91Ó°ÊÓ

Understanding the null and alternative hypotheses is crucial in hypothesis testing. They are competing statements about the true nature of the population. When a political candidate considers supporting casino gambling based on public approval, setting up these hypotheses helps guide the decision-making process.

The null hypothesis (H0) represents a default position that there is no effect or no difference. It is a statement of no change, and in our example, it suggests that two-thirds of American adults approve of casino gambling. The alternative hypothesis (H1), on the other hand, is what the researcher wants to prove. It posits that there is an effect or difference—in this case, that more than two-thirds of adults are in favor.

To test these hypotheses, we gather evidence in the form of sample data. If the data significantly contradict H0, we might reject it in favor of H1. If not, we would fail to find evidence to support H1 and therefore not reject H0.

The sample proportion is a statistic that estimates a population proportion from sample data. It's represented by the symbol \( \hat{p} \). In our candidate's investigation on casino gambling, 1035 out of 1523 sampled American adults approved of casino gambling, leading to a sample proportion (\( \hat{p} \)) of approximately 0.6796.

Calculating \( \hat{p} \) is straightforward: divide the number of 'successes' (those who approve of gambling, in this context) by the total number of observations in the sample. This ratio provides an estimate of the population proportion, which is necessary to determine if the sample provides strong enough evidence to support the alternative hypothesis.

The z-test is a statistical procedure used to determine whether there is a significant difference between sample data and a known or assumed population parameter. In hypothesis testing for proportions, the z-test helps in comparing the sample proportion (\( \hat{p} \)) to the claimed proportion (p). The formula for the z-test statistic is:\[ Z = (\hat{p} - p)/\sqrt{ \frac{p(1-p)}{n} } \]

In the gambling example, the test statistic informs us about the likelihood that the observed sample proportion could have occurred if the null hypothesis were true. A value of z = 0.71 was calculated indicating how many standard deviations our sample proportion is from the null hypothesis value. If the z-test yields a high enough statistic, it suggests that the sample provides evidence against the null hypothesis.

A p-value is a probability that measures the evidence against the null hypothesis. It's calculated from the test statistic and it tells us how likely it is to observe a statistic as extreme as the test statistic under the null hypothesis. The lower the p-value, the stronger the evidence against H0.

In our exercise, with a calculated p-value of approximately 0.24, we assess the strength of the evidence against the null hypothesis that two-thirds of adults approve of casino gambling. Since the p-value is a probability, a result of 0.24 suggests that there is a 24% chance of obtaining a test statistic as extreme as 0.71, if the true population proportion were actually two-thirds.

Statistical significance tells us whether the findings from our sample are likely to reflect a real effect in the population, rather than just random chance. A common benchmark is the significance level (\( \alpha \)), often set at 0.05. It acts as a threshold for determining whether the p-value is low enough to reject the null hypothesis.

In hypothesis testing, if the p-value is less than \( \alpha \), there is statistically significant evidence against H0. However, in the given exercise, because our p-value of 0.24 is much greater than the significance level of 0.05, we do not have sufficient evidence to reject H0. Therefore, the sample does not provide statistically significant evidence that more than two-thirds of American adults approve of casino gambling.