In his book Outliers,author Malcolm Gladwell argues that more

American-born baseball players have birth dates in the months immediately following July 31 because that was the age cutoff date for nonschool baseball leagues. The table below lists months of births for a sample of American-born baseball players and foreign-born baseball players. Using a 0.05 significance level, is there sufficient evidence to warrant rejection of the claim that months of births of baseball players are independent of whether they are born in America? Do the data appear to support Gladwell’s claim?

	Born in America	Foreign Born
Jan.	387	101
Feb.	329	82
March	366	85
April	344	82
May	336	94
June	313	83
July	313	59
Aug.	503	91
Sept.	421	70
Oct.	434	100
Nov.	398	103
Dec.	371	82

The data for the months of births for a sample of American-born baseball players and foreign-born baseball players is provided.

The level of significance is 0.05.

Theexpected frequencyis computed as,

\(E = \frac{{\left( {{\rm{row}}\;{\rm{total}}} \right)\left( {{\rm{column}}\;{\rm{total}}} \right)}}{{\left( {{\rm{grand}}\;{\rm{total}}} \right)}}\)

The table with row and column total is represented as,

Theexpected frequency tableis represented as,

The expected value is larger than 5.

Assuming that the subjects are randomly selected, the requirements of the test are satisfied.

The hypotheses are stated below,

\({H_0}:\)The months of births of baseball players are independent of whether they were born in America.

\({H_1}:\)The months of births of baseball players are dependent on whether they are born in America.

The value of the test statistic is computed as,

\[\begin{aligned}{c}{\chi ^2} = \sum {\frac{{{{\left( {O - E} \right)}^2}}}{E}} \\ = \frac{{{{\left( {387 - 397.2093} \right)}^2}}}{{397.2093}} + \frac{{{{\left( {329 - 334.5349} \right)}^2}}}{{334.5349}} + ... + \frac{{{{\left( {82 - 84.2790} \right)}^2}}}{{84.2790}}\\ = 20.0539\\ \approx 20.054\end{aligned}\]

Therefore, the value of the test statistic is 20.054.

The degrees of freedom are computed as,

\(\begin{aligned}{c}\left( {r - 1} \right)\left( {c - 1} \right) = \left( {2 - 1} \right)\left( {12 - 1} \right)\\ = 11\end{aligned}\)

Therefore, the degrees of freedom are 11.

From the chi-square table, the critical value corresponding to 11 degrees of freedom and at 0.05 level of significance is 19.675.

Therefore, the critical value is 19.675.

From the table, the p-value is obtained as 0.045.

Since the critical value (19.675) is less than the value of the test statistic (20.054). In this case, the null hypothesis is rejected.

Therefore, the decision is to reject the null hypothesis.

There isinsufficient evidence to favor the claim of the studythat months of births of baseball players are independent of whether they are born in America.

Thus, the players’ birth months are dependent on their places of birth (America or not).

Thus, the data support the claim of the author.