Question

assume that females have pulse rates that are normally distributed with a mean of μ = 75.0 beats per minute and a standard deviation of σ = 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 less than 79 beats per minute. the probability is 0.6255 (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 less than 79 beats per minute. the probability is (round to four decimal places as needed.)

Explanation:

Step1: Identify the formula for the z - score of the sample mean

For the sampling distribution of the sample mean $\bar{x}$, the z - score is given by $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$, where $\mu$ is the population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size. Here, $\mu = 75$, $\sigma=12.5$, $\bar{x} = 79$, and $n = 4$.

Step2: Calculate the z - score

Substitute the values into the formula: $z=\frac{79 - 75}{\frac{12.5}{\sqrt{4}}}=\frac{4}{\frac{12.5}{2}}=\frac{4}{6.25}=0.64$.

Step3: Find the probability

We want to find $P(\bar{X}<79)$, which is equivalent to $P(Z < 0.64)$ using the standard normal distribution table. Looking up the value of $P(Z < 0.64)$ in the standard - normal table, we get $P(Z < 0.64)=0.7389$.

Answer:

$0.7389$