The article “Breast-Feeding Rates Up Early” (USA Today, Sept. 14, 2010) summarizes a survey of mothers whose babies were born in 2009. The Center for Disease Control sets goals for the proportion of mothers who will still be breast-feeding their babies at various ages. The goal for 12 months after birth is 0.25 or more. Suppose that the survey used a random sample of 1,200 mothers and that you want to use the survey data to decide if there is evidence that the goal is not being met. Let p denote the proportion of all mothers of babies born in 2009 who were still breast-feeding at 12 months.
a. Describe the shape, center, and spread of the sampling distribution of  for random samples of size 1,200 if the null hypothesis H0: p = 0.25 is true.
b. Would you be surprised to observe a sample proportion of  = 0.24 for a sample of size 1,200 if the null hypothesis H0: p = 0.25 were true? Explain why or why not.
c. Would you be surprised to observe a sample proportion as small as  = 0.20 for a sample of size 1,200 if the null hypothesis H0: p = 0.25 were true? Explain why or why not.
d. The actual sample proportion observed in the study was  = 0.22. Based on this sample proportion, is there convincing evidence that the goal is not being met, or is the observed sample proportion consistent with what you would expect to see when the null hypothesis is true? Support your answer with a probability calculation.