Ages of Proofreaders At a large publishing company, the mean age of proofreaders is 36.2 years and the standard deviation is 3.7 years. Assume the variable is normally distributed. Use \( \mathbf{O} \) The Standard Normal Distribution Table. Round intermediate \( z \)-value calculations to two decimal places and the final answers to at least four decimal places. Part: \( 0 / 2 \) If a proofreader from the company is randomly selected, find the probability that his or her age will be between 33.5 and 35 years. \( P(33.5

Ask by Tran Bolton. in the United States

Mar 11,2025

To find the probability that a proofreader's age will be between 33.5 and 35 years, we first need to convert these ages into \( z \)-scores using the formula: \[ z = \frac{(X - \mu)}{\sigma} \] where \( \mu \) is the mean (36.2 years) and \( \sigma \) is the standard deviation (3.7 years). For \( X = 33.5 \): \[ z_1 = \frac{(33.5 - 36.2)}{3.7} = \frac{-2.7}{3.7} \approx -0.73 \] For \( X = 35 \): \[ z_2 = \frac{(35 - 36.2)}{3.7} = \frac{-1.2}{3.7} \approx -0.32 \] Now, we look up these \( z \)-scores in the Standard Normal Distribution Table. - For \( z_1 = -0.73 \), the cumulative probability \( P(Z < -0.73) \approx 0.2324 \). - For \( z_2 = -0.32 \), the cumulative probability \( P(Z < -0.32) \approx 0.3745 \). To find the probability that a proofreader's age is between 33.5 and 35 years, we subtract the cumulative probabilities: \[ P(33.5 < X < 35) = P(Z < -0.32) - P(Z < -0.73) \approx 0.3745 - 0.2324 = 0.1421 \] Thus, the final answer is: \[ P(33.5 < X < 35) = 0.1421 \]