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
Importance of the Central Limit Theorem The Central Limit Theorem (CLT) states that the distribution of the sample mean X from any population with mean and standard deviation is approximately normal N(, (^2)/(n)) for large sample size n (typically n ≥ 30), regardless of the population's distribution shape. This is crucial because it enables the use of normal probabilities and z-scores for inference on from sample data, even for skewed or unknown population distributions. Step 1: Identify parameters. Population mean: = 75 minutes (midpoint of 60 and 90 minutes). Population standard deviation: = 2 minutes. Sample size: n=40. Step 2: Compute standard error of the mean (by CLT). Formula: _X = ()/(sqrt(n)). Substitute: _X = (2)/(sqrt(40)). Simplify: sqrt(40) = 2sqrt(10) ≈ 6.3246, so _X ≈ (2)/(6.3246) ≈ 0.3162 minutes. Exact: _X = (1)/(sqrt(10)) minutes. Step 3: Standardize the bounds to z-scores (by CLT, X ≈ N(75, 0.3162^2)). Lower bound formula: z_1 = (60 - 75)/(_X). Substitute: z_1 = (-15)/(0.3162) ≈ -47.43. Upper bound formula: z_2 = (90 - 75)/(_X). Substitute: z_2 = (15)/(0.3162) ≈ 47.43. Step 4: Compute probability. P(60 < X < 90) = P(-47.43 < Z < 47.43) where Z N(0,1). For |z| > 5, (z) ≈ 1, so P ≈ 1 (or 100%). 1