Determine whether the statement is true or false. If the statement is false, rewrite it as a true statement.

If $\overline{x}$ is the mean of a large $(n \gt 30)$ simple random sample from a population with mean $\mu$ and standard deviation $\sigma$, then $\overline{x}$ is approximately normal with $\sigma_{\overline{x}} = \dfrac{\sigma}{\sqrt{n}}$ .

The U.S. Fish and Wildlife Service reported that the mean length of six-year-old rainbow trout in the Arolik River in Alaska is $481$ millimeters with a standard deviation of $41$ millimeters. Assume these lengths are normally distributed.

$\textbf{a.}\hspace{10pt}$ Find the $58$th percentile of the lengths.

$\textbf{b.}\hspace{10pt}$ Find the $76$th percentile of the lengths.

$\textbf{c.}\hspace{10pt}$ Find the first quartile of the lengths.

$\textbf{d.}\hspace{10pt}$ A size limit is to be put on trout that are caught. What should the size limit be so that $15\%$ of six-year-old trout have lengths shorter than the limit?

According to a report by the U.S. Fish and Wildlife Service, the mean length of six-year-old rainbow trout in the Arolik River in Alaska is $481$ millimeters with a standard deviation of $41$ millimeters. Assume these lengths are normally distributed.

$\textbf{a.}\hspace{10pt}$ What proportion of six-year-old rainbow trout are less than $450$ millimeters long?

$\textbf{b.}\hspace{10pt}$ What proportion of six-year-old rainbow trout are between $400$ and $500$ millimeters long?

$\textbf{c.}\hspace{10pt}$ Is it unusual for a six-year-old rainbow trout to be less than $400$ millimeters long?

The Centers for Disease Control and Prevention reported that diastolic blood pressures of adult women in the United States are approximately normally distributed with mean $80.5$ and standard deviation $9.9$.

$\textbf{a.}\hspace{10pt}$ Find the $30$th percentile of the blood pressures.

$\textbf{b.}\hspace{10pt}$ Find the $67$th percentile of the blood pressures.

$\textbf{c.}\hspace{10pt}$ Find the third quartile of the blood pressures.

In a recent study, the Centers for Disease Control and Prevention reported that diastolic blood pressures of adult women in the United States are approximately normally distributed with mean $80.5$ and standard deviation $9.9$.

$\textbf{a.}\hspace{10pt}$ What proportion of women have blood pressures lower than $70$?

$\textbf{b.}\hspace{10pt}$ What proportion of women have blood pressures between $75$ and $90$?

$\textbf{c.}\hspace{10pt}$ A diastolic blood pressure greater than $90$ is classified as hypertension (high blood pressure). What proportion o women have hypertension?

$\textbf{d.}\hspace{10pt}$ Is it unusual for a woman to have a blood pressure lower than $65$?

A normal population has mean $\mu = 46$ and standard deviation $\sigma = 9$. What is the $19$th percentile of the population?

A normal population has mean $\mu = 12$ and standard deviation $\sigma = 3$. What is the $40$th percentile of the population?

A normal population has mean $\mu = 47$ and standard deviation $\sigma = 3$.

$\textbf{a.}\hspace{10pt}$ What proportion of the population is between $40$ and $50$?

$\textbf{b.}\hspace{10pt}$ What is the probability that a randomly chosen value will be between $50$ and $55$?

A normal population has mean $\mu = 20$ and standard deviation $\sigma = 4$.

$\textbf{a.}\hspace{10pt}$ What proportion of the population is less than $18$?

$\textbf{b.}\hspace{10pt}$ What is the probability that a randomly chosen value will be greater than $25$?

Use the Cumulative Normal Distribution table to find the $z$-score with an area of $0.74$ to its left.

A normal population has mean $\mu = 242$ and standard deviation $\sigma = 31$.

$\textbf{a.}\hspace{10pt}$ What proportion of the population is greater than $233$?

$\textbf{b.}\hspace{10pt}$ What is the probability that a randomly chosen value will be less than $249$?

Speeds of automobiles on a certain stretch of freeway at 11:00 P.M. are normally distributed with mean $65$ mph. Twenty percent of the cars are traveling at speeds between $55$ and $65$ mph. What percentage of the cars are going faster than $75$ mph?

Look at the normal curve below, and find $μ$, $μ + σ$, and $σ$.

Heights of men in a certain city are normally distributed with mean $70$ inches. Sixteen percent of the men are more than $73$ inches tall. What percentage of the men are between $67$ and $70$ inches tall?