Chapter 1.1, Problem 34E

(a)

To determine

To make: a two-way table of player versus outcome

Answer to Problem 34E

	Outcomes
Players	Hit	No Hit
Joe	$40+80=120$	$60+320=380$
Moe	$120+10=130$	$280+90=370$

Explanation of Solution

Given:

Player	Pitcher	Hits	At-bats
Joe	Right Left	$40$   $80$	$100$   $400$
Moe	Right Left	$120$   $10$	$400$   $100$

Calculation:

We have a data on the performance of two players Joe and Moe against right-handed and left-handed pitchers.

As per the requirement, consider the above given table with no hits where no hits is the value obtained by subtracting the number of hits from the number of at-bats.

Player	Pitcher	Hits	AT-bats	No Hits
Joe	Right	$40$	$100$	$100-40=60$
	Left	$80$	$400$	$400-80=320$
Moe	Right	$120$	$400$	$400-120=280$
	Left	$10$	$100$	$100-10=90$

Using above table, we construct a two-way table of player(Joe or Moe) versus outcome(hit or no hit).

	Outcomes
Players	Hit	No Hit
Joe	$40+80=120$	$60+320=380$
Moe	$120+10=130$	$280+90=370$

Conclusion:

Therefore, the two-way table is drawn.

(b)

To determine

To show: Simpson’s paradox holds:

Answer to Problem 34E

The given illustration holds Simpson’s paradox.

Explanation of Solution

Here Moe’s overall batting average is good( $130$ hits of Moe vs. $120$ hits of Joe).

For left pitchers Joe faced for $400$ times and Moe has faced left handers $100$ times only.

By definitions, the proportion of hits for Joe in left pitcher is given as $\frac{80}{100}=20\%$

By definitions, the proportion of hits for Moe in left pitcher is given as $\frac{10}{100}=10\%$ .

For left pitchers we observed that Joe has given $20\%$ of hits and Moe has given $10\%$ of hits. Similarly, we observed that for right pitcher, the proportion of hits by Joe is given as

  $\frac{40}{100}=40\%$ and the proportion of hits by Moe is given as $\frac{120}{400}=30\%$ .

The Simpson’s Paradox says that sometimes as association between two categorical variables gets reversed when we consider third variable. From the above we observed that the Moe’s batting average is good, as we consider both the pitchers combined but the Joe’s batting average is good, as we consider both the pitchers individually. Therefore the given illustration holds Simpson’s paradox.

Conclusion:

Therefore the given illustration holds Simpson’s paradox.

(c)

To determine

To explain: why the given scenario happens to Joe and Moe.

Answer to Problem 34E

This could have happened because Joe has faced left pitcher $4$ times more than Moe and is able to hit $20\%$ of them as compare to $10\%$ by Moe.

Explanation of Solution

Calculation:

The manager doesn’t believe that one Joe can hit better against both left-handed and right-handed yet have overall lower batting score. The data is shown below:

Player	Pitcher	Hits	AT-bats	$\%$ of Hits AT-Bats
Joe	Right	$40$	$100$	$40\%$
	Left	$80$	$400$	$20\%$
Moe	Right	$120$	$400$	$30\%$
	Left	$10$	$100$	$10\%$

This could have happened because Joe has faced left pitcher $4$ times more than Moe and is able to hit $20\%$ of them as compare to $10\%$ by Moe. On an average Moe is able to hit $4\times 10\%=40\%$ of total at-bats $(400times)$ .

Joe’s Right handed Hits $(40\%)$ is greater than Moe’s Right handed Hits $(30\%)$ and joe;s Left handed Hits $(20\%)$ is greater than Moe’s Left handed Hits $(10\%)$ .

Therefore, Joe has lower average score is that he has faced left pitcher $80\%$ of times and Moe has faced right pitcher $80\%$ times.

Conclusion:

Therefore, Joe has lower average score is that he has faced left pitcher $80\%$ of times and Moe has faced right pitcher $80\%$ times.