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Question: A wildlife researcher wants to know how many fish in a lake have been tagged by previous researchers. Use a 90% confidence interval to estimate the minimum number of fish that need to be caught and examined in order to yield results that are normally distributed with a standard deviation of 3 and a margin of error of ±0.6 fish with tags.
To calculate the minimum number of fish that need to be caught and examined, we will be using the formula for the sample size in a confidence interval estimation as follows: n = (Z^2 * σ^2) / E^2 where n is the sample size, Z is the Z-score corresponding to the desired confidence level, σ is the known standard deviation, and E is the margin of error. Since we want a 90% confidence interval, we need to find the Z-score corresponding to it. For a 90% confidence interval, the Z-score is 1.645 (you can find this from a standard normal distribution table, or by using a calculator that takes into account the confidence level). Given the standard deviation (σ) of 3 and a margin of error (E) of ±0.6 fish with tags, we can plug these values into the formula: n = (1.645^2 * 3^2) / (0.6)^2 n = (2.706 * 9) / 0.36 n = 24.354 / 0.36 n ≈ 67.65 Since we cannot catch a fraction of a fish, we'll round up to the nearest whole number, so the minimum number of fish that need to be caught and examined is approximately 68.
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