Final answer:
To find the probability that the mean weight for a sample of 40 trout exceeds 405.5 grams, we can use the Central Limit Theorem. According to the Central Limit Theorem, if we have a large enough sample size, the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution. The probability that the mean weight for a sample of 40 trout exceeds 405.5 grams is approximately 0.0257.
Step-by-step explanation:
To find the probability that the mean weight for a sample of 40 trout exceeds 405.5 grams, we can use the Central Limit Theorem. According to the Central Limit Theorem, if we have a large enough sample size, the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution.
In this case, the sample size is 40, which is reasonably large. So we can assume that the sampling distribution of the sample mean will be approximately normal. The mean weight for the sample of 40 trout can be calculated as follows:
mean = mean of the population = 402.7 grams
The standard deviation of the sample mean can be calculated using the formula:
standard deviation = standard deviation of the population / sqrt(sample size)
Substituting the values, we get:
standard deviation = 8.8 grams / sqrt(40)
Now, we can convert the problem into a standard normal distribution problem by using the formula:
Z = (x - mean) / standard deviation
Substituting the values, we get:
Z = (405.5 - 402.7) / (8.8 / sqrt(40))
Calculating the value of Z, we find:
Z = 1.97
The area to the right of Z = 1.97 in the standard normal distribution is approximately 0.0257. Therefore, the probability that the mean weight for a sample of 40 trout exceeds 405.5 grams is approximately 0.0257.