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

The hypotheses tested are \(H_0: p = \frac{2}{3}\) and \(H_A: p > \frac{2}{3}\). The P-value given is 0.013, which is less than the significance level (\(\alpha=0.05\)), therefore we reject the null hypothesis and support the alternative hypothesis. This provides us with convincing evidence that the proportion of American adults who oppose reinstatement of the draft is greater than two-thirds.

Step-by-step solution

Formulate the Hypotheses

The null hypothesis (\(H_0\)) is that the population proportion is two-thirds (\(\frac{2}{3}\)), which means there is no change. In contrast, the alternative hypothesis (\(H_A\)) is that the population proportion is greater than two-thirds (\(\frac{2}{3}\)). In formulas, we write these as: \(H_0: p = \frac{2}{3}\) and \(H_A: p > \frac{2}{3}\). Here, 'p' represents the population proportion.

Use the Given P-value and Compare with Significance level (\(\alpha\))

A P-value of 0.013 is given, which means the probability of obtaining a sample like ours, or more extreme, if the null hypothesis is true. The significance level \(\alpha\) is also given as 0.05. This is the threshold below which we reject the null hypothesis. Since the P-value is less than \(\alpha\) (0.013 < 0.05), we reject the null hypothesis.

Draw Conclusion

Drawing from the previous steps, it can be decided that there is enough evidence to support the alternative hypothesis. Thus, there is convincing evidence that the proportion of American adults who oppose reinstatement of the draft is greater than two-thirds.

Key concepts

Null and Alternative Hypotheses

Understanding the null and alternative hypotheses is fundamental to hypothesis testing, a core component of statistical analysis. The null hypothesis, symbolized by H₀, is a statement suggesting that there is no effect or no difference, and it represents a skeptical perspective or a baseline that you want to test against. For example, if we're looking at the proportion of people opposing the reinstatement of a military draft, the null hypothesis would assert that the true population proportion is two-thirds, reflecting no significant sentiment in the population beyond this specified ratio.In contrast, the alternative hypothesis, denoted by H_A or H₁, is the statement we are trying to find evidence for in our hypothesis test. It is the assertion that there is an effect or a difference. In our exercise, the alternative hypothesis proposes that the true population proportion of those opposing the draft is greater than two-thirds.It is crucial to clearly define these hypotheses because they set the stage for the testing process and determine how the evidence will be interpreted.

P-value Significance

The P-value is a pivotal concept in statistical hypothesis testing. It provides a method for quantifying the evidence against the null hypothesis. When we perform a hypothesis test, we calculate the probability of obtaining sample outcomes at least as extreme as the one we observed assuming the null hypothesis is true. This probability is the P-value. It is a measure of the strength of the evidence: a small P-value suggests that our sample result is unlikely under the null hypothesis, and thus provides strong evidence against it.The P-value is compared to a predetermined significance level, denoted by α. This significance level is the cut-off point we choose to decide whether or not to reject the null hypothesis. A common choice for α is 0.05. If the P-value is lower than α, as in our example where the P-value of 0.013 is less than α of 0.05, we reject the null hypothesis. It is important to approach P-values with a thoughtful interpretation—while a low P-value indicates that obtaining the sample data was unlikely if the null hypothesis were true, it does not necessarily mean the alternative hypothesis is true without a doubt. It simply suggests there is statistical evidence to support it.

Population Proportion

When discussing hypothesis testing in the context of a population, the term population proportion often arises. It represents the fraction of individuals in the population that have a certain characteristic or attribute. In our example, the population proportion, denoted as p, would be the ratio of American adults who oppose the reinstatement of a military draft. Estimating the population proportion involves taking a random sample and using the number of individuals in the sample with the characteristic of interest. This sample proportion is then used to infer the true population proportion.A key aspect of working with population proportions is ensuring that the sample is random and representative because biases in sampling can lead to inaccurate estimates. Thoughtful consideration is also given to the size of the sample—larger sample sizes generally provide more accurate estimates of the population proportion.Interpreting the population proportion in the context of hypothesis testing allows us to compare the estimated proportion from the sample with the hypothesized population proportion under the null hypothesis. This comparison, through the calculation of a test statistic and the resulting P-value, tells us whether the observed sample proportion is significantly different from the hypothesized proportion.