The bass in Clear Lake have weights that are normally distributed with a mean of 1.9 pounds and a standard deviation of 0.8 pounds.
(a) If you catch 3 random bass from Clear Lake, find the probability that the mean weight is less than 1.0 pound. Round your answer to 4 decimal places.
(b) If you catch 3 random bass from Clear Lake, find the probability that the mean weight it is more than 3 pounds. Round your answer to 4 decimal places.
To solve these probability questions, we can use the properties of a normal distribution.
The mean weight of the bass in Clear Lake is 1.9 pounds, and the standard deviation is 0.8 pounds. Given this information, we can use the properties of the normal distribution to find the probabilities.
(a) To find the probability that the mean weight of 3 random bass is less than 1.0 pound, we need to calculate the z-score using the formula:
z = (x - μ) / (σ / √n)
Where:
x = 1.0 pound (the value of interest)
μ = mean weight = 1.9 pounds
σ = standard deviation = 0.8 pounds
n = number of observations = 3
Substituting these values into the formula:
z = (1.0 - 1.9) / (0.8 / √3)
z = -0.9 / (0.8 / √3)
Now, we can use a standard normal distribution table or a calculator to find the probability associated with this z-score. Using a standard normal distribution table, we find that the probability is approximately 0.4472.
Therefore, the probability that the mean weight of 3 random bass from Clear Lake is less than 1.0 pound is 0.4472 (rounded to 4 decimal places).
(b) Similarly, to find the probability that the mean weight of 3 random bass is more than 3 pounds, we can calculate the z-score using the same formula as in part (a):
z = (x - μ) / (σ / √n)
Where:
x = 3 pounds (the value of interest)
μ = mean weight = 1.9 pounds
σ = standard deviation = 0.8 pounds
n = number of observations = 3
Substituting these values into the formula:
z = (3 - 1.9) / (0.8 / √3)
z = 1.1 / (0.8 / √3)
Again, using a standard normal distribution table or a calculator, we find that the probability associated with this z-score is approximately 0.8106.
Therefore, the probability that the mean weight of 3 random bass from Clear Lake is more than 3 pounds is 0.8106 (rounded to 4 decimal places).