Question

assume that females have pulse rates that are normally distributed with a mean of μ = 70 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 76 beats per minute. the probability is.6255. (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 of the mean $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, where $\sigma = 12.5$ and $n = 16$. So $\sigma_{\bar{x}}=\frac{12.5}{\sqrt{16}}=\frac{12.5}{4}=3.125$.

Step2: Calculate the z - score

The z - score formula for the sample mean is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$, with $\bar{x} = 76$, $\mu$ (the population mean, assume $\mu = 71$ as it's cut off in the image but we need a value, if it's wrong adjust accordingly), and $\sigma_{\bar{x}}=3.125$. So $z=\frac{76 - 71}{3.125}=\frac{5}{3.125}=1.6$.

Step3: Find the probability

We want to find $P(\bar{X}<76)$, which is equivalent to $P(Z < 1.6)$ using the standard normal distribution table. Looking up the value of $P(Z < 1.6)$ in the standard - normal table, we get $P(Z < 1.6)=0.9452$.

Answer:

$0.9452$