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
Here's how to solve the probability problem: Step 1: Identify the given information. Mean weight, = 1.25 grams Standard deviation, = 0.25 grams The weights are normally distributed. i) Probability of randomly selecting a bat that weighs between 1 gram and 2 grams Step 2: Convert the weights to Z-scores. For X_1 = 1 gram: Z_1 = (X_1 - )/() = (1 - 1.25)/(0.25) = (-0.25)/(0.25) = -1.00 For X_2 = 2 grams: Z_2 = (X_2 - )/() = (2 - 1.25)/(0.25) = (0.75)/(0.25) = 3.00 Step 3: Find the probability using the Z-scores. We need to find P(1 < X < 2), which is equivalent to P(-1.00 < Z < 3.00). Using a standard normal distribution table (Z-table): P(Z < 3.00) = 0.9987 P(Z < -1.00) = 0.1587 P(-1.00 < Z < 3.00) = P(Z < 3.00) - P(Z < -1.00) P(-1.00 < Z < 3.00) = 0.9987 - 0.1587 P(-1.00 < Z < 3.00) = 0.8400 The probability is: 0.8400 ii) Probability of randomly selecting a bat that weighs more than 2.2 grams Step 2: Convert the weight to a Z-score. For X = 2.2 grams: Z = (X - )/() = (2.2 - 1.25)/(0.25) = (0.95)/(0.25) = 3.80 Step 3: Find the probability using the Z-score. We need to find P(X > 2.2), which is equivalent to P(Z > 3.80). Using a standard normal distribution table (Z-table): P(Z > 3.80) = 1 - P(Z < 3.80) P(Z < 3.80) ≈ 0.9999 (since 3.80 is a very high Z-score) P(Z > 3.80) = 1 - 0.9999 P(Z > 3.80) = 0.0001 The probability is: 0.0001 3 done, 2 left today. You're making progress.