91影视

When we start any hypothesis test, we begin with a null hypothesis, often abbreviated as \(H_0\). This represents a statement of no effect or no difference. In our problem, the null hypothesis suggests that there is no difference in the proportions of Catholics and seculars who favor capital punishment for persons under the age of 18.

In more formal terms, we write this as \( H_0: p_1 = p_2 \), where \(p_1\) is the proportion of Catholics who favor capital punishment, and \(p_2\) is the proportion for seculars. This hypothesis is essentially saying that both groups have the same opinion regarding this issue.

We use the null hypothesis to set up our test; if the evidence strongly contradicts it, we will reject it in favor of an alternative hypothesis.

After setting up the null hypothesis, we establish the alternative hypothesis, noted as \(H_a\). This describes what we would believe if the null hypothesis were proven false. In our case, it suggests that there is a significant difference between the two proportions.

Formally, we write it as \( H_a: p_1 eq p_2 \). This indicates that the proportions of Catholics and seculars who favor capital punishment are significantly different.

The decision to reject the null hypothesis in favor of the alternative one depends on the results of the test statistic we calculate during the hypothesis testing process.

In our exercise, we are comparing the proportions of two distinct groups: Catholics and seculars. Proportions are a way of expressing parts of a whole and are calculated by dividing the number of individuals with a particular characteristic by the total number of individuals in the group.

Let's compute the proportions for each group:

• For Catholics: \( p_1 = \frac{180}{580} \approx 0.3103 \)
• For seculars: \( p_2 = \frac{238}{600} \approx 0.3967 \)

These sample proportions form the basis of our hypothesis test.

The next step involves combining these proportions to compute a pooled proportion, which we use to find the standard error of our comparison.

The standard error (SE) is a measure of the variability or spread of the sampling distribution of a statistic, in this case, the difference between two proportions. It helps us understand how much the sample proportions might vary from the true population proportions.

Using the pooled proportion \( p = 0.3542 \), we compute the standard error as follows:

\[ SE = \sqrt{p (1-p) \left( \frac{1}{n_1} + \frac{1}{n_2}\right)} \]

With our pooled proportion \( p \approx 0.3542 \), and sample sizes for Catholics \( n_1 = 580 \) and seculars \( n_2 = 600 \), we find:

\[ SE \approx 0.0291 \]

This standard error quantifies the uncertainty in our estimate of the difference in proportions.

A test statistic is a standardized value that reflects the difference between the sample proportions we observe and what the null hypothesis expects. We use the standard error to compute this value. For our exercise, the test statistic \(Z\) is calculated as follows:

\[ Z = \frac{(p_1 - p_2)}{SE} \]

Substituting our values:

\[ Z = \frac{0.3103 - 0.3967}{0.0291} \approx -2.97 \]

This \(Z\) value tells us how many standard errors our observed difference is away from the hypothesized difference of zero.

We then compare this value to the critical values from the \(Z\) distribution corresponding to our chosen significance level \(\alpha\). If our test statistic falls beyond these critical values, we reject the null hypothesis and conclude that there is a significant difference between the proportions.