Views on the Death Penalty The Pew Research Group conducted...

Views on the Death Penalty The Pew Research Group conducted a poll in which they asked, "Are you in favor of, or opposed to, executing persons as a general policy when the crime was committed while under the age of $18 ? "$ Of the 580 Catholics surveyed, 180 indicated they favored capital punishment; of the 600 seculars (those who do not associate with a religion) surveyed, 238 favored capital punishment. Is there a significant difference in the proportion of individuals in these groups in favor of capital punishment for persons under the age of $18 ?$ Use the $\alpha=0.01$ level of significance.

Key Concepts

Pooled Proportion Estimate

When comparing two proportions under the assumption of the null hypothesis (that the proportions are equal), a pooled proportion is calculated by combining the successes and total sample sizes from both groups. This pooled estimate is used to compute the standard error and helps in standardizing the difference between the sample proportions.

Test Statistic (Z-score) and p-value

The test statistic for comparing two proportions often follows a normal (Z) distribution under the null hypothesis. The Z-score is computed by standardizing the difference between the sample proportions using the estimated standard error. The p-value is then obtained from this Z-score, which indicates the probability of observing a test statistic as extreme as the one computed if the null hypothesis were true.

Significance Level (Alpha)

The significance level, denoted by alpha (?), is the threshold for determining whether the observed effect is statistically significant. It represents the probability of rejecting the null hypothesis when it is actually true (Type I error). Common values are 0.05 and 0.01; in this case, ? is 0.01, indicating strong evidence is required to conclude a significant difference.

Hypothesis Testing for Two Proportions

This is a statistical method used to compare the proportions of a specific outcome in two independent groups. The goal is to determine if any observed difference in the sample proportions reflects a true difference in the population proportions or if it could have occurred by chance during sampling.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis typically states that there is no difference between the group proportions (i.e., any observed difference is due to sampling variability), while the alternative hypothesis asserts that there is a significant difference. Clearly defining these hypotheses is critical for setting up the test and guiding the interpretation of the results.

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