In a survey of a group of men, the heights in the \( 20-29 \) age group were normally distributed, with a mean of 67.2 inches and a standard deviation of 2.0 inches. A study paticipant 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 0.4602 (Round ta fouf decimal places as needed) (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 0.4590 . (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.)

To find the probability that a study participant has a height that is more than 70 inches, we can use the properties of the normal distribution. Since we have the mean (67.2) and standard deviation (2.0), we can calculate the z-score for 70 inches using the formula: \[ z = \frac{(X - \mu)}{\sigma} = \frac{(70 - 67.2)}{2.0} = 0.9 \] Next, we look up this z-score in the standard normal distribution table or use a calculator. The probability corresponding to a z-score of 0.9 is about 0.8176. Since we want the probability of being more than 70 inches tall, we subtract this value from 1: \[ P(X > 70) = 1 - P(Z < 0.9) = 1 - 0.8176 = 0.1824 \] Therefore, the probability that the study participant selected at random is more than 70 inches tall is approximately **0.1824** (rounded to four decimal places). --- The heights of individuals can vary widely, and understanding the normal distribution helps showcase how probabilities can fluctuate around the mean. Fun fact: the classic "68-95-99.7" rule tells us that about 68% of people fall within one standard deviation of the mean, 95% within two, and a whopping 99.7% within three! If you’re fascinated by normal distributions and want to dive deeper, consider exploring additional topics such as the Central Limit Theorem or the applications of normal distributions in real life, like quality control, finance, or even testing! They are everywhere, influencing decisions big and small!