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
b) (i) Outline 4 characteristics of the Normal distribution. • The normal distribution is bell-shaped and symmetrical about its mean. • The mean, median, and mode are all equal and located at the center of the distribution. • The total area under the curve is equal to 1 (or 100%). • The tails of the curve are asymptotic, meaning they extend indefinitely in both directions, approaching the horizontal axis but never touching it. b) (ii) Calculate the number of recruits with an IQ between 100 and 115. Step 1: Identify the given parameters. Total number of recruits (N) = 20,000 Mean () = 108 Variance (^2) = 49 Standard deviation () = sqrt(49) = 7 Step 2: Standardize the IQ values using the z-score formula. The z-score formula is Z = (X - )/(). For X_1 = 100: Z_1 = (100 - 108)/(7) = (-8)/(7) ≈ -1.14 For X_2 = 115: Z_2 = (115 - 108)/(7) = (7)/(7) = 1.00 Step 3: Find the probability P(100 < X < 115) using the z-scores. This is equivalent to finding P(-1.14 < Z < 1.00). Using a standard normal distribution (Z-table): P(Z < 1.00) = 0.8413 P(Z < -1.14) = 0.1271 The probability is: P(-1.14 < Z < 1.00) = P(Z < 1.00) - P(Z < -1.14) P(-1.14 < Z < 1.00) = 0.8413 - 0.1271 = 0.7142 Step 4: Calculate the number of recruits. Number of recruits = Total recruits × Probability Number of recruits = 20,000 × 0.7142 Number of recruits = 14284 The number of recruits with an IQ between 100 and 115 is 14284.