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.

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 80100=20%

By definitions, the proportion of hits for Moe in left pitcher is given as 10100=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

  40100=40% and the proportion of hits by Moe is given as 120400=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.

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 4×10%=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.