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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 1000 American adults, 430 answered "yes" to the following question: "If the military draft were reinstated, would you favor drafting women as well as men?" The data were used to test \(H_{0}: p=0.5\) versus \(H_{a}: p<0.5,\) and the null hypothesis was rejected. (Hint: See discussion following Example \(10.5 .\) ) a. Based on the result of the hypothesis test, what can you conclude about the proportion of American adults who fayor drafting women if a military draft were reinstated? b. Is it reasonable to say that the data provide strong support for the alternative hypothesis? c. Is it reasonable to say that the data provide strong evidence against the null hypothesis?

Step by step solution

01

We are given a random sample of 1000 American adults, and 430 of them answered "yes" to the question, favoring drafting women as well as men. The sample proportion \(\hat{p}\) is calculated as follows: \[\hat{p} = \frac{430}{1000} = 0.43\] #Step 2: Conclusion about population proportion#

Based on the result of the hypothesis test, which rejected the null hypothesis, we can conclude that the proportion of American adults who favor drafting women if a military draft were reinstated is less than 50%, as the alternative hypothesis suggests. To be more specific, our sample proportion indicates that around 43% of American adults support drafting women. #Step 3: Support for the alternative hypothesis#

02

Considering that the null hypothesis was rejected, there is evidence in favor of the alternative hypothesis. The fact that the proportion of American adults who favor drafting women is 43%, which is less than 50%, provides support for the alternative hypothesis \(H_{a}: p < 0.5\). However, it is important to note that the strength of this support cannot be determined solely using the information provided in this exercise. #Step 4: Evidence against the null hypothesis#

Since the null hypothesis was rejected based on the hypothesis test, it is reasonable to say that the data provide evidence against it. While we can't provide an exact measure of evidence strength, it is still notable that given the sample size of 1000 and observing 43% of American adults supporting the drafting of women, we have enough evidence to reject the null hypothesis, implying the proportion is less than 50%. In conclusion, we found that the proportion of American adults who favor drafting women in the military is less than 50%, as the null hypothesis was rejected. The data provide support for the alternative hypothesis, but the strength of this support cannot be determined with the given information. However, the data provide evidence against the null hypothesis, and it is reasonable to reject it based on the sample size and observed sample proportion.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Proportion

The sample proportion is a critical concept in statistics. It represents the proportion of individuals in a sample with a particular attribute or characteristic. In this context, the sample proportion \(^p\) is the fraction of American adults in the sample who favor drafting women if a military draft is reinstated.
To calculate the sample proportion, you divide the number of favorable outcomes by the total number of observations in the sample.
In this exercise:

• Number of adults favoring drafting women: 430
• Total sample size: 1000

Thus, the sample proportion is computed as:
\[ \hat{p} = \frac{430}{1000} = 0.43 \] This means that 43% of the sampled adults support drafting women alongside men.
Sample proportions are often used as estimates for population proportions, especially in hypothesis testing, where they help to make inferences about the entire population.

Null Hypothesis

In hypothesis testing, the null hypothesis serves as the initial claim or assumption that we seek to test. It generally represents the status quo or no effect situation.
For this scenario, the null hypothesis \(H_{0}\) is that the true population proportion \(p\) is equal to 0.5 (or 50%). This implies that half of all American adults are in favor of drafting both genders equally if a draft were reintroduced.
Null hypotheses are often expressed using equality because they specify the absence of an effect or difference. It operates as a benchmark against which the alternative hypothesis is tested.

Alternative Hypothesis

The alternative hypothesis represents the statement we suspect might be true and are attempting to find evidence for through our data.
In this problem, the alternative hypothesis \(H_{a}\) is that the proportion of American adults who support drafting women is less than 50%, \(p < 0.5\).
Identifying the alternative hypothesis correctly is crucial as it guides the direction of the test, often dictated by questions of interest. It suggests a one-tailed test as it is concerned with proportions being specifically less than 0.5.
By rejecting the null hypothesis in favor of the alternative, we assert that the evidence points to fewer than half of the population supporting the idea.

Statistical Conclusion

A statistical conclusion is the final statement from a hypothesis test that explains the outcome based on the collected data.
In this exercise, the null hypothesis was rejected. This suggests there is statistically significant evidence to support the alternative hypothesis, indicating the proportion of adults in favor of drafting women is indeed less than 50%.
Such conclusions are drawn from analyzing whether the sample data provide enough evidence to oppose the null hypothesis.
However, it's important to acknowledge that the strength of evidence can vary, and the absence of specific statistical values (like p-values) in this scenario means we can't declare the strength of this conclusion definitively. Typically, additional information such as the significance level would aid in determining confidence in these results.