Questions: 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 a TI-83 Plus/TI-84 Plus calculator and round the final answers to at least four decimal places. If a proofreader from the company is randomly selected, find the probability that his or her age will be between 35.4 and 36.9 years. [ P(35.4<X<36.9)= ]

Solution Steps

We are given that the mean age of proofreaders is μ = 36.2 years and the standard deviation is σ = 3.7 years. We want to find the probability that a randomly selected proofreader's age is between 35.4 and 36.9 years. We can use the z-score formula to standardize these values:

z = (x - μ) / σ

For x = 35.4: z1 = (35.4 - 36.2) / 3.7 = -0.8 / 3.7 ≈ -0.2162

For x = 36.9: z2 = (36.9 - 36.2) / 3.7 = 0.7 / 3.7 ≈ 0.1892

Using a z-table or calculator, we can find the probabilities corresponding to these z-scores:

P(Z ≤ -0.2162) ≈ 0.4142 P(Z ≤ 0.1892) ≈ 0.5750

The probability that a randomly selected proofreader's age is between 35.4 and 36.9 is the difference between the probabilities we found in Step 2:

P(35.4 < X < 36.9) = P(-0.2162 < Z < 0.1892) = P(Z ≤ 0.1892) - P(Z ≤ -0.2162) ≈ 0.5750 - 0.4142 ≈ 0.1608

Final Answer: The probability is approximately 0.1608.