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The null and alternative hypotheses are foundational components of hypothesis testing, a method used to determine if there is enough evidence in a sample of data to infer that a certain condition is true for the entire population.

The null hypothesis (\( H_0 \)), is a statement that there is no effect or no difference, and it represents a skeptical perspective or a claim to be tested. In our scenario, the null hypothesis is that two-thirds or less of U.S. adults approve of casino gambling, or \( P \leq \frac{2}{3} \).

The alternative hypothesis (\( H_1 \) or \( H_a \) ), is a statement that indicates the presence of an effect or a difference. It's what you aim to support. For the provided scenario, the alternative hypothesis tests whether more than two-thirds (\( P > \frac{2}{3} \) ) of the population approve of casino gambling.

Establishing these hypotheses is a crucial first step in the process of hypothesis testing, as they define the focus of the study and dictate the direction of the statistical analysis.

A test statistic is a standardized value that is calculated from sample data during a hypothesis test. It's used to decide whether to reject the null hypothesis.

In the context of our gambling poll example, the Z-score serves as the test statistic. It measures how many standard deviations an element is from the mean. The formula for computing the Z-score in a proportion test is: \[ Z = \frac{ p - P_0 }{ \text{sqrt} [ \frac{ P_0(1 - P_0) }{ n } ] } \], where 'p' is the sample proportion, \( P_0 \) is the assumed population proportion under the null hypothesis, and 'n' is the sample size.

The calculated Z-score is then compared to the critical value to determine whether the results are statistically significant—providing evidence to support or reject the null hypothesis.

The sample proportion is the percentage of individuals in the sample who exhibit a particular attribute of interest. It is denoted by \( p \) and is a key component in many statistical hypotheses tests, including the Z-test for proportions.

In our casino gambling example, the sample proportion is the number of adults who approve of casino gambling (\( 1035 \) adults) divided by the total number of adults in the sample (\( 1523 \) adults), so \( p = \frac{1035}{1523} \approx 0.6797 \). This sample proportion is what is utilized to calculate the test statistic, which is a crucial step in determining whether the null hypothesis can be rejected based on the data collected from the sample.

The critical value is a threshold that determines the boundary or cut-off point for rejecting the null hypothesis. It's derived from the desired level of significance (\( \alpha \) ), which is the probability of rejecting the null hypothesis when it is actually true (also known as a type I error).

In hypothesis testing, we compare the test statistic against the critical value. If the test statistic falls within the critical region (beyond the critical value), we reject the null hypothesis. For our one-sided test, the critical value at a 5% significance level is 1.645. This indicates that if the calculated Z-score is greater than 1.645, the evidence is in favor of the alternative hypothesis that more than two-thirds of U.S. adults approve of casino gambling.