Math Problem Statement
The wildlife department has been feeding a special food to rainbow trout fingerlings in a pond. Based on a large number of observations, the distribution of trout weights is normally distributed with a mean of 402.7 grams and a standard deviation of 8.8 grams. What is the probability that the mean weight for a sample of 40 trout exceeds 405.5 grams?
Solution
We are asked to find the probability that the mean weight of a sample of 40 trout exceeds 405.5 grams. Given the population parameters, we can apply the Central Limit Theorem (CLT) to address this problem.
Given:
- Population mean () = 402.7 grams
- Population standard deviation () = 8.8 grams
- Sample size () = 40
- Sample mean threshold = 405.5 grams
Step 1: Calculate the standard error of the mean (SE)
The standard error (SE) of the sample mean is given by the formula:
Substituting the values:
Step 2: Convert the problem into a Z-score
The Z-score is calculated using the formula:
Where:
- is the sample mean (405.5 grams)
- is the population mean (402.7 grams)
- is the standard error calculated above.
Substitute the values:
Step 3: Find the probability corresponding to the Z-score
Using a Z-table or a standard normal distribution calculator, the probability of getting a Z-score greater than 2.01 is approximately 0.0222. This represents the area under the standard normal curve to the right of Z = 2.01.
Conclusion:
The probability that the mean weight of a sample of 40 trout exceeds 405.5 grams is approximately 0.0222 or 2.22%.
Do you need more details or further clarification?
Here are 5 related questions to expand on this topic:
- How would the probability change if the sample size were increased to 100?
- What is the probability that the mean weight is less than 400 grams?
- Can we find the probability that a single trout weighs more than 405.5 grams?
- What role does the Central Limit Theorem play in this type of problem?
- How would the answer change if the standard deviation of the population were different?
Tip: Remember, as the sample size increases, the standard error decreases, leading to a more accurate estimate of the population mean.
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Math Problem Analysis
Mathematical Concepts
Statistics
Normal Distribution
Central Limit Theorem
Probability
Formulas
Standard Error of the Mean (SE) = σ / √n
Z-score = (x̄ - μ) / SE
Theorems
Central Limit Theorem
Suitable Grade Level
Grades 10-12
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