Question

This was the only information given in the question

In 2012, the Detroit Tigers and the San Francisco Giants met in baseball's World Series.  That year, the Tigers had a won-lost record of 88-74 and the Giants had a won-lost record of 94-68.   The question is whether the Giants were in fact the superior team based on their won-lost record.  So, test the claim that the 'true' proportion of their games won by the Giants was in fact higher than it was for the Tigers in 2012. Assume the only alternate hypothesis of interest is where the Giants are the superior team.   

The p-value of the hypothesis test is: ?????



Which of the following are correct conclusions?

At a 10% significance level, it would appear that the Giants were in fact the superior team.

At a 1% significance level, it would appear that the Giants were in fact the superior team.

At a 5% significance level, it would appear that the Giants were in fact the superior team.

At a 15% significance level, it would appear that the Giants were in fact the superior team.

It would seem that this 6-win difference over the course of the season is not statistically significant.

Answer 1

For sample 1, we have that the sample size is N 1=168, the number of favorable cases is X1=94, so then the sample proportion is p^1=X1/N1=94/168=0.5595

For sample 2, we have that the sample size is N2=162, the number of favorable cases is X2=88, so then the sample proportion is p^2=X2/N2=88/162=0.5432

The value of the pooled proportion is computed as p¯=X1+X2​​/N1+N2 =94+88/168+162=0.5515

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:p1​=p2​

Ha:p1​>p2​

This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted.

Test Statistics

The z-statistic is computed as follows:

z=p1^-p2^/sqrt(p¯​(1−p¯​)(1/n1​+1/n2​)​)

=0.5595-0.5432/sqrt(0.5515⋅(1−0.5515)(1/168+1/162))​=0.298

Decision about the null hypothesis

Using the P-value approach: The p-value is p=0.3829

Not statistically significant.