Suppose you are a fan of baseball, which has two different leagues in the US, the National League and the American League. The winners of both the leagues play in the World Series Championship each year. You want to test whether both the leagues are equally strong, or whether one of the leagues is stronger than the other. Assume that each year the AL champion wins the world series with probability p, and the NL champions win with probability 1−p, where p is an unknown number in (0, 1). Starting from 1903, the world series championship has been played 115 times, and the AL champions won it 66 times, and the NL champions won it 49 times. Based on this data, carry out the following tests of hypothesis: (a) H0:p=12versus H1:p&gt;12 at ≤vel α=.1. (b) H0:p=12 versus H1:p≠12 at ≤vel α=.1.

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Suppose you are a fan of baseball, which has two different leagues in the US, the National League and the American League. The winners of both the leagues play in the World Series Championship each year. You want to test whether both the leagues are equally strong, or whether one of the leagues is stronger than the other. Assume that each year the AL champion wins the world series with probability p, and the NL champions win with probability 1−p, where p is an unknown number in (0, 1).  Starting from 1903, the world series championship has been played 115 times, and the AL champions won it 66 times, and the NL champions won it 49 times. Based on this data, carry out the following tests of hypothesis:  (a) H0:p=12versus H1:p&amp;gt;12 at ≤vel α=.1.  (b) H0:p=12 versus H1:p≠12 at ≤vel α=.1.

oces3y · Accepted Answer

Step 1   a) The following information is provided: The sample size is N=115, the number of favorable cases is X=66, and the sample proportion is p―=XN=66115=0.5739, and the significance level is α=0.1    Null and Alternative Hypotheses   The following null and alternative hypotheses need to be tested:   Ho:p=0.5    Ha:p&amp;gt;0.5    This corresponds to a right-tailed test, for which a z-test for one population proportion needs to be used.   Rejection Region   Based on the information provided, the significance level is α=0.1, and the critical value for a right-tailed test is zc=1.28   The rejection region for this right-tailed test is R={z:z&amp;gt;1.28}  Step 2   Test Statistics   The z-statistic is computed as follows:  z=p―−p0p01−p0n=0.5739−0.50.51−0.5115=1.585  Decision about the null hypothesis  Since it is observed that z=1.585&amp;gt;zc=1.28, it is then concluded that the null hypothesis is rejected.   Using the P-value approach: The p-value is p=0.0565, and since p=0.0565&amp;lt;0.1, it is concluded that the null hypothesis is rejected.   Conclusion   It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population proportion p is greater than p0​, at the α=0.1 significance level.  b) Null and Alternative Hypotheses   The following null and alternative hypotheses need to be tested:   Ho:p=0.5    Ha:p≠0.5    This corresponds to a two-tailed test, for which a z-test for one population proportion needs to be used.   Rejection Region   Based on the information provided, the significance level is α=0.1, and the critical value for a right-tailed test is   Critical value Zc:±1.6449   Critical region:   (−∞,−1.6449]∪[1.6449,∞)   Test Statistics   The z-statistic is computed as follows:  z=p―−p0p01−p0n=0.5739−0.50.51−0.5115=1.585  Decision about the null hypothesis   Since it is observed that z=1.585 does not fall in critical region, it is then concluded that the null hypothesis is not  rejected.   Conclusion   It is concluded that the null hypothesis Ho is rejected. Therefore, there is not enough evidence to claim that the population proportion p is not equal to p0​, at the α=0.1 significance level.

Suppose you are a fan of baseball, which has two different leagues in

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