Assume that females have pulse rates that are normally distributed with a mean of $\mu=74.0$ beats per minute and a standard deviation of $\sigma=12.5$ beats per minute. Complete parts (a) through (c) below. a. If 1 adult female is randomly selected, find the probability that her pulse rate is between 67 beats per minute and 81 beats per minute. The probability is 0.4245 (Round to four decimal places as needed.) b. If 4 adult females are randomly selected, find the probability that they have pulse rates with a mean between 67 beats per minute and 81 beats per minute. The probability is $\square$. (Round to four decimal places as needed.) an example Get more help - Clear all Check answer

Step 1 ：Given that the pulse rates of females are normally distributed with a mean of \( \mu = 74.0 \) beats per minute and a standard deviation of \( \sigma = 12.5 \) beats per minute.

Step 2 ：For part a, we are asked to find the probability that a randomly selected adult female has a pulse rate between 67 and 81 beats per minute. We can use the z-score formula to find this probability. The z-score is calculated as \( z = \frac{X - \mu}{\sigma} \), where X is the value we're interested in, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. For X = 67, the z-score is \( z_1 = \frac{67 - 74}{12.5} = -0.56 \). For X = 81, the z-score is \( z_2 = \frac{81 - 74}{12.5} = 0.56 \). The probability that a randomly selected adult female has a pulse rate between 67 and 81 beats per minute is approximately 0.4245.

Step 3 ：For part b, we are asked to find the probability that the mean pulse rate of 4 randomly selected adult females is between 67 and 81 beats per minute. We need to adjust the standard deviation because we're dealing with a sample mean rather than an individual value. The standard deviation of a sample mean is \( \sigma_b = \frac{\sigma}{\sqrt{n}} \), where n is the sample size. In this case, \( n = 4 \), so \( \sigma_b = \frac{12.5}{\sqrt{4}} = 6.25 \). We can then use the z-score formula again to find the probability. For X = 67, the z-score is \( z_{1b} = \frac{67 - 74}{6.25} = -1.12 \). For X = 81, the z-score is \( z_{2b} = \frac{81 - 74}{6.25} = 1.12 \). The probability that the mean pulse rate of 4 randomly selected adult females is between 67 and 81 beats per minute is approximately 0.7373.

Step 4 ：Final Answer: The probability that a randomly selected adult female has a pulse rate between 67 and 81 beats per minute is approximately \( \boxed{0.4245} \). The probability that the mean pulse rate of 4 randomly selected adult females is between 67 and 81 beats per minute is approximately \( \boxed{0.7373} \).