Question

A number of initiatives on the topic of legalized gambling have appeared on state ballots. Suppose that a political candidate has decided to support legalization of casino gambling if he is convinced that more than twothirds of U.S. adults approve of casino gambling. USA Today (June 17,1999 ) reported the results of a Gallup poll in which 1523 adults (selected at random from households with telephones) were asked whether they approved of casino gambling. The number in the sample who approved was 1035 . Does the sample provide convincing evidence that more than two-thirds approve?

Short answer

The answer depends on the computed value of the test statistic and the p-value. If the p-value is less than the elected level of significance (for example, 0.05), then there is convincing evidence that more than two-thirds of U.S. adults approve of casino gambling.

Step-by-step solution

State the Hypotheses

The null hypothesis (H0) assumes the default position, which is typically a statement of no effect or no difference. In this case, it's that two-thirds or fewer adults approve of casino gambling. The alternative hypothesis (H1) is what you might believe to be true or are testing for, here it's that more than two-thirds approve. So,\n\nNull hypothesis (H0): p ≤ 2/3\n\nAlternative Hypothesis (H1): p > 2/3\n\nwhere p is the proportion of U.S. adults who approve of casino gambling.

Compute the sample proportion and Test Statistic

The sample proportion \( \hat{p} \) is calculated by dividing the number of individuals who approved (1035) by the total number in the sample (1523)\n\n\( \hat{p} = 1035 / 1523 = 0.680 \n\nThe test statistic (Z) for a test of proportion is calculated by\n\nZ = ( \( \hat{p} - p_{0} \) ) / \( \sqrt {p_{0} (1 - p_{0} ) / n } \n\nwhere \( p_{0} \) is the proportion under the null hypothesis, in this case two thirds or, 0.67 (approx), and n is the sample size.\n\nZ = (0.680 - 0.67) / \( \sqrt { (0.67 * (1 - 0.67)) / 1523 } \)

Find the P-value

The P-value is the probability of obtaining a test statistic as extreme as the one computed, given that the null hypothesis is true. This can be found by looking the Z statistic value up in a standard normal (Z) table or using statistical software.

Draw a Conclusion

Compare the p-value to your chosen significance level (for instance, 0.05) to decide whether to reject the null hypothesis. If the p-value is less than or equal to the significance level, reject the null hypothesis and conclude that the data provides convincing evidence that more than two-thirds of U.S. adults approve of casino gambling.