Calculation: It is noted from the referred Exercises 2.70 and 2.71 that the predicted mean strength of the dominant arm (y^1) is 17.716 cm4/1000 for a non-baseball player and the predicted mean strength of the dominant arm (y^1) is 22.854 cm4/1000 for a baseball player who uses exercise strength. In both the cases, the non-dominant arm strength is 16 cm4/1000.

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

y^2−y^1={Mean arm strength for   a baseball player }−{Mean arm strength for  a non-baseball player }=22.854−17.716=5.138

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 y^2>y^1.

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

Dominant=2.74+(0.936×nondominant)

and for a baseball player is:

Dominant=0.886+(1.373×nondominant)

From the above part (a), the dominant arm strengths when the non-dominant strength is 16 cm4/1000 are obtained as 17.716 cm4/1000 for a non-baseball player and 22.854 cm4/1000 for a baseball player.

The dominant arm strengths provided the non-dominant arm strength as 12 cm4/1000 are calculated as follows:

For a non-baseball player, it is represented as y^3 and computed as:

Dominant=2.74+(0.936×nondominant)y^3=2.74+(0.936×12)y^3=13.972

For a baseball player, it is represented as y^4 and computed as:

Dominant=0.886+(1.373×nondominant)y^4=0.886+(1.373×12)y^4 =17.362

The dominant arm strengths provided the non-dominant arm strength as 20 cm4/1000 are calculated as follows:

For a non-baseball player, it is represented as y^5 and computed as:

Dominant=2.74+(0.936×nondominant)y^5=2.74+(0.936×20)y^5=21.46

For a baseball player, it is represented as y^6 and computed as:

Dominant=0.886+(1.373×nondominant)y^6=0.886+(1.373×20)y^6 =28.346

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

Calculation: The arm strengths for the non-dominant arm strengths 12 cm4/1000 and 20 cm4/1000 are obtained from the above part as follows:

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

For non-dominant arm strength 12 cm4/1000.

y^4 −y^3={Mean arm strength for   a baseball player }−{Mean arm strength for  a non-baseball player }=17.362−13.972=3.39

For non-dominant arm strength 20 cm4/1000.

y^6 −y^5={Mean arm strength for   a baseball player }−{Mean arm strength for  a non-baseball player }=28.346−21.46=6.886 cm4/1000

Hence, the differences are 3.39 cm4/1000 for 12 cm4/1000 and 6.886 cm4/1000 for 20 cm4/1000.

y^1=17.716 cm4/1000

y^2=22.854 cm4/1000

y^3=13.972

y^4 =17.362

Also,

y^5=21.46

y^6 =28.346

The differences in the estimated strengths have been calculated as:

y^2−y^1=5.138 cm4/1000

y^4 −y^3=3.39 cm4/1000

y^6 −y^5=6.886 cm4/1000

where, y^1, y^3, and y^5 represent the dominant arm strengths of the non-baseball players with non-dominant arm strengths as 16 cm4/1000, 12 cm4/1000, and 20 cm4/1000, respectively, and y^2, y^4, and y^6 represent the dominant arm strengths of the baseball players with non-dominant arm strengths as 16 cm4/1000, 12 cm4/1000, and 20 cm4/1000, respectively. The (y^2−y^1), (y^4−y^3), and (y^6−y^5) 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:

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 cm4/1000, 12 cm4/1000, and 20 cm4/1000 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.

y^4 −y^3=3.39,

y^2−y^1=5.138, and

y^6 −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.