Question

A number of initiatives on the topic of legalized gambling have appeared on state ballots. A political candidate has decided to support legalization of casino gambling if he is convinced that more than two-thirds of American adults approve of casino gambling. Suppose that 1,035 of the people in a random sample of 1,523 American adults said they approved of casino gambling. Is there convincing evidence that more than two-thirds approve?

Short answer

No, the available data does not provide convincing evidence that more than two-thirds of American adults approve of casino gambling.

Step-by-step solution

Identify Hypotheses

The null hypothesis (\(H_0\)) is that the proportion of American adults (\(P\)) who approve of casino gambling is two-thirds (\(\frac{2}{3}\)), \(P = \frac{2}{3}\). The alternative hypothesis (\(H_1\)) is that the proportion \(P\) is greater than two-thirds, \(P > \frac{2}{3}\). The level of significance for this hypothesis test is assumed to be 0.05.

Calculate Sample Proportion

The sample proportion (\(\hat{P}\)) is the number of people in the sample who approved of casino gambling divided by the total number of people in the sample: \(\hat{P} = \frac{1035}{1523} = 0.6795856360718831.\) This is the observed proportion in the sample.

Compute the Standard Error and Test Statistic

The standard error (SE) is calculated as \(\sqrt{\frac{\hat{P}(1-\hat{P})}{N}}\), where \(N\) is the sample size. So, \(SE = \sqrt{\frac{0.68*(1-0.68)}{1523}} = 0.012485913666376535\). The test statistic (Z) is given by \(Z = \frac{\hat{P}-P}{SE}\). Therefore, \(Z = \frac{0.68 - \frac{2}{3}}{0.0125} = 0.32754160100740544\).

Calculate the p-value

Since this is a one-tailed test, the p-value is the probability of observing a z-score as extreme as the test statistic, or more extreme, given the null hypothesis is true. By using a Z-table or calculator, p-value = P(Z > 0.33), which approximates to 0.3712.

Compare p-value to Significance Level

As the p-value (0.37) is greater than the significance level (0.05), there is not enough evidence to reject the null hypothesis.