This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
This problem involves a normally distributed random variable and can be solved using the Empirical Rule. We are given: Mean () = 100 Standard deviation () = 10 We need to find the probability that an observed value of this random variable is less than 90 or greater than 110. This can be written as P(Y < 90 or Y > 110). Step 1: Express the values 90 and 110 in terms of the mean and standard deviation. 90 = 100 - 10 = - 110 = 100 + 10 = + So, we are looking for the probability that Y is less than one standard deviation below the mean or greater than one standard deviation above the mean. Step 2: Apply the Empirical Rule. The Empirical Rule states that for a normal distribution, approximately 68% of the data falls within 1 standard deviation of the mean. This means P( - < Y < + ) ≈ 0.68. In our case, P(90 < Y < 110) ≈ 0.68. Step 3: Calculate the probability of being outside this interval. The probability that Y is less than 90 or greater than 110 is the complement of the probability that Y is between 90 and 110. P(Y < 90 or Y > 110) = 1 - P(90 < Y < 110) P(Y < 90 or Y > 110) ≈ 1 - 0.68 P(Y < 90 or Y > 110) ≈ 0.32 Comparing this result with the given options: a) 0.15 b) 0.32 c) 0.68 d) 0.84 e) 0.95 The calculated probability matches option (b). The final answer is b) 0.32. Last free one today — make it count tomorrow, or type /upgrade for unlimited.