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 conclusion depends on the calculated test statistic and the corresponding p-value. If the p-value is less than 0.05, we conclude that there is sufficient evidence suggesting a majority feeling. Otherwise, we cannot say there is sufficient evidence.

Step-by-step solution

State the hypotheses

First, state the null hypothesis \(H_0: p=.5\). This represents the stance that no majority opinion exists, i.e. 50% of fans feel the rule should be altered. On the other hand, the alternative hypothesis \(H_1: p>.5\) implies a majority of fans are in favor of altering the rule.

Calculate the test statistic

The test statistic \(z\) is given by \[z=(\hat{p}-p_0)/(\sqrt{(p_0*(1-p_0))/n})\]. Here, \(\hat{p}\) is the sample proportion, \(p_0\) is the value under the null hypothesis, and \(n\) is the number of observations. Substitute the given values into the equation to get \(z=\frac{272/394 - 0.5}{\sqrt{(0.5 * (1-0.5))/394}}\).

Look up the p-value

The p-value is the probability of observing the current result (or a more extreme result) if the null hypothesis is true. The p-value can be found in a standard z-table or through software. A typical threshold value for the p-value to reject the null hypothesis is 0.05.

Draw a conclusion

If the p-value is less than 0.05, we reject the null hypothesis and conclude that there is sufficient evidence to suggest a majority of baseball fans are in favor of changing the designated hitter rule. Otherwise, we do not have sufficient evidence to support the claim.