In a recent issue of Consumer Reports, Consumers Union reported on their investigation of bacterial contamination in packages of name brand chicken sold in supermarkets. Packages of Tyson and Perdue chicken were purchased. Laboratory tests found campylobacter contamination in 35 of the 75 Tyson packages and 22 of the 75 Perdue packages.

problem 1. Find 95% confidence intervals for the proportion of Tyson packages with contamination and the proportion of Perdue packages with contamination (use 3 decimal places in your answers).
problem 2. The confidence intervals in problem 1 overlap. What does this suggest about the difference in the proportion of Tyson and Perdue packages that have bacterial contamination?
problem 3. Find the 95% confidence interval for the difference in the proportions of Tyson and Perdue chicken packages that have bacterial contamination (use 3 decimal places in your answers).
problem 4. What does this interval suggest about the difference in the proportions of Tyson and Perdue chicken packages with bacterial contamination?
problem 5. The results in problems 2 and 4 seem contradictory. Which method is correct: doing two-sample inference, or doing one-sample inference twice?
problem 6. Why don't the results agree?