Question

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

Short answer

There is sufficient evidence to warrant rejection of the claim that months of births of baseball players are independent of whether they are born in America.

Step-by-step solution

Given information

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.

Compute the expected frequencies

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,

	Born in America	Foreign Born	Column total
Jan.	387	101	488
Feb.	329	82	411
March	366	85	451
April	344	82	426
May	336	94	430
June	313	83	396
July	313	59	372
Aug.	503	91	594
Sept.	421	70	491
Oct.	434	100	534
Nov.	398	103	501
Dec.	371	82	453
Row Total	4515	1032	5547

Theexpected frequency tableis represented as,

	Born in America	Foreign Born
Jan.	397.2093	90.7907
Feb.	334.5349	76.46512
March	367.093	83.90698
April	346.7442	79.25581
May	350	80
June	322.3256	73.67442
July	302.7907	69.2093
Aug.	483.4884	110.5116
Sept.	399.6512	91.3488
Oct.	434.6512	99.3488
Nov.	407.7907	93.2093
Dec.	368.7209	84.27907

The expected value is larger than 5.

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

State the null and alternate hypothesis

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.

Compute the test statistic

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.

Compute the degrees of freedom

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.

Compute the critical value

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.

State the decision

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.

State the conclusion

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.