Question

assume that females have pulse rates that are normally distributed with a mean of $mu = 73.0$ beats per minute and 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 less than 76 beats per minute
the probability is 0.5948
(round to four decimal places as needed.)

b. if 16 adult females are randomly selected, find the probability that they have pulse rates with a mean less than 76 beats per minute.
the probability is
(round to four decimal places as needed.)

Explanation:

Step1: Calculate the standard error

The standard error formula is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$. Given $\sigma = 12.5$ and $n = 16$, we have $\sigma_{\bar{x}}=\frac{12.5}{\sqrt{16}}=\frac{12.5}{4}=3.125$.

Step2: Calculate the z - score

The z - score formula is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$. Here, $\bar{x} = 76$, $\mu=73$, so $z=\frac{76 - 73}{3.125}=\frac{3}{3.125}=0.96$.

Step3: Find the probability

Using the standard normal distribution table (or a calculator with a normal - distribution function), we find $P(Z < 0.96)$. Looking up the value in the standard normal table, $P(Z < 0.96)=0.8315$.

Answer:

$0.8315$