Question

In an AP-AOL sports poll (Associated Press, December 18,2005 ), 272 of 394 randomly selected baseball fans stated that they thought the designated hitter rule should either be expanded to both baseball leagues or eliminated. Based on the given information, is there sufficient evidence to conclude that a majority of baseball fans feel this way?

Short answer

The steps involve hypothesis testing where a null hypothesis and alternative hypothesis is defined based on the given statement. A test statistic is calculated and the obtained p-value is compared with the significance level to arrive at a conclusion if there is substantial evidence to believe that a majority of baseball fans feel a certain way about the designated hitter rule.

Step-by-step solution

Establish the Null and Alternative Hypotheses

In order to conduct a hypothesis test, first establish the null and alternative hypotheses. The null hypothesis (\(H_0\)) can be that the proportion of baseball fans who believe the designated hitter rule should either be expanded to both leagues or eliminated is less than or equal to 0.5 (50%), that means, there's not a majority. The alternative hypothesis (\(H_1\)), which we are trying to gather evidence to support, can be that the proportion is greater than 0.5 (50%), meaning there's a majority.

Test Statistic Calculation

Now calculate the test statistic, which is the sample proportion minus the population proportion in the null hypothesis, divided by the standard error. The sample proportion (\( \hat{p} \)) is 272 out of 394, which is approximately 0.690. The standard error (\( SE \)) of a proportion under the Central Limit Theorem is \(\sqrt{\frac{{(P)(1-P)}}{n}}\) where \(P\) is the assumed population proportion and \(n\) is the sample size. Here, \(P\) is 0.5 and \(n\) is 394, thus the standard error is approximately 0.025.

P-Value Calculation

After obtaining the test statistic, the next step is to calculate the P-value. The P-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated under the null hypothesis. Given that the test statistic follows a normal distribution, we can utilize a standard normal Z-table or software to get the P-value. As the test is one-tailed (greater than 0.5), find the one-tailed P-value.

Conclusion

Finally, make a conclusion based on the P-value. In general, if the P-value is less than the significance level (0.05), reject the null hypothesis. If the P-value is greater than the significance level, do not reject the null hypothesis.