Postpartum depression and anxiety (PPD), is a common medical condition affecting mothers and their families after the birth of a baby. The CDC estimated that $13 \%$ of women who have recently given birth suffer from PPD. However, this research only reflected self-reported cases. Therefore, one group dedicated to helping women and their families with PPD believes that the true percentage of women who suffer from PPD is much higher. The group conducts a simple random sample of 103 women who had given birth in the last year and discovers that 21 of them report having PPD. Based on this evidence, can the group claim that the true percentage of women who have PPD is greater than $13 \%$ ? Use a 0.05 level of significance.
Step 3 of 3 : Draw a conclusion and interpret the decision.
Answer
We fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.05 level of significance that the true percentage of women who have PPD is greater than $13 \%$.
We fail to reject the null hypothesis and conclude that there is sufficient evidence at a 0.05 level of significance that the true percentage of women who have PPD is greater than $13 \%$.
We reject the null hypothesis and conclude that there is sufficient evidence at a 0.05 level of significance that the true percentage of women who have PPD is greater than $13 \%$.
We reject the null hypothesis and conclude that there is insufficient evidence at a 0.05 level of significance that the true percentage of women who have PPD is greater than $13 \%$.
Answer
\(\boxed{\text{We reject the null hypothesis and conclude that there is sufficient evidence at a 0.05 level of significance that the true percentage of women who have PPD is greater than 13%}}\)
Step 1 ：Define the null hypothesis as the true proportion of women who suffer from PPD is 13% (0.13), and the alternative hypothesis as the true proportion is greater than 13%.
Step 2 ：Given a sample size of 103 women, and 21 of them report having PPD, calculate the sample proportion.
Step 3 ：Perform a z-test to determine if we can reject the null hypothesis at a 0.05 level of significance.
Step 4 ：Calculate the p-value. If the p-value is less than the significance level (0.05), we can reject the null hypothesis.
Step 5 ：\(n = 103\)
Step 6 ：\(x = 21\)
Step 7 ：\(p_{null} = 0.13\)
Step 8 ：\(\alpha = 0.05\)
Step 9 ：\(p_{sample} = 0.20388349514563106\)
Step 10 ：\(z = 2.2296399737490824\)
Step 11 ：\(p_{value} = 0.012885676919781486\)
Step 12 ：Since the p-value is less than the significance level, we reject the null hypothesis.
Step 13 ：\(\boxed{\text{We reject the null hypothesis and conclude that there is sufficient evidence at a 0.05 level of significance that the true percentage of women who have PPD is greater than 13%}}\)
