In a recent issue of Consumer Reports, Consumers Unionreported on their investigation of bacterial contamination inpackages of name brand chicken sold in supermarkets.

Packages of Tyson and Perdue chicken werepurchased. Laboratory tests found campylobacter contamination in 35of the 75 Tyson packages and 22 of the 75 Perduepackages.

Question 1. Find 90% confidence intervals forthe proportion of Tyson packages with contamination andthe proportion of Perdue packages with contamination (use3 decimal places in your answers).

lower bound of Tyson interval
upper bound of Tyson interval
lower bound of Perdue interval
upper bound of Perdue interval

Question 2. The confidence intervals in question 1overlap. What does this suggest about the difference in theproportion of Tyson and Perdue packages that havebacterial contamination? One submission only; noexceptions

The overlap suggests that there is no significant difference inthe proportions of packages of Tyson and Perduechicken with bacterial contamination.

Even though there is overlap, Tyson's sample proportionis higher than Perdue's so clearly Tyson has thegreater true proportion of contaminatedchicken.    

Question 3. Find the 90% confidence intervalfor the difference in the proportions of Tyson andPerdue chicken packages that have bacterial contamination(use 3 decimal places in your answers).

lower bound of confidence interval
upper bound of confidence interval

Question 4. What does this interval suggest aboutthe difference in the proportions of Tyson andPerdue chicken packages with bacterial contamination?One submission only; no exceptions

Tyson's sample proportion is higher thanPerdue's so clearly Tyson has the greater trueproportion of contaminated chicken.

Natural sampling variation is the only reason thatTyson appears to have a higher proportion of packages withbacterial contamination.    

We are 90% confident that the interval in question 3 capturesthe true difference in proportions, so it appears thatTyson chicken has a greater proportion of packages withbacterial contamination than Perdue chicken.

Question 5. The results in questions 2 and 4seem contradictory. Which method is correct: doing two-sampleinference, or doing one-sample inference twice? Onesubmission only; no exceptions

one-sample inference twice

two-sample inference    

Question 6. Why don't the results agree?2 submission only; no exceptions

Different methods were used in the two samples to detectbacterial contamination.

The one- and two-sample procedures for analyzing the data areequivalent; the results differ in this problem only because ofnatural sampling variation.    

If you attempt to use two confidence intervals to assess adifference between proportions, you are adding standard deviations.But it's the variances that add, not the standard deviations. Thetwo-sample difference-of-proportions procedure takes this intoaccount.

Tyson chicken is sold in less sanitarysupermarkets.