"Antibiotic Resistance Puzzle" reports that children in a certain city "are resistant to common antibiotics at a rate that is double the national average [. . .] To look at local resistance rates, the Children's Hospital researchers tested 708 cultures of Group A strep that were obtained from pediatric patients between September 2001 and May 2002. Overall, 68 (or 9.6%) of the strep samples were resistant to macrolides. [. . .] Studies conducted elsewhere indicate a national macrolide resistance rate of about 5%."13
a. Based on a sample size of 708 and a sample proportion of 0.10 (0.096 rounded to the nearest hundredth), use the formula for approximate standard deviation, , to show that the error (standard deviation of sample proportion) is roughly 0.01.
b. To justify use of a normal approximation, the observed numbers in and out of the category of interest must be at least 10. What are those two numbers in this case?
c. Report an approximate 95% confidence interval for the proportion of all Group A strep cultures from the area that would be resistant to macrolides.
d. Does your confidence interval contain the national resistance rate, 0.05, or does it contain only values less than 0.05, or contain only values greater than 0.05?
e. Does your interval suggest that all Group A strep cultures from that city are more resistant to macrolides than all Group A strep cultures in the United States? Explain.
f. Is a representative sample of size 708 small enough relative to the population size to ensure that the formula for standard deviation of sample proportion in part (a) is approximately correct?