Question

A consumer advocacy group published a study of labeling of seafood sold in three U.S. states. The group purchased 207 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. The study found that 14 of the 26 ​"red snapper" packages tested were a different kind of fish. Assume that the study used a simple random sample. Complete parts a through c below.

​a) Are the conditions for creating a confidence interval​ satisfied? Explain.

B) Construct a 95% confidence level interval for the proportion of "red snapper" packages that were a different kind of fish.

C) ​Explain what the confidence interval from part​ (b) says about​ "red snapper" sold in these three states. Select the correct choice below and fill in the answer boxes within your choice.

Answer 1

Ans:

a)Yes conditions are met,as number of success or failures are greater than 10.

b)

sample proportion=14/26=0.538

95% cnfidence interval for p

=0.538+/-1.96*sqrt(0.538*(1-0.538)/26)

=0.538+/-0.192

=(0.347, 0.730)

c)We are 95% confident that true proportion of "red snapper" packages that were a different kind of fish is between 0.347 and 0.730.

Answer 2

a) To create a confidence interval for the proportion of "red snapper" packages that were a different kind of fish, certain conditions need to be satisfied. These conditions are:

1. Random Sampling: The study mentioned that a simple random sample was used, which ensures that each "red snapper" package has an equal chance of being selected for the study.

2. Independent Observations: The samples must be independent of each other. Since the study collected seafood from various kinds of food stores and restaurants in three different states, it is reasonable to assume that the samples are independent.

3. Sample Size: The sample size should be large enough for the Central Limit Theorem to apply. The study used 207 samples, which is reasonably large, so this condition is satisfied.

4. Success-Failure Condition: The number of successes (packages of a different kind of fish) and failures (packages of actual red snapper) in the sample should be both greater than or equal to 10. In this case, 14 packages were different kinds of fish (success) and 12 packages were actual red snapper (failure), both of which are greater than 10.

Since all the conditions are satisfied, we can proceed to construct a confidence interval.

b) To construct a 95% confidence interval for the proportion of "red snapper" packages that were a different kind of fish, we can use the formula for the confidence interval for a proportion:

$\text{Confidence Interval}=\widehat{�}\pm �\times \sqrt{\frac{\widehat{�}(1-\widehat{�})}{�}}$Confidence Interval=p^±Z×np^(1−p^)

where:

• $\widehat{�}$p^ is the sample proportion of packages that were a different kind of fish (14/26 in this case).

• $�$n is the sample size (26 in this case).

• $�$Z is the critical value corresponding to the desired confidence level (95% in this case).

To find the critical value$�$Z, we can look it up in the standard normal distribution table or use a calculator. For a 95% confidence level, the critical value is approximately 1.96.

Now, we can plug the values into the formula:

$\text{Confidence Interval}=\frac{14}{26}\pm 1.96\times \sqrt{\frac{\frac{14}{26}\times \frac{12}{26}}{26}}$Confidence Interval=2614±1.96×262614×2612

$\text{Confidence Interval}=0.5385\pm 1.96\times 0.1114$Confidence Interval=0.5385±1.96×0.1114

$\text{Confidence Interval}=(0.3198,0.7572)$Confidence Interval=(0.3198,0.7572)

c) The 95% confidence interval (0.3198, 0.7572) tells us that we are 95% confident that the true proportion of "red snapper" packages that were a different kind of fish lies within this range. In other words, we estimate that the proportion of mislabeled "red snapper" packages in these three states is between 31.98% and 75.72%. Since the confidence interval does not include 100%, it suggests that there is a significant chance that "red snapper" packages may be mislabeled, raising concerns about the accuracy of the labeling of "red snapper" sold in these three states.