In a survey of a group of men, the heights in the 20 - 29 age group were normally distributed, with a mean of 67.4 inches and a standard deviation of 3.0 inches. A study participant is randomly selected. Complete parts (a) through (d) below. (a) Find the probability that a study participant has a height that is less than 67 inches. The probability that the study participant selected at random is less than 67 inches tall is (b) Find the probability that a study participant has a height that is between 67 and 70 inches. The probability that the study participant selected at random is between 67 and 70 inches tall is \( \square \). (Round to four decimal places as needed.) (c) Find the probability that a study participant has a height that is more than 70 inches. The probability that the study participant selected at random is more than 70 inches tall is \( \square \). (Round to four decimal places as needed.) (d) Identify any unusual events. Explain your reasoning. Choose the correct answer below. A. The events in parts (a), (b), and (c) are unusual because all of their probabilities are less than 0.05 . The events in parts (a) and (c) are unusual because its probabilities are less than 0.05 . C. There are no unusual events because all the probabilities are greater than 0.05. D. The event in part (a) is unusual because its probability is less than 0.05 .

To solve the problems, we will use the properties of the normal distribution. Given that the heights are normally distributed with a mean (\(\mu\)) of 67.4 inches and a standard deviation (\(\sigma\)) of 3.0 inches, we can standardize the heights using the z-score formula: \[ z = \frac{X - \mu}{\sigma} \] where \(X\) is the height we are interested in. ### (a) Find the probability that a study participant has a height that is less than 67 inches. 1. Calculate the z-score for 67 inches: \[ z = \frac{67 - 67.4}{3.0} = \frac{-0.4}{3.0} \approx -0.1333 \] 2. Use the z-table or a calculator to find the probability corresponding to \(z = -0.1333\): \[ P(Z < -0.1333) \approx 0.4477 \] Thus, the probability that a study participant selected at random is less than 67 inches tall is approximately **0.4477**. ### (b) Find the probability that a study participant has a height that is between 67 and 70 inches. 1. Calculate the z-scores for 67 inches and 70 inches: - For 67 inches: \[ z_{67} = \frac{67 - 67.4}{3.0} \approx -0.1333 \] - For 70 inches: \[ z_{70} = \frac{70 - 67.4}{3.0} = \frac{2.6}{3.0} \approx 0.8667 \] 2. Find the probabilities: - For \(z = -0.1333\): \[ P(Z < -0.1333) \approx 0.4477 \] - For \(z = 0.8667\): \[ P(Z < 0.8667) \approx 0.8078 \] 3. Calculate the probability between 67 and 70 inches: \[ P(67 < X < 70) = P(Z < 0.8667) - P(Z < -0.1333) \approx 0.8078 - 0.4477 = 0.3601 \] Thus, the probability that a study participant selected at random is between 67 and 70 inches tall is approximately **0.3601**. ### (c) Find the probability that a study participant has a height that is more than 70 inches. 1. We already calculated the z-score for 70 inches: \[ z_{70} \approx 0.8667 \] 2. Find the probability: \[ P(Z > 0.8667) = 1 - P(Z < 0.8667) \approx 1 - 0.8078 = 0.1922 \] Thus, the probability that a study participant selected at random is more than 70 inches tall is approximately **0.1922**. ### (d) Identify any unusual events. Explain your reasoning. To determine if any of the events are unusual, we typically consider an event unusual if its probability is less than 0.05. - Probability from part (a): \(0.4477\) (not unusual) - Probability from part (b): \(0.3601\) (not unusual) - Probability from part (c): \(0.1922\) (not unusual) Since none of the probabilities are less than 0.05, the correct answer is: **C. There are no unusual events because all the probabilities are greater than 0.05.**