Ap Statistics 5th Edition Chapter 6 Random Variables Answers
Problem 27
Consider selecting a household in rural Thailand at random. Define the random variable $x$ to be
$x=$ number of individuals living in the selected household
Based on information in an article that appeared in the Journal of Applied Probability (2011: $173-188$ ), the probability distribution of $x$ is as follows:
$\begin{array}{llllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\end{array}$$p(x) \quad .140 .175 .220 \quad .260 \quad .155 .025 .015 \quad .005 .004 .001$
Calculate the mean value of the random variable $x$. (Hint: See Example 7.9.)
Problem 28
The probability distribution of $x$, the number of defective tires on a randomly selected automobile checked at a certain inspection station, is given in the following table:
$\begin{array}{ccccll}x & 0 & 1 & 2 & 3 & 4 \\ p(x) & .54 & .16 & .06 & .04 & .20\end{array}$
Problem 29
Consider the following probability distribution for $y=$ the number of broken eggs in a carton:
$\begin{array}{lllllll}y & 0 & 1 & 2 & 3 & 4\end{array}$
$\begin{array}{lll}p(y) & .65 & 20 & 10\end{array}$
$\begin{array}{ll}.04 & 0\end{array}$
a. Calculate and interpret $\mu_{y}$.
b. In the long run, for what percentage of cartons is the number of broken eggs less than $\mu_{y} ?$ Does this surprise you?
c. Why doesn't $\mu_{y}=(0+1+2+3+4) / 5=2.0 ?$ Explain.
Problem 30
Referring to the previous exercise, use the result of Part (a) along with the fact that a carton contains 12 eggs to determine the mean value of $z=$ the number of unbroken eggs. (Hint: $z$ can be written as a linear function of $y$; see Example $7.15 .$ )
Problem 31
Exercise $7.8$ gave the following probability distribution for $x=$ the number of courses for which a randomly selected student at a certain university is registered:
$$
\begin{array}{cccccccc}
x & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
p(x) & .02 & .03 & .09 & .25 & .40 & .16 & .05
\end{array}
$$
It can be easily verified that $\mu=4.66$ and $\sigma=1.20$.
a. Because $\mu-\sigma=3.46$, the $x$ values 1,2, and 3 are more than 1 standard deviation below the mean. What is the probability that $x$ is more than 1 standard deviation below its mean? (Hint: See Example 7.13.)
b. What $x$ values are more than 2 standard deviations away from the mean value (either less than $\mu-2 \sigma$ or greater than $\mu+2 \sigma) ?$
c. What is the probability that $x$ is more than 2 standard deviations away from its mean value?
Problem 32
Example $7.11$ gave the probability distributions of $x=$ number of flaws in a randomly selected glass panel for two suppliers of glass used in the manufacture of flat screen TVs. If the manufacturer wanted to select a single supplier for glass panels, which of these two suppliers would you recommend? Justify your choice based on consideration of both center and variability. (Hint:
See Example 7.11.)
Problem 33
Consider a large ferry that can accommodate cars and buses. The toll for cars is $\$ 3$, and the toll for buses is \$10. Let $x$ and $y$ denote the number of cars and buses, respectively, carried on a single trip. Cars and buses are accommodated on different levels of the ferry, so the number of buses accommodated on any trip is independent of the number of cars on the trip. Suppose that $x$ and $y$ have the following probability distributions:
$\begin{array}{ccccccc}x & 0 & 1 & 2 & 3 & 4 & 5 \\ p(x) & .05 & 10 & .25 & .30 & .20 & .10\end{array}$ $\begin{array}{cccc}y & 0 & 1 & 2 \\ p(y) & .50 & 30 & .20\end{array}$
a. Compute the mean and standard deviation of $x$.
b. Compute the mean and standard deviation of $y$.
c. Compute the mean and variance of the total amount of money collected in tolls from cars.
d. Compute the mean and variance of the total amount of money collected in tolls from buses.
e. Compute the mean and variance of $z=$ total number of vehicles (cars and buses) on the ferry.
f. Compute the mean and variance of $w=$ total amount of money collected in tolls.
Problem 34
Suppose that for a given computer salesperson, the probability distribution of $x=$ the number of systems sold in 1 month is given by the following table:
$\begin{array}{ccccccccc}x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ p(x) & .05 & 10 & 12 & 30 & .30 & .11 & .01 & .01\end{array}$
a. Find the mean value of $x$ (the mean number of systems sold).
b. Find the variance and standard deviation of $x$. How would you interpret these values?
c. What is the probability that the number of systems sold is within 1 standard deviation of its mean value?
d. What is the probability that the number of systems sold is more than 2 standard deviations from the mean? (Hint: See Example $7.13 .$ )
Problem 35
A local television station sells 15 -second, 30 -second, and 60 -second advertising spots. Let $x$ denote the length of a randomly selected commercial appearing on this station, and suppose that the probability distribution of $x$ is given by the following table:
$\begin{array}{llll}x & 15 & 30 & 60\end{array}$
$\begin{array}{llll}p(x) & .1 & .3 & .6\end{array}$
a. Find the average length for commercials appearing on this station.
b. If a 15 -second spot sells for $\$ 500$, a 30 -second spot for $\$ 800$, and a 60 -second spot for $\$ 1000$, find the average amount paid for commercials appearing on this station. (Hint: Consider a new variable, $y=\mathrm{cost}$, and then find the probability distribution and mean value of $y .$.)
Problem 36
An author has written a book and submitted it to a publisher. The publisher offers to print the book and gives the author the choice between a flat payment of $\$ 10,000$ and a royalty plan. Under the royalty plan the author would receive $\$ 1$ for each copy of the book sold. The author thinks that the following table gives the probability distribution of the variable $x=$ the number of books that will be sold:
$\begin{array}{lllll}x & 1000 & 5000 & 10,000 & 20,000\end{array}$
$p(x)$
$\begin{array}{llll}.05 & .30 & .40 & .25\end{array}$
Which payment plan should the author choose? Why?
Problem 37
A grocery store has an express line for customers purchasing five or fewer items. Let $x$ be the number of items purchased by a randomly selected customer using this line. Give examples of two different assignments of probabilities such that the resulting distributions have the same mean but quite different standard deviations.
Problem 38
An appliance dealer sells three different models of upright freezers having $13.5,15.9$, and $19.1$ cubic
Problem 39
- To assemble a piece of furniture, a wood peg must be inserted into a predrilled hole. Suppose that the diameter of a randomly selected peg is a random variable with mean $0.25$ inch and standard deviation $0.006$ inch and that the diameter of a randomly selected hole is a random variable with mean $0.253$ inch and standard deviation $0.002$ inch. Let $x_{1}=$ peg diameter, and let $x_{2}=$ denote hole diameter.
a. Why would the random variable $y$, defined as $y=x_{2}-x_{1}$, be of interest to the furniture manufacturer?
b. What is the mean value of the random variable $y$ ?
c. Assuming that $x_{1}$ and $x_{2}$ are independent, what is the standard deviation of $y ?$
d. Is it reasonable to think that $x_{1}$ and $x_{2}$ are independent? Explain.
e. Based on your answers to Parts (b) and (c), do you think that finding a peg that is too big to fit in the predrilled hole would be a relatively common or a alotiw. $=$
Problem 40
A multiple-choice exam consists of 50 questions. Each question has five choices, of which only one is correct. Suppose that the total score on the exam is computed as
$$
y=x_{1}-\frac{1}{4} x_{2}
$$
where $x_{1}=$ number of correct responses and $x_{2}=$ number of incorrect responses. (Calculating a total score by subtracting a term based on the number of incorrect responses is known as a correction for guessing and is designed to discourage test takers from choosing answers at random.)
a. It can be shown that if a totally unprepared student answers all 50 questions by just selecting one of the five answers at random, then $\mu_{x_{1}}=10$ and $\mu_{x_{2}}=40$. What is the mean value of the total score, $y$ ? Does this surprise you? Explain. (Hint: See Example 7.16.)
Problem 41
Consider a game in which a red die and a blue die are rolled. Let $x_{R}$ denote the value showing on the uppermost face of the red die, and define $x_{B}$ similarly for the blue die.
a. The probability distribution of $x_{R}$ is
$\begin{array}{lllllll}x_{R} & 1 & 2 & 3 & 4 & 5 & 6\end{array}$
$\begin{array}{lllllll}p\left(x_{\mathrm{B}}\right) & 1 / 6 & 1 / 6 & 1 / 6 & 1 / 6 & 1 / 6 & 1 / 6\end{array}$
Find the mean, variance, and standard deviation of $x_{R}$
b. What are the values of the mean, variance, and standard deviation of $x_{B} ?$ (You should be able to answer this question without doing any additional calculations.)
c. Suppose that you are offered a choice of the following two games:
Game 1: Costs $\$ 7$ to play, and you win $y_{1}$ dollars, where $y_{1}=x_{R}+x_{B}$.
Game 2: Doesnt cost anything to play initially, but you "win" $3 y_{2}$ dollars, where $y_{2}=x_{R}-x_{B} .$ If $y_{2}$ is negative, you must pay that amount; if it is positive, you receive that amount.
For Game 1 , the net amount won in a game is $w_{1}=y_{1}-7=x_{R}+x_{B}-7$. What are the mean and standard deviation of $w_{1} ?$
d. For Game 2 , the net amount won in a game is $w_{2}=3 y_{2}=3\left(x_{R}-x_{B}\right)$. What are the mean and standard deviation of $w_{2} ?$
e. Based on your answers to Parts (c) and (d), if you had to play, which game would you choose and why?
Problem 42
The states of Ohio, Iowa, and Idaho are often confused, probably because the names sound so similar. Each year, the State Tourism Directors of these three states drive to a meeting in one of the state capitals to discuss strategies for attracting tourists to their states so that the states will become better known.
The location of the meeting is selected at random from the three state capitals. The shortest highway distance from Boise, Idaho to Columbus, Ohio passes through Des Moines, Iowa. The highway distance from Boise to Des Moines is 1350 miles, and the distance from Des Moines to Columbus is 650 miles. Let $d_{1}$ represent the driving distance from Columbus to the meeting, with $d_{2}$ and $d_{3}$ representing the distances from Des Moines and Boise, respectively.
a. Find the probability distribution of $d_{1}$ and display it in a table.
b. What is the expected value of $d_{1} ?$
Ap Statistics 5th Edition Chapter 6 Random Variables Answers
Source: https://www.numerade.com/books/chapter/random-variables-and-probability-distributions-7/?section=48470