Question

The poll referenced in the previous exercise ("Military Draft Study," AP- Ipsos, June 2005) also included the following question: "If the military draft were reinstated, would you favor or oppose drafting women as well as men?" Forty-three percent of the 1000 people responding said that they would favor drafting women if the draft were reinstated. Using a \(.05\) significance level, carry out a test to determine if there is convincing evidence that fewer than half of adult Americans would favor the drafting of women.

Short answer

There's convincing evidence that less than half of adult Americans would favor the drafting of women.

Step-by-step solution

Stating the hypotheses

The null hypothesis \(H_0\) is that the true proportion we’re testing is half (0.5), while the alternative hypothesis \(H_a\) is that the true proportion is less than half, meaning that less than half of adult Americans would favor the drafting of women.

Collect and summarize the data

Here, the sample proportion \(\hat{p}\) is \(0.43\). The sample size \(n\) is \(1000\). The sample proportionis a useful statistic for estimating the population proportion.

Calculate the test statistic

For this, a test statistic \(z\) is calculated using the formula: \( z = \frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\). With \( p_0 = 0.5\) (the null hypothesis), \(n= 1000\) and \(\hat{p} = 0.43\), the result will be \( z = \frac{0.43 - 0.5}{\sqrt{\frac{0.5 * 0.5}{1000}}} = -4.47\).

Compute the P-value

The level of significance for this test is \(0.05\). The P-value is the probability that the z-score is less than \( -4.47\) under the null hypothesis, which can be looked up in a z-distribution table or calculated using statistical software. The P-value is virtually 0, which is less than the level of significance.

Conclusion

Given that the P-value is less than the significance level, the null hypothesis is rejected, meaning that there's convincing evidence that less than half of adult Americans would favor the drafting of women.