Question

The article "Poll Finds Most Oppose Return to Draft, Wouldn't Encourage Children to Enlist" (Associated Press, December 18,2005 ) reports that in a random sample of 1,000 American adults, 700 indicated that they oppose the reinstatement of a military draft. Suppose you want to use this information to decide if there is convincing evidence that the proportion of American adults who oppose reinstatement of the draft is greater than two-thirds. a. What hypotheses should be tested in order to answer this question? b. The \(P\) -value for this test is \(0.013 .\) What conclusion would you reach if \(\alpha=0.05 ?\)

Short answer

a. The hypotheses to be tested are: \(H_0: p = 0.6667\) (null hypothesis) and \(H_a: p > 0.6667\) (alternative hypothesis). b. As 0.013 < 0.05, the null hypothesis is rejected and the conclusion is that the proportion of American adults who oppose reinstatement of the draft is greater than two-thirds.

Step-by-step solution

Formulate the Hypotheses

First, create the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)). The null hypothesis is that the proportion of American adults who oppose reinstatement of the draft is equal to two-thirds, or \(0.6667\). This can be written as \(H_0: p = 0.6667\). The alternative hypothesis is that the proportion is greater than two-thirds. This can be written as \(H_a: p > 0.6667\).

Calculate Test Statistic

Next, calculate the test statistic. The formula for the test statistic in a hypothesis test for a proportion is \(Z = (p - p_0) / sqrt((p_0*(1 - p_0)) / n)\) where \(p\) is the sample proportion, \(p_0\) is the proportion under the null hypothesis, and \(n\) is the sample size. Here, the sample proportion (\(p\)) is 700/1000 = 0.7, the proportion under the null hypothesis (\(p_0\)) is two-thirds or 0.6667, and the sample size (\(n\)) is 1000.

Determine the P-value

The P-value for this test is given in the prompt as 0.013. The P-value is the probability of observing a result as extreme as the one observed (or more extreme) if the null hypothesis is true. It measures the strength of evidence against the null hypothesis.

Clear Conclusion Based on Alpha

Finally, compare the P-value to the significance level (\(α\)). If the P-value is less than or equal to \(α\), then the null hypothesis is rejected in favor of the alternative. Here, the P-value of 0.013 is less than the significance level of 0.05, so the null hypothesis is rejected. This suggests that the proportion of American adults who oppose reinstatement of the draft is indeed greater than two-thirds.

Key concepts

Null Hypothesis

In the realm of hypothesis testing, the null hypothesis, denoted as \(H_0\), serves as a default position. It asserts that there is no effect or no difference, and in the context of our example, it proposes that the proportion of American adults opposing the reinstatement of a military draft is exactly two-thirds, or \(0.6667\). In this scenario, \(H_0: p = 0.6667\) is the null hypothesis.

The significance of the null hypothesis lies in its role as the claim that is initially assumed to be true, and it can only be rejected if the evidence suggests otherwise. Setting a clear null hypothesis is crucial as it establishes a benchmark against which the actual findings of the study are compared.

Alternative Hypothesis

Conversely, the alternative hypothesis, symbolized as \(H_a\) or \(H_1\), articulates a researcher's assertion about the population parameter that bears substantive interest. It is what you are attempting to demonstrate through your study or experiment.

For our example with American adults' opposition to the draft, the alternative hypothesis is that the proportion who are against is greater than two-thirds, formally stated as \(H_a: p > 0.6667\). The alternative hypothesis shapes the direction of the statistical test and is considered only if the null hypothesis is rejected.

P-value

The P-value stands as a vital concept in hypothesis testing. It represents the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the assumption that the null hypothesis is true. It's a bridge between your data and the hypotheses.

So, when you obtain a P-value of \(0.013\), as in the given exercise, it suggests that there is only a 1.3% chance of seeing the collected data, or something more extreme, if the null hypothesis were correct. Small P-values, typically below a predetermined significance level, are indicative of strong evidence against the null hypothesis.

Statistical Significance

Statistical significance is the likelihood that the relationship observed in your data is not due to random chance. It is often assessed by comparing the P-value to an alpha level (denoted as \(\alpha\)), which represents the threshold for significance.

In the context of our exercise, choosing an \(\alpha\) level of \(0.05\) sets up a criterion under which we would deem the results significant enough to reject the null hypothesis. Since the P-value in the exercise, \(0.013\), is lower than our alpha level, we conclude that the findings are statistically significant. This leads to rejecting the null hypothesis in favor of the alternative: a larger proportion of American adults oppose the draft than initially posited.