Question

assume that females have pulse rates that are normally distributed with a mean of μ = 72.0 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 0.7123.
(round to four decimal places as needed.)
b. if 25 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: Calculate the z - score for part b

The formula for the z - score of the sample mean is $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$, where $\bar{x} = 79$, $\mu=72$, $\sigma = 12.5$, and $n = 25$.
$z=\frac{79 - 72}{\frac{12.5}{\sqrt{25}}}$

Step2: Simplify the z - score expression

First, $\sqrt{25}=5$, then $\frac{12.5}{\sqrt{25}}=\frac{12.5}{5}=2.5$.
So $z=\frac{79 - 72}{2.5}=\frac{7}{2.5}=2.8$

Step3: Find the probability using the z - table

We want $P(\bar{X}<79)$, which is equivalent to $P(Z < 2.8)$ from the standard normal distribution. Looking up the value in the standard - normal table, $P(Z < 2.8)=0.9974$

Answer:

$0.9974$