Baseball paradox Most baseball hitters perform differently against right-handed and left-handed pitching. Consider two players, Joe and Moe, both of whom bat right-handed. The table below records their performance against right-handed and left-handed pitchers:

(a) Use these data to make a two-way table of player (Joe or Moe) versus outcome (hit or no hit).

(b) Show that Simpson’s paradox holds: one player has a higher overall batting average, but the other player hits better against both left-handed and right-handed pitching.

(c) The manager doesn’t believe that one player can hit better against both left-handers and right-handers yet have a lower overall batting average. Explain in

simple language why this happens to Joe and Moe.

Developing a table based on given information:

Data concerning two categorical variables are organized in a two-way table of counts. Two-way tables are frequently used to summarize vast volumes of data by categorizing outcomes.

We have 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.

Moe's overall batting average is good in this situation ($130$ hits for Moe vs. $120$ hits for Joe). Joe has faced left-handed pitchers $400$ times, while Moe has only faced left-handed pitchers$100$ times. Joe's share of hits in the left pitcher is defined as $\frac{80}{100}=20$ percent by definition. The proportion of hits for Moe in the left pitcher is provided as $\frac{10}{100}=10$ percent by definition. Joe has given up $20\%$ of hits to left-handed pitchers, while Moe has given up $10\%$ of hits to left-handed pitchers. Similarly, we discovered that Joe's proportion of hits is $\frac{40}{100}=40$ percent, while Moe's proportion of hits is $\frac{120}{400}=30$ percent for the right pitcher. Simpson's paradox states that when we consider a third variable, the relationship between two category variables can be reversed. We can see from the above that Moe's batting average is good when both pitchers are included, but Joe's batting average is good when both pitchers are considered individually. As a result, the illustration contains Simpson's dilemma. As a result, the graphic depicts Simpson's paradox.

The manager does not feel that a single Joe can hit better against both left and right-handed batters while yet having a lower overall batting average. The following information is provided:

This could have occurred because Joe has faced more left-handed pitchers than Moe and is able to hit $20\%$of them, compared to $10\%$for Moe. On average, Moe is able to hit $4\times 10\%=40\%$of his total at-bats ($400$times)

Joe's right-handed hits are $40\%$more than Moe's right-handed hits are $30\%$, while Joe's left-handed hits are $20\%$more than Moe's left-handed hits are $10\%$As a result, Joe's average score is lower than Moe's since he has faced the left pitcher $80\%$of the time whereas Moe has faced the right pitcher $80\%$of the time.