Recreational fishing is permitted at a particular lake. However, there are restrictions on the size of any fish caught from the lake. Suppose that the weight of any given fish caught from the lake is uniformly distributed between 7 and 33 kilograms. Only fish that are among the heaviest 21% are permitted to be kept and all other fish must be returned to the lake.What should the cut-off weight be for fish caught from the lake (below which they must be returned to the lake)? Do not include any units in your answer.
Question
Recreational fishing is permitted at a particular lake. However, there are restrictions on the size of any fish caught from the lake. Suppose that the weight of any given fish caught from the lake is uniformly distributed between 7 and 33 kilograms. Only fish that are among the heaviest 21% are permitted to be kept and all other fish must be returned to the lake.What should the cut-off weight be for fish caught from the lake (below which they must be returned to the lake)? Do not include any units in your answer.
Solution
Step 1: Find the range of weights that correspond to the heaviest 21% of fish caught from the lake.
Since the weight of any given fish caught from the lake is uniformly distributed between 7 and 33 kilograms, we can find the range of weights that correspond to the heaviest 21% of fish by finding the weight that corresponds to the 79th percentile of the distribution.
Step 2: Find the weight that corresponds to the 79th percentile of the distribution.
To find the weight that corresponds to the 79th percentile of the distribution, we can use a standard normal distribution table or a calculator with a normal distribution function.
Using a standard normal distribution table, we can find that the z-score corresponding to the 79th percentile is approximately 0.81.
Step 3: Convert the z-score to a weight value.
To convert the z-score to a weight value, we can use the formula:
z = (x - μ) / σ
where z is the z-score, x is the weight value, μ is the mean of the distribution (which is the midpoint of the range, or (7 + 33) / 2 = 20), and σ is the standard deviation of the distribution (which is (33 - 7) / sqrt(12) ≈ 6.055).
Solving for x, we get:
x = z * σ + μ x = 0.81 * 6.055 + 20 x ≈ 24.64
Therefore, the cut-off weight for fish caught from the lake (below which they must be returned to the lake) should be approximately 24.64 kilograms.
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