In a recent issue of Consumer Reports, Consumers Union reported on their investigation of bacterial contamination…

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.

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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. Determine 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

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.     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 inferenceone-sample inference
twice     

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

Tyson chicken is sold in less sanitary supermarkets.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.The one- and
two-sample procedures for analyzing the data are equivalent; the
results differ in this problem only because of natural sampling
variation.