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.
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This problem involves comparing probabilities related to a normal distribution, specifically for an individual observation versus a sample mean. Given: • Population mean time, = 46.2 minutes. • Population standard deviation, = 8 minutes. We need to determine which of two events is more likely: 1. A randomly selected adult completes the test in less than 43 minutes. 2. The mean time for 50 randomly selected adults to complete the test is less than 43 minutes. Part 1: Probability for a single randomly selected adult Step 1: Define the random variable and parameters. Let X be the time it takes for a single randomly selected adult to complete the test. X follows a normal distribution with = 46.2 and = 8. We want to find P(X < 43). Step 2: Standardize the value X=43 to a Z-score. The formula for a Z-score is Z = (X - )/(). Z_1 = (43 - 46.2)/(8) Z_1 = (-3.2)/(8) Z_1 = -0.4 Step 3: Find the probability using the standard normal distribution table. P(X < 43) = P(Z < -0.4) From the Z-table, P(Z < -0.4) ≈ 0.3446. Part 2: Probability for the mean of 50 randomly selected adults Step 1: Define the random variable for the sample mean and its parameters. Let X be the mean time for a sample of n=50 randomly selected adults. According to the Central Limit Theorem, X follows a normal distribution with: • Mean of the sample means, _X = = 46.2 minutes. • Standard deviation of the sample means (standard error), _X = ()/(sqrt(n)). _X = (8)/(sqrt(50)) _X = (8)/(7.0710678) _X ≈ 1.1314 We want to find P(X < 43). Step 2: Standardize the sample mean value X=43 to a Z-score. The formula for the Z-score of a sample mean is Z = X - _X_X. Z_2 = (43 - 46.2)/(1.1314) Z_2 = (-3.2)/(1.1314) Z_2 ≈ -2.828 Step 3: Find the probability using the standard normal distribution table. P(X < 43) = P(Z < -2.83) (rounding Z-score to two decimal places for table lookup) From the Z-table, P(Z < -2.83) ≈ 0.0023. Part 3: Compare the probabilities Comparing the two probabilities: • P(X < 43) ≈ 0.3446 • P(X < 43) ≈ 0.0023 Since 0.3446 > 0.0023, it is more likely that a randomly selected adult will complete the test in less than 43 minutes. This is because the distribution of sample means is much narrower (smaller standard error) than the distribution of individual observations, making extreme values for the mean less probable. The final answer is It is more likely that a randomly selected adult will complete the test in less than 43 minutes. 3 done, 2 left today. You're making progress.