Final answer:
To calculate the probabilities, we need to find the area under the probability density function curve between the given values. The mean of a continuous uniform distribution is (lower bound + upper bound) / 2. The standard deviation is (upper bound - lower bound) / √12.
Step-by-step explanation:
To calculate the probabilities, we need to find the area under the probability density function curve between the given values.
1. P(x ≤ 30): This is the area under the curve from 15 to 30. Since it is a uniform distribution, the area under the curve is the same as the width of the interval divided by the total width. So, P(x ≤ 30) = (30-15) / (35-15) = 15 / 20 = 0.75.
2. P(x ≤ 25): This is the area under the curve from 15 to 25. So, P(x ≤ 25) = (25-15) / (35-15) = 10 / 20 = 0.5.
3. P(x ≤ 20): This is the area under the curve from 15 to 20. So, P(x ≤ 20) = (20-15) / (35-15) = 5 / 20 = 0.25.
4. P(x = 23): Since it is a continuous distribution, the probability of a single value is 0. Therefore, P(x = 23) = 0.
B. The mean of a continuous uniform distribution is given by (lower bound + upper bound) / 2. So, the mean of this distribution is (15 + 35) / 2 = 25. The standard deviation of a continuous uniform distribution is given by (upper bound - lower bound) / √12. So, the standard deviation of this distribution is (35-15) / √12 ≈ 5.77.