A survey of 1,000 adult Americans included the following question: "If the military draft were reinstated, would you favor or oppose drafting women as well as men?" Forty-three percent responded that they would favor drafting women if the draft were reinstated. Using a significance level of 5%, determine if there is convincing evidence that less than half of adult Americans favor drafting women.
Use a one-sample z-test for proportions.
With your data:
z = (.43 - .50)/√(.50)(.50)/1000) = -4.358
Reject the null and accept the alternate hypothesis (p < .50).
To determine if there is convincing evidence that less than half of adult Americans favor drafting women, we can conduct a hypothesis test.
Here's how to do it:
Step 1: Define the null hypothesis and alternative hypothesis:
- Null hypothesis (H0): The proportion of adult Americans who favor drafting women is equal to or greater than 0.5 (p ≥ 0.5).
- Alternative hypothesis (Ha): The proportion of adult Americans who favor drafting women is less than 0.5 (p < 0.5).
Step 2: Determine the test statistic and the critical value:
- Since the sample size is large (n = 1000) and the data is dichotomous (favor or oppose), we can use the z-test for a proportion.
- The test statistic is calculated using the formula:
z = (p̂ - p) / √(p * (1-p) / n)
where p̂ is the sample proportion, p is the hypothesized population proportion, and n is the sample size.
- The critical value for a one-tailed test with a 5% significance level is z = -1.645 (we use the negative value because we are testing for less than 0.5).
Step 3: Calculate the test statistic:
- The sample proportion (p̂) is given as 43% or 0.43.
- The hypothesized population proportion (p) is 0.5.
- The sample size (n) is 1000.
Using these values in the formula, we have:
z = (0.43 - 0.5) / √(0.5 * (1-0.5) / 1000)
z = (-0.07) / √(0.25 / 1000)
z ≈ -2.8
Step 4: Compare the test statistic with the critical value:
- Since the test statistic (-2.8) is less than the critical value (-1.645), we have evidence to reject the null hypothesis.
Step 5: Interpret the result:
- Based on the survey data, there is convincing evidence that less than half (p < 0.5) of adult Americans favor drafting women if the military draft were reinstated.
In conclusion, the survey results provide significant evidence that less than half of adult Americans favor drafting women if the draft were reinstated, with a p-value less than 0.05.