Question

In Exercises 5–20, conduct the hypothesis test and provide the test statistic and the P-value and, or critical value, and state the conclusion.

Baseball Player Births In his book Outliers, author Malcolm Gladwell argues that more baseball players have birth dates in the months immediately following July 31, because that was the age cutoff date for nonschool baseball leagues. Here is a sample of frequency counts of months of birth dates of American-born Major League Baseball players starting with January: 387, 329, 366, 344, 336, 313, 313, 503, 421, 434, 398, 371. Using a 0.05 significance level, is there sufficient evidence to warrant rejection of the claim that American-born Major League Baseball players are born in different months with the same frequency? Do the sample values appear to support Gladwell’s claim?

Short answer

There is enough evidence to conclude thatbaseball players are not born with the same frequency in different months of the year.

Sample data does not support the author’s claim.

Step-by-step solution

Given information

The frequencies of baseball players born in different months are provided.

Check the requirements

Let O denote the observed frequencies of the players born in the 12 months.

Jan\(\left( {{O_1}} \right)\)	Feb\(\left( {{O_2}} \right)\)	March\(\left( {{O_3}} \right)\)	April\(\left( {{O_4}} \right)\)	May\(\left( {{O_5}} \right)\)	June\(\left( {{O_6}} \right)\)
387	329	366	344	336	313
July\(\left( {{O_7}} \right)\)	Aug\(\left( {{O_8}} \right)\)	Sep\(\left( {{O_9}} \right)\)	OcT\(\left( {{O_{10}}} \right)\)	Nov\(\left( {{O_{11}}} \right)\)	Dec\(\left( {{O_{12}}} \right)\)
313	503	421	434	398	371

The sum of all observed frequencies is computed below:

\(\begin{aligned}{c}n = 387 + 329 + ...... + 371\\ = 4515\end{aligned}\)

Let E denote the expected frequencies.

It is given that the number of births is expected to occur with equal frequency in all of the 12 months.

The expected frequency for each of the 12 months is the same and is equal to:

\(\begin{aligned}{c}E = \frac{{4515}}{{12}}\\ = 376.25\end{aligned}\)

As the expected value is greater than 5, the requirements for the test are satisfied.

State the hypotheses

The null hypothesis for conducting the given test is as follows:

\({H_0}:\)The frequency of baseball players’ births is equal in different months of the year.

The alternative hypothesis is as follows:

\({H_a}:\)The frequency of baseball players’ births is not equal in different months of the year.

The test is right-tailed.

Conduct the test

The table below shows the necessary calculations:

Months	O	E	\(\left( {O - E} \right)\)	\({\left( {O - E} \right)^2}\)	\(\frac{{{{\left( {O - E} \right)}^2}}}{E}\)
January	387	376.25	10.75	115.5625	0.307143
February	329	376.25	-47.25	2232.563	5.933721
March	366	376.25	-10.25	105.0625	0.279236
April	344	376.25	-32.25	1040.063	2.764286
May	336	376.25	-40.25	1620.063	4.305814
June	313	376.25	-63.25	4000.563	10.63272
July	313	376.25	-63.25	4000.563	10.63272
August	503	376.25	126.75	16065.56	42.69917
September	421	376.25	44.75	2002.563	5.322425
October	434	376.25	57.75	3335.063	8.863953
November	398	376.25	21.75	473.0625	1.257309
December	371	376.25	-5.25	27.5625	0.073256

The value of the test statistic is equal to:

\[\begin{aligned}{c}{\chi ^2} = \sum {\frac{{{{\left( {O - E} \right)}^2}}}{E}} \\ = 0.307143 + 5.933721 + ... + 0.073256\\ = 93.07176\end{aligned}\]

Thus,\({\chi ^2} = 93.072\).

Let k be the number of months, which is 12.

The degrees of freedom for\({\chi ^2}\)is computed below:

\(\begin{aligned}{c}df = k - 1\\ = 12 - 1\\ = 11\end{aligned}\)

State the decision

The critical value of\({\chi ^2}\)at\(\alpha = 0.05\)with 11 degrees of freedom is equal to 19.675.

The p-value is equal to 0.000.

Since the test statistic value is greater than the critical value and the p-value is less than 0.05, the null hypothesis is rejected.

State the conclusion

There is enough evidence to conclude thatbaseball players are not born with the same frequency in different months of the year.

The sample data for verifying the author’s claim,

August	503
September	421
October	434
November	398
December	371
Total	2127

January	387
February	329
March	366
April	344
May	336
June	313
July	313
Total	2388

The sum is higher in case of births before july 31.

Thus, it does not support the claim of the author.