## Continuous Random Variables

First, this section talks about how to describe continuous distributions and compute related probabilities, including some basic facts about the normal distribution. Then, it covers how to compute probabilities related to any normal random variable and gives examples of using $z$-score transformations. Finally, it defines tail probabilities and illustrates how to find them.

### Basic

1. $X$ is a normally distributed random variable with mean $57$ and standard deviation $6$. Find the probability indicated.

a. $P(X < 59.5)$
b. $P(X < 46.2)$
c. $P(X > 52.2)$

d. $P(X > 70)$

3. $X$ is a normally distributed random variable with mean $112$ and standard deviation $15$. Find the probability indicated.

a. $P(100 < X < 125)$
b. $P(91 < X < 107)$
c. $P(118 < X < 160)$

5. $X$ is a normally distributed random variable with mean $500$ and standard deviation $25$. Find the probability indicated.

a. $P(X < 400)$
b. $P(466 < X < 625)$

7. $X$ is a normally distributed random variable with mean $15$ and standard deviation $1$. Use Figure 12.2 "Cumulative Normal Probability" to find the first probability listed. Find the second probability using the symmetry of the density curve. Sketch the density curve with relevant regions shaded to illustrate the computation.

a. $P(X < 12), P(X > 18)$
b. $P(X < 14), P(X > 16)$
c. $P(X < 11.25), P(X > 18.75)$
d. $P(X < 12.67), P(X > 17.33)$

9. $X$ is a normally distributed random variable with mean $67$ and standard deviation $13$. The probability that $X$ takes a value in the union of intervals $(-\infty, 67-a] \cup[67+a, \infty)$ will be denoted $P(X \leq 67-a$ or $X \geq 67+a)$. Use Figure 12.2 "Cumulative Normal Probability" to find the following probabilities of this type. Sketch the density curve with relevant regions shaded to illustrate the computation. Because of the symmetry of the density curve you need to use Figure 12.2 "Cumulative Normal Probability" only one time for each part.

a. $P(X < 57$ or $X > 77)$
b. $P(X < 47$ or $X > 87)$
c. $P(X < 49$ or $X > 85)$
d. $P(X < 37$ or $X > 97)$

### Applications

11. The amount $X$ of beverage in a can labeled $12$ ounces is normally distributed with mean $12.1$ ounces and standard deviation $0.05$ ounce. A can is selected at random.

a. Find the probability that the can contains at least $12$ ounces.
b. Find the probability that the can contains between $11.9$ and $12.1$ ounces.

13. The systolic blood pressure $X$ of adults in a region is normally distributed with mean $112 \mathrm{~mm} \mathrm{Hg}$ and standard deviation $15 \mathrm{~mm} \mathrm{Hg}$. A person is considered "prehypertensive" if his systolic blood pressure is between $120$ and $130 \mathrm{~mm}\mathrm{Hg}$. Find the probability that the blood pressure of a randomly selected person is prehypertensive.

15. Heights $X$ of adult men are normally distributed with mean $69.1$ inches and standard deviation $2.92$ inches. Juliet, who is $63.25$ inches tall, wishes to date only men who are taller than she but within $6$ inches of her height. Find the probability that the next man she meets will have such a height.

17. A regulation golf ball may not weigh more than $1.620$ ounces. The weights $X$ of golf balls made by a particular process are normally distributed with mean $1.361$ ounces and standard deviation $0.09$ ounce. Find the probability that a golf ball made by this process will meet the weight standard.

19. The amount of non-mortgage debt per household for households in a particular income bracket in one part of the country is normally distributed with mean $\ 28,350$ and standard deviation $\ 3,425$. Find the probability that a randomly selected such household has between $\ 20,000$ and $\ 30,000$ in non-mortgage debt.

21. The distance from the seat back to the front of the knees of seated adult males is normally distributed with mean $23.8$ inches and standard deviation $1.22$ inches. The distance from the seat back to the back of the next seat forward in all seats on aircraft flown by a budget airline is $26$ inches. Find the proportion of adult men flying with this airline whose knees will touch the back of the seat in front of them.

23. The useful life of a particular make and type of automotive tire is normally distributed with mean $57,500$, miles and standard deviation $950$ miles.

a. Find the probability that such a tire will have a useful life of between $57,000$ and $58,000$ miles.
b. Hamlet buys four such tires. Assuming that their lifetimes are independent, find the probability that all four will last between $57,000$ and $58,000$ miles. (If so, the best tire will have no more than $1,000$ miles left on it when the first tire fails.) Hint: There is a binomial random variable here, whose value of $p$ comes from part (a).

25. The lengths of time taken by students on an algebra proficiency exam (if not forced to stop before completing it) are normally distributed with mean $28$ minutes and standard deviation $1.5$ minutes.

a. Find the proportion of students who will finish the exam if a $30$-minute time limit is set.
b. Six students are taking the exam today. Find the probability that all six will finish the exam within the $30$-minute limit, assuming that times taken by students are independent. Hint: There is a binomial random variable here, whose value of $p$ comes from part (a).

27. A regulation hockey puck must weigh between $5.5$ and $6$ ounces. In an alternative manufacturing process the mean weight of pucks produced is $5.75$ ounce. The weights of pucks have a normal distribution whose standard deviation can be decreased by increasingly stringent (and expensive) controls on the manufacturing process. Find the maximum allowable standard deviation so that at most $0.005$ of all pucks will fail to meet the weight standard. (Hint: The distribution is symmetric and is centered at the middle of the interval of acceptable weights).