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

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

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

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.

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

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 detect bacterial contamination.

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.

Tyson chicken is sold in less sanitary supermarkets.

Answer 1

Normal approximation to the binomial calculation:

Tyson packages

Standard error of the mean = SEM = √x(N-x)/N³ = 0.058

α = (1-CL)/2 = 0.025

Standard normal deviate for α = Z_α = 1.960

Proportion of positive results = P = x/N = 0.467

Lower bound = P - (Z_α*SEM) = 0.354

Upper bound = P + (Z_α*SEM) = 0.580

Perdue packages

Standard error of the mean = SEM = √x(N-x)/N³ = 0.053

α = (1-CL)/2 = 0.025

Standard normal deviate for α = Z_α = 1.960

Proportion of positive results = P = x/N = 0.293

Lower bound = P - (Z_α*SEM) = 0.190

Upper bound = P + (Z_α*SEM) = 0.396

2)

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

3)We need to construct the 95% confidence interval for the difference between population proportions p₁ -p_2.

We have been provided with the following information about the sample proportions:

Favorable Cases 1 (X1​) =	35
Sample Size 1 (N1​) =	75
Favorable Cases 2 (X2​) =	22
Sample Size 2 (N2) =	75

The sample proportion 1 is computed as follows, based on the sample size N​₁=75 and the number of favorable cases X₁=35:

=35/75=0.467

The sample proportion 2 is computed as follows, based on the sample size N2​=75 and the number of favorable cases X₂​=22:

=22/75=0.293

The critical value for α=0.05 is = 1.96. The corresponding confidence interval is computed as shown below:

Therefore, based on the data provided, the 95% confidence interval for the difference between the population proportions p₁ -p₂ is 0.02<p<0.326, which indicates that we are 95% confident that the true difference between population proportions is contained by the interval (0.02, 0.326)

0 .020 lower bound of confidence interval

0.326 upper bound of confidence interval

4)

We are 95% 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.