In December 2011, Consumer Reports published their study of...

In December 2011, Consumer Reports published their study of labeling of seafood sold in New York, New Jersey, and Connecticut. They purchased 190 pieces of seafood from various kinds of food stores and restaurants in the three states and genetically compared the pieces to standard gene fragments that can identify the species. Laboratory results indicated that $22 \%$ of these packages of seafood were mislabeled, incompletely labeled, or misidentified by store or restaurant employees. a) Construct a $95 \%$ confidence interval for the proportion of all seafood packages in those three states that are mislabeled or misidentified. b) Explain what your confidence interval says about seafood sold in these three states. c) A 2009 report by the Government Accountability Board says that the Food and Drug Administration has spent very little time recently looking for seafood fraud. Suppose an official said, "That's only 190 packages out of the billions of pieces of seafood sold in a year. With the small number tested, I don't know that one would want to change one's buying habits." (An official was quoted similarly in a different but similar context). Is this argument valid? Explain.

In December 2011, Consumer Reports published their study of labeling of seafood sold in New York, New Jersey, and Connecticut. They purchased 190 pieces of seafood from various kinds of food stores and restaurants in the three states and genetically compared the pieces to standard gene fragments that can identify the species. Laboratory results indicated that $22 \%$ of these packages of seafood were mislabeled, incompletely labeled, or misidentified by store or restaurant employees.
a) Construct a $95 \%$ confidence interval for the proportion of all seafood packages in those three states that are mislabeled or misidentified.
b) Explain what your confidence interval says about seafood sold in these three states.
c) A 2009 report by the Government Accountability Board says that the Food and Drug Administration has spent very little time recently looking for seafood fraud. Suppose an official said, "That's only 190 packages out of the billions of pieces of seafood sold in a year. With the small number tested, I don't know that one would want to change one's buying habits." (An official was quoted similarly in a different but similar context). Is this argument valid? Explain.

Key Concepts

Extrapolation from Small Samples

This involves the caution needed when generalizing findings from a limited number of observations to a much larger population. While statistical methods account for sampling variability, conclusions drawn from small samples should be carefully interpreted, as they may not capture all the variability present in the entire population. This concept is central when addressing claims that a small sample is insufficient to alter broader consumer behavior despite its statistical significance.

Confidence Interval for a Proportion

This concept involves calculating a range using the sample proportion, which is extended by adding and subtracting a margin of error computed typically as the product of a critical value (such as the z-score for a desired level of confidence) and the standard error of the proportion. This interval estimates the range within which the true population proportion is likely to fall with a specified degree of confidence.

Interpretation of a Confidence Interval

A confidence interval provides a probabilistic statement about where the true population parameter lies. For example, a 95% confidence interval implies that if we were to take many samples and build an interval from each, approximately 95% of those intervals would contain the true population proportion. It is a tool for understanding the precision and reliability of an estimate made from sample data.

Sampling and Representativeness

This concept addresses the importance of having a sample that accurately reflects the population from which it is drawn. The validity of inferences about a population largely depends on how representative the sample is. Even if the sample size is small, if it is randomly selected and representative, the results can provide useful insights into the population.

Transcript

00:01 In december 2011, consumer reports published their study of labeling of seafood sold in new york, new jersey, and connecticut.

00:06 They purchased 190 pieces of seafood from various kinds of food stores and restaurants in the three states and genetically compared the pieces to standard gene fragments that can identify the species.

00:17 Lab results indicated that 22 % of these packages of seafood were mislabeled, incompletely labeled, or misidentified by store restaurant employees.

00:25 We want to construct a 95 % confidence interval for the proportion of all seafood packages in the local.

00:30 Those three states that are mislabeled or misidentified.

00:37 Here's what we know.

00:38 We know that n is 190, that p hat is 0 .22, and that we want a c of 0 .95.

00:50 And so for our confidence of 0 .95, we're going to use a z score of 0 .025, half of it.

01:03 So using the normal probability table in the appendix back of the book, you'll see that that core corresponds to a z score of 1 .96.

01:11 And so we can then find the margin of error.

01:14 It's using that number times the square root of p hat, which is 0 .22 times 1 minus 0 .22 over n, which is 190.

01:33 So 1 .96 times the square root of 0 .22 times 1 minus 0 .22 over 190 gives us 0 .0589.

01:51 So the boundaries of the confidence interval are going to be p hat minus and plus our margin of error.

02:00 So p hat is 0 .22...

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