In a survey of a group of men, the heights in the 20-29 age group were normally distributed, with a mean of 67.1 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 (Round to four decimal places as needed) (b) Find the probability that a study participant has a height that is between 67 and 72 inches. The probability that the study participant selected at random is betwe (Round to four decimal places as needed) (c) Find the probability that a study participant has a height that is more than 72 inches. The probability that the study participant selected at random is more than 72 inches tall is (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) and (c) are unusual because its probabilities are less than 0.05 . B. There are no unusual events because all the probabilities are greater than 0.05 . C. The event in part (a) is unusual because its probability is less than 0.05 .
Solution
Step 1 ：The problem is asking for the probability of a study participant having a height less than 67 inches. This is a problem of normal distribution. We know that the mean height is 67.1 inches and the standard deviation is 3.0 inches.
Step 2 ：We can use the z-score formula to calculate the z-score for a height of 67 inches. The z-score is a measure of how many standard deviations an element is from the mean.
Step 3 ：Using the formula \(Z = \frac{X - \mu}{\sigma}\), where \(X\) is the height, \(\mu\) is the mean, and \(\sigma\) is the standard deviation, we find that the z-score is approximately -0.033.
Step 4 ：After finding the z-score, we can look up the corresponding probability in the z-table. The probability associated with a z-score of -0.033 is approximately 0.487.
Step 5 ：Final Answer: The probability that a study participant selected at random is less than 67 inches tall is approximately \(\boxed{0.487}\).
