({\text{cm}}^{\text{4}}\text{/1000}\right)$	Baseball Player $\left({\text{cm}}^{\text{4}}\text{/1000}\right)$	Differences
Non-Dominant Arm Strength	$12{\text{cm}}^{\text{4}}\text{/1000}$	${\widehat{y}}_3=\text{13}\text{.972}$	${\widehat{y}}_4\text{=17}\text{.362}$	${\widehat{y}}_4-{\widehat{y}}_3=3.39$
	$16{\text{cm}}^{\text{4}}\text{/1000}$	${\widehat{y}}_1=17.716$	${\widehat{y}}_2=22.854$	${\widehat{y}}_2-{\widehat{y}}_1=5.138$
	$20{\text{cm}}^{\text{4}}\text{/1000}$	${\widehat{y}}_5=21.46$	${\widehat{y}}_6\text{=28}\text{.346}$	${\widehat{y}}_6-{\widehat{y}}_5=6.886$

(d)

Section 1:

To determine

To explain: The summary of results obtained in part (c) of exercise 2.72.

Answer to Problem 72E

Solution: The results show that the baseball throwing exercise has resulted in an improvement in the dominant arm strengths of the baseball players as compared to the non-baseball players for all the three cases. That is, the difference between the two is positive for all the three cases.

Explanation of Solution

The results obtained in the above part are as follows:

		Dominant arm Strength
		Non-Baseball Player $\left({\text{cm}}^{\text{4}}\text{/1000}\right)$	Baseball Player $\left({\text{cm}}^{\text{4}}\text{/1000}\right)$	Difference
Non-Dominant Arm Strength	$12{\text{cm}}^{\text{4}}\text{/1000}$	${\widehat{y}}_3=\text{13}\text{.972}$	${\widehat{y}}_4\text{=17}\text{.362}$	${\widehat{y}}_4-{\widehat{y}}_3=3.39$
	$16{\text{cm}}^{\text{4}}\text{/1000}$	${\widehat{y}}_1=17.716$	${\widehat{y}}_2=22.854$	${\widehat{y}}_2-{\widehat{y}}_1=5.138$
	$20{\text{cm}}^{\text{4}}\text{/1000}$	${\widehat{y}}_5=21.46$	${\widehat{y}}_6\text{=28}\text{.346}$	${\widehat{y}}_6-{\widehat{y}}_5=6.886$

From the table obtained, it is ascertained that the difference in the dominant arm strengths of baseball and the non-baseball players for all the three cases of non-dominant arm strengths as $16{\text{cm}}^{\text{4}}\text{/1000}$, $12{\text{cm}}^{\text{4}}\text{/1000,}$ and $20{\text{cm}}^{\text{4}}\text{/1000}$ are positive. This implies that the baseball throwing exercise results in an improvement in the arm strengths of the players. Thus, the use of the baseball throwing exercise is more beneficial for the players than using the control exercise.

Section 2:

To determine

To explain: The reason for the non-similarity of the three differences obtained.

Answer to Problem 72E

Solution: The non-similarity is due to a positive relation between the dominant arm strength and the non-dominant arm strength. The more the value of non-dominant arm strength, the more will be the dominant arm strength and hence, the greater will be the difference.

Explanation of Solution

The differences for all the three cases non-dominant arm strengths as $16{\text{cm}}^{\text{4}}\text{/1000}$, $12{\text{cm}}^{\text{4}}\text{/1000,}$ and $20{\text{cm}}^{\text{4}}\text{/1000}$ are:

${\widehat{y}}_4-{\widehat{y}}_3=3.39,$

${\widehat{y}}_2-{\widehat{y}}_1=5.138,\text{and}$

${\widehat{y}}_6-{\widehat{y}}_5=6.886,$

respectively. The point to be noted here is that when the non-dominant arm strength increases, the value of the dominant arm strength also improves, thus, the difference in the dominant arm strengths of baseball and the non-baseball players also improves. That is, there is a positive relation amongst the non-dominant arm strength and the difference thus obtained. The more the value of non-dominant strength, the more will be the dominant strength and the more will be the difference between the two. That is why, there is a non-similarity of the differences in all the three cases.