Question

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.

Question 1. Find 90% 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).

_____ 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 1 overlap. What does this suggest about the difference in the proportion of Tyson and Perdue packages that have bacterial contamination? One submission only; no exceptions

The overlap suggests that there is no significant difference in the proportions of packages of Tyson and Perdue chicken with bacterial contamination.

Even though there is overlap, Tyson's sample proportion is higher than Perdue's so clearly Tyson has the greater true proportion of contaminated chicken.

Question 3. Find the 90% 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).

_____ lower bound of confidence interval

_____ upper bound of confidence interval

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

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

Natural sampling variation is the only reason that Tyson appears to have a higher proportion of packages with bacterial contamination.

Tyson's sample proportion is higher than Perdue's so clearly Tyson has the greater true proportion of contaminated chicken.

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

two-sample inference

one-sample inference twice

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

The one- and two-sample procedures for analyzing the data are equivalent; the results differ in this problem only because of natural sampling variation.

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

Different methods were used in the two samples to detect bacterial contamination.

Tyson chicken is sold in less sanitary supermarkets.

Answer 1

#1.

n = 75

p = 35/75 = 0.467

z-value of 90% CI = 1.645

SE = sqrt(p*(1-p)/n)

= sqrt(0.467*(1-0.467)/75) = 0.058

ME = z*SE = 0.095

Lower Limit = p - ME= 0.372

Upper Limit = p + ME = 0.561

90% CI (0.3719 , 0.5614 )

For Perdue

p = 22/75 = 0.293

SE = sqrt(p*(1-p)/n) = 0.053

ME = z*SE = 0.086

Lower Limit = p - ME = 0.207

Upper Limit = p + ME = 0.380

90% CI (0.2069 , 0.3798 )

0.372 lower bound of Tyson interval

0.561 upper bound of Tyson interval

0.207 lower bound of Perdue interval

0.380 upper bound of Perdue interval

#2.

The overlap suggests that there is no significant difference in the proportions of packages of Tyson and Perdue chicken with bacterial contamination.

#3.

pcap = (35 + 22)/150 = 0.3800

SE = sqrt(0.38*0.62*(1/75 + 1/75)) = 0.0793

p1cap - p2cap = 0.1733

ME = sigma*z = 0.1283

Lower limit = (p1cap - p2cap)-ME = 0.0451

Upper limit = (p1cap - p2cap)+ME = 0.3016

90% CI (0.0451 , 0.3016 )

#4.

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

#5.

two-sample inference

#6.

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