Final answer:
To construct a 90% confidence interval for the mislabeling of seafood, one would calculate the sample proportion, find the standard error using the sample size and proportion, and then apply the critical value for the 90% confidence level to determine the range of values that should capture the true proportion of mislabeled seafood.
Step-by-step explanation:
To construct a 90% confidence interval for the proportion of all seafood sold in the country that is mislabeled or misidentified, we will use the sample proportion and the standard error of the proportion. In the given scenario, we have 540 samples of seafood, and 34% were found to be mislabeled. Therefore, the sample proportion (π) is 0.34.
The standard error (SE) of the proportion is calculated using the formula SE = √[π(1 - π) / n], where π is the sample proportion and n is the size of the sample. Substituting the given values, SE = √[0.34(1 - 0.34) / 540].
For a 90% confidence interval, we need to find the critical value (z*) that corresponds to the middle 90% of the standard normal distribution. This value can be found using statistical tables or software and is approximately 1.645 for a 90% confidence interval.
The confidence interval (CI) is then calculated as:
CI = π ± z* · SE
Substituting the values, we have the confidence interval.
Confidence intervals are a statistical tool used to estimate the reliability of an estimate; the wider the interval, the less precise the estimate. Repeated samples would contain the true proportion approximately 90% of the time if the confidence level is set at 90%.