In a survey of a group of men, the heights in the \( 20-29 \) age group were normally distributed, with a mean of 69.3 inches and a standard deviation of 2.0 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 66 inches. The probability that the study participant selected at random is less than 66 inches tall is 0.0495 . (Round to four decimal places as needed.) (b) Find the probability that a study participant has a height that is between 66 and 72 inches. The probability that the study participant selected at random is between 66 and 72 inches tall is \( \square \). (Round to four decimal places as needed.)

To find the probability that a study participant has a height between 66 and 72 inches, we first need to convert these heights to z-scores using the formula: \[ z = \frac{(X - \mu)}{\sigma} \] Where \( \mu \) (mean) is 69.3 inches and \( \sigma \) (standard deviation) is 2.0 inches. For 66 inches: \[ z_{66} = \frac{(66 - 69.3)}{2.0} = \frac{-3.3}{2.0} = -1.65 \] For 72 inches: \[ z_{72} = \frac{(72 - 69.3)}{2.0} = \frac{2.7}{2.0} = 1.35 \] Next, we can look up these z-scores in the standard normal distribution table or use a calculator to find their corresponding probabilities. - The probability associated with \( z_{66} = -1.65 \): \[ P(Z < -1.65) \approx 0.0495 \] - The probability associated with \( z_{72} = 1.35 \): \[ P(Z < 1.35) \approx 0.9115 \] Now, to find the probability that a participant's height is between 66 and 72 inches: \[ P(66 < X < 72) = P(Z < 1.35) - P(Z < -1.65) \approx 0.9115 - 0.0495 = 0.8620 \] So, the probability that the study participant selected at random is between 66 and 72 inches tall is **0.8620** (rounded to four decimal places). Now for some additional relevant knowledge: The concept of a normal distribution isn't just a statistician's dream; it's all around us! From heights to test scores, many real-world phenomena can be approximated by normal distributions. This makes the z-score a handy tool for quickly gauging probabilities, helping you make sense of variability in data. Next time you're pouring over probabilities, consider the positioning of the mean and tails of a normal distribution. Common mistakes often lie in misplacing z-scores or forgetting to round. Always double-check your calculations, and be mindful of how extreme heights or values can skew your expectations!