Question

assume that females have pulse rates that are normally distributed with a mean of $mu = 72.0$ 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 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.8997.
(round to four decimal places as needed.)
c. why can the normal distribution be used in part (b), even though the sample size does not exceed 30?
a. since the distribution is of sample means, not individuals, the distribution is a normal distribution for any sample size.
b. since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size.
c. since the distribution is of individuals, not sample means, the distribution is a normal distribution for any sample size.
d. since the mean pulse rate exceeds 30, the distribution of sample means is a normal distribution for any sample size.

Explanation:

Step1: Standardize for part a

We use the z - score formula $z=\frac{x-\mu}{\sigma}$. Here, $\mu = 72.0$, $\sigma=12.5$, and $x = 76$. So $z=\frac{76 - 72.0}{12.5}=\frac{4}{12.5}=0.32$. Then we find $P(X<76)=P(Z < 0.32)$ from the standard - normal table, and $P(Z < 0.32)=0.6255$.

Step2: Standardize for part b

For the sampling distribution of the sample mean $\bar{X}$, the mean of the sampling distribution of the sample mean is $\mu_{\bar{X}}=\mu = 72.0$ and the standard deviation is $\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}$, where $n = 16$ and $\sigma=12.5$. So $\sigma_{\bar{X}}=\frac{12.5}{\sqrt{16}}=\frac{12.5}{4}=3.125$. The z - score is $z=\frac{\bar{x}-\mu_{\bar{X}}}{\sigma_{\bar{X}}}=\frac{76 - 72.0}{3.125}=\frac{4}{3.125}=1.28$. Then $P(\bar{X}<76)=P(Z < 1.28)$ from the standard - normal table, and $P(Z < 1.28)=0.8997$.

Step3: Answer part c

The Central Limit Theorem states that if the original population is normally distributed, the sampling distribution of the sample mean $\bar{X}$ is normally distributed for any sample size $n$.

Answer:

a. $0.6255$
b. $0.8997$
c. B. Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size.