Question 1

Calculate the specified probability
-Suppose that $W$ is a random variable. Given that $P ( W \leq 3 ) = 0.1$, find $P ( W > 3 )$.

Accepted Answer

A) 0 
B)  3 
C)  $0.1$ 
D)  $0.9$ 
A) 0 
B)  3 
C)  $0.1$ 
D)  $0.9$

Question 2

Use random-variable notation to represent the event.
-Suppose that two balanced dice are rolled. Let X denote the absolute value of the difference of the two numbers. Use random-variable notation to represent the event that the absolute value of the
Difference of the two numbers is 2.

Accepted Answer

A)  $\{ X = 2 \}$ 
B)  $P \{ X = 2 \}$ 
C)  $| X | = 2$ 
D)  $\{ ( 1,3 ) , ( 2,4 ) , ( 3,5 ) , ( 4,6 ) , ( 3,1 ) , ( 4,2 ) , ( 5,3 ) , ( 6,4 ) \}$ 
A)  $\{ X = 2 \}$ 
B)  $P \{ X = 2 \}$ 
C)  $| X | = 2$ 
D)  $\{ ( 1,3 ) , ( 2,4 ) , ( 3,5 ) , ( 4,6 ) , ( 3,1 ) , ( 4,2 ) , ( 5,3 ) , ( 6,4 ) \}$

Question 3

Find the standard deviation of the binomial random variable.
-A company manufactures batteries in batches of 26 and there is a 3% rate of defects. Find the standard deviation for the random variable X, the number of defects per batch.

Accepted Answer

A) 0.883 
B) 0.853 
C) 0.87 
D) 0.867 
A) 0.883 
B) 0.853 
C) 0.87 
D) 0.867

Question 4

Provide an appropriate response.
-Which of the random variables described below is/are discrete random variables? The random variable X represents the number of heads when a coin is flipped 20 times.The random variable Y represents the number of calls received by a car tow service in a year. The random variable Z represents the weight of a randomly selected student.

Accepted Answer

A) X, Y, and Z 
B) X and Y 
C) X only 
D) Y only 
A) X, Y, and Z 
B) X and Y 
C) X only 
D) Y only

Question 5

Provide an appropriate response.
-For any discrete random variable, the possible values of the random variable form a
finite set of numbers.

Accepted Answer

A) True 
 B)False

Question 6

Use the Poisson Distribution to find the indicated probability. Round to three decimal places when necessary.
-A computer salesman averages 1.9 sales per week. Use the Poisson distribution to find the probability that in a randomly selected week the number of computers sold is

Accepted Answer

A) 3. 0.188 
B) 0.214 
C) 0.171 
D) 0.325 
A) 3. 0.188 
B) 0.214 
C) 0.171 
D) 0.325

Question 7

Find the mean of the Poisson random variable.
-The number of calls received by a car towing service in an hour has a Poisson distribution with parameter $\lambda = 1.33$. Let $X$ denote the number of calls received by the service in a randomly selected hour. Find the mean of X.

Accepted Answer

A) $0.665$ 
B)  $1.33$ 
C)  $1.77$ 
D)  $1.153$ 
A) $0.665$ 
B)  $1.33$ 
C)  $1.77$ 
D)  $1.153$

Question 8

16% of the employees of a certain company cycle to work. Three employees are selected at random from the company and asked whether or not they cycle to work. Considering a success to be "cycles to work", formulate the process of observing whether each of the three employees cycles to work as a sequence of three Bernoulli trials. Complete the table below by showing each possible outcome together with its probability. Display the probabilities to three decimal places. List the outcomes in which exactly one of the three employees cycles to work. Without using the binomial probability formula, find the probability that exactly one of the three employees cycles to work. $$\begin{array} { r | r }  \text { Outcome } & \text { Probability } \ \hline \text { sss } & ( 0.16 ) ( 0.16 ) ( 0.16 ) = 0.004 \end{array}$$

Accepted Answer

Each trial consists of observing whether

Question 9

Find the indicated binomial probability. Round to five decimal places when necessary.
-A cat has a litter of 7 kittens. Find the probability that exactly 5 of the little furballs are female. Assume that male and female births are equally likely.

Accepted Answer

A) 0.16406 
B) 0.32813 
C) 0.00781 
D) 0.65625 
A) 0.16406 
B) 0.32813 
C) 0.00781 
D) 0.65625

Question 10

Determine the possible values of the random variable.
-Suppose that two balanced dice are rolled. Let Y denote the product of the two numbers. What are the possible values of the random variable Y?

Accepted Answer

A)  $1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,30,36$ 
B)  $( 1,1 ) , ( 1,2 ) , ( 1,3 ) , ( 1,4 ) , ( 1,5 ) , ( 1,6 ) , ( 2,1 ) , ( 2,2 ) , ( 2,3 ) , ( 2,4 ) , ( 2,5 ) , ( 2,6 ) , ( 3,1 ) , ( 3,2 ) , ( 3,3 ) , ( 3$, $4 ) , ( 3,5 ) , ( 3,6 ) , ( 4,1 ) , ( 4,2 ) , ( 4,3 ) , ( 4,4 ) , ( 4,5 ) , ( 4,6 ) , ( 5,1 ) , ( 5,2 ) , ( 5,3 ) , ( 5,4 ) , ( 5,5 ) , ( 5,6 ) , ( 6,1 )$, $( 6,2 ) , ( 6,3 ) , ( 6,4 ) , ( 6,5 ) , ( 6,6 )$ 
C)  $0,1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,30,36$ 
D)  $2,3,4,5,6,8,10,12,15,18,20,24,30$ 
A)  $1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,30,36$ 
B)  $( 1,1 ) , ( 1,2 ) , ( 1,3 ) , ( 1,4 ) , ( 1,5 ) , ( 1,6 ) , ( 2,1 ) , ( 2,2 ) , ( 2,3 ) , ( 2,4 ) , ( 2,5 ) , ( 2,6 ) , ( 3,1 ) , ( 3,2 ) , ( 3,3 ) , ( 3$, $4 ) , ( 3,5 ) , ( 3,6 ) , ( 4,1 ) , ( 4,2 ) , ( 4,3 ) , ( 4,4 ) , ( 4,5 ) , ( 4,6 ) , ( 5,1 ) , ( 5,2 ) , ( 5,3 ) , ( 5,4 ) , ( 5,5 ) , ( 5,6 ) , ( 6,1 )$, $( 6,2 ) , ( 6,3 ) , ( 6,4 ) , ( 6,5 ) , ( 6,6 )$ 
C)  $0,1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,30,36$ 
D)  $2,3,4,5,6,8,10,12,15,18,20,24,30$

Question 11

Find the specified probability distribution of the binomial random variable.
-A multiple choice test consists of four questions. Each question has five possible answers of which only one is correct. A student guesses on every question. Find the probability distribution of X, the
Number of questions she answers correctly.

Accepted Answer

A) 
$\begin{array}{c|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.4096 \
1 & 0.4096 \
2 & 0.1536 \
3 & 0.0256 \
4 & 0.0016
\end{array}$ 
B) 
$\begin{array}{c|r}
x & P(X=x) \
\hline 1 & 0.4096 \
2 & 0.1536 \
3 & 0.0256 \
4 & 0.0016
\end{array}$ 
C) 
$\begin{array}{l|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.4096 \
1 & 0.4096 \
2 & 0.1024 \
3 & 0.0768 \
4 & 0.0016
\end{array}$ 
D) 
$\begin{array}{c|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0 \
1 & 0.2 \
2 & 0.04 \
3 & 0.008 \
4 & 0.0016
\end{array}$ 
A) 
$\begin{array}{c|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.4096 \
1 & 0.4096 \
2 & 0.1536 \
3 & 0.0256 \
4 & 0.0016
\end{array}$ 
B) 
$\begin{array}{c|r}
x & P(X=x) \
\hline 1 & 0.4096 \
2 & 0.1536 \
3 & 0.0256 \
4 & 0.0016
\end{array}$ 
C) 
$\begin{array}{l|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.4096 \
1 & 0.4096 \
2 & 0.1024 \
3 & 0.0768 \
4 & 0.0016
\end{array}$ 
D) 
$\begin{array}{c|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0 \
1 & 0.2 \
2 & 0.04 \
3 & 0.008 \
4 & 0.0016
\end{array}$

Question 12

Find the specified probability.
-The number of loaves of rye bread left on the shelf of a local bakery at closing (denoted by the random variable X)varies from day to day. Past records show that the probability distribution of X
Is as shown in the following table. Find the probability that there will be at least three loaves left
Over at the end of any given day. $\begin{array}{cccccccc}
\mathrm{x} & 0 & 1 & 2 & 3 & 4 & 5 & 6 \
\hline \mathrm{P}(\mathrm{X}=\mathrm{x}) & 0.20 & 0.25 & 0.20 & 0.15 & 0.10 & 0.08 & 0.02
\end{array}$

Accepted Answer

A) $0.35$ 
B)  $0.65$ 
C)  $0.20$ 
D)  $0.15$ 
A) $0.35$ 
B)  $0.65$ 
C)  $0.20$ 
D)  $0.15$

Question 13

Find the mean of the binomial random variable. Round to two decimal places when necessary.
-The probability that a person has immunity to a particular disease is 0.3. Find the mean for the random variable X, the number who have immunity in samples of size

Accepted Answer

A) 24. 12 
B) 0.3 
C) 16.8 
D) 7.2 
A) 24. 12 
B) 0.3 
C) 16.8 
D) 7.2

Question 14

Find the standard deviation of the Poisson random variable. Round to three decimal places.
-Suppose X has a Poisson distribution with parameter ʎ = 1.300. Find the standard deviation of X.

Accepted Answer

A) 1.690 
B) 1.14 
C) 1.300 
D) 1.000 
A) 1.690 
B) 1.14 
C) 1.300 
D) 1.000

Question 15

Use the Poisson Distribution to find the indicated probability. Round to three decimal places when necessary.
-The number of calls received by a mountain search and rescue team in a day has a Poisson distribution with parameter $\lambda$ = 0.70. Find the probability that on a randomly selected day, they
Will receive fewer than two calls.

Accepted Answer

A) 0.156 
B) 0.348 
C) 0.122 
D) 0.844 
A) 0.156 
B) 0.348 
C) 0.122 
D) 0.844

Question 16

Find the indicated probability. Round to four decimal places.
-A car insurance company has determined that 9% of all drivers were involved in a car accident last year. Among the 10 drivers living on one particular street, 3 were involved in a car accident last
Year. If 10 drivers are randomly selected, what is the probability of getting 3 or more who were
Involved in a car accident last year?

Accepted Answer

A) 0.9548 
B) 0.0452 
C) 0.4435 
D) 0.0541 
A) 0.9548 
B) 0.0452 
C) 0.4435 
D) 0.0541

Question 17

Determine the possible values of the random variable.
-Suppose that two balanced dice, a red die and a green die, are rolled. Let Y denote the value of G - R where G represents the number on the green die and R represents the number on the red die.
What are the possible values of the random variable Y?

Accepted Answer

A)  $- 5 , - 4 , - 3 , - 2 , - 1,0,1,2,3,4,5$ 
B)  $0,1,2,3,4,5$ 
C)  $0,1,2,3,4,5,6$ 
D)  $- 6 , - 5 , - 4 , - 3 , - 2 , - 1,0,1,2,3,4,5,6$ 
A)  $- 5 , - 4 , - 3 , - 2 , - 1,0,1,2,3,4,5$ 
B)  $0,1,2,3,4,5$ 
C)  $0,1,2,3,4,5,6$ 
D)  $- 6 , - 5 , - 4 , - 3 , - 2 , - 1,0,1,2,3,4,5,6$

Question 18

Obtain the probability distribution of the random variable.
-When two balanced dice are rolled, 36 equally likely outcomes are possible as shown below. $ (1,1)(1,2)(1,3)(1,4)(1,5)(1,6) $
$ (2,1)(2,2)(2,3)(2,4)(2,5)(2,6) $
$ (3,1)(3,2)(3,3)(3,4)(3,5)(3,6) $
$ (4,1)(4,2)(4,3)(4,4)(4,5)(4,6) $
$ (5,1)(5,2)(5,3)(5,4)(5,5)(5,6) $
$ (6,1)(6,2)(6,3)(6,4)(6,5)(6,6) $

Let $ X $ denote the absolute value of the difference of the two numbers. Find the probability distribution of X. Give the probabilities as decimals rounded to three decimal places.

Accepted Answer

A) 
$\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.167 \
1 & 0.167 \
2 & 0.167 \
3 & 0.167 \
4 & 0.167 \
5 & 0.167
\end{array}$ 
B) 
$\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 1 & 0.278 \
2 & 0.222 \
3 & 0.167 \
4 & 0.111 \
5 & 0.056
\end{array}$ 
C) 
$\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.167 \
1 & 0.251 \
2 & 0.222 \
3 & 0.167 \
4 & 0.111 \
5 & 0.056 \
6 & 0.027
\end{array}$ 
D) 
$\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.167 \
1 & 0.278 \
2 & 0.222 \
3 & 0.167 \
4 & 0.111 \
5 & 0.056
\end{array}$ 
A) 
$\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.167 \
1 & 0.167 \
2 & 0.167 \
3 & 0.167 \
4 & 0.167 \
5 & 0.167
\end{array}$ 
B) 
$\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 1 & 0.278 \
2 & 0.222 \
3 & 0.167 \
4 & 0.111 \
5 & 0.056
\end{array}$ 
C) 
$\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.167 \
1 & 0.251 \
2 & 0.222 \
3 & 0.167 \
4 & 0.111 \
5 & 0.056 \
6 & 0.027
\end{array}$ 
D) 
$\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.167 \
1 & 0.278 \
2 & 0.222 \
3 & 0.167 \
4 & 0.111 \
5 & 0.056
\end{array}$

Question 19

Determine the required probability by using the Poisson approximation to the binomial distribution. Round to threedecimal places.
-The probability that a call received by a certain switchboard will be a wrong number is 0.01. Use the Poisson approximation to the binomial distribution to find the probability that among 150 calls
Received by the switchboard, there are no wrong numbers.

Accepted Answer

A) 0.223 
B) 0.335 
C) 0.245 
D) 0.777 
A) 0.223 
B) 0.335 
C) 0.245 
D) 0.777

Question 20

Find the specified probability distribution of the binomial random variable.
-38% of the murder trials in one district result in a guilty verdict. Five murder trials are selected at random from the district. Determine the probability distribution of X, the number of trials among
The five selected in which the defendant is found guilty.

Accepted Answer

A) 
$\begin{array}{c|c}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.0916 \
1 & 0.2938 \
2 & 0.3441 \
3 & 0.2109 \
4 & 0.0516 \
5 & 0.0079
\end{array}$ 
B) 
$\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.0916 \
1 & 0.2808 \
2 & 0.3221 \
3 & 0.2329 \
4 & 0.0646 \
5 & 0.0079
\end{array}$ 
C) 
$\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.0916 \
1 & 0.2808 \
2 & 0.3441 \
3 & 0.2109 \
4 & 0.0646 \
5 & 0.0079
\end{array}$ 
D) 
$\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 1 & 0.38 \
2 & 0.1444 \
3 & 0.0549 \
4 & 0.0209 \
5 & 0.0079
\end{array}$ 
A) 
$\begin{array}{c|c}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.0916 \
1 & 0.2938 \
2 & 0.3441 \
3 & 0.2109 \
4 & 0.0516 \
5 & 0.0079
\end{array}$ 
B) 
$\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.0916 \
1 & 0.2808 \
2 & 0.3221 \
3 & 0.2329 \
4 & 0.0646 \
5 & 0.0079
\end{array}$ 
C) 
$\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.0916 \
1 & 0.2808 \
2 & 0.3441 \
3 & 0.2109 \
4 & 0.0646 \
5 & 0.0079
\end{array}$ 
D) 
$\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 1 & 0.38 \
2 & 0.1444 \
3 & 0.0549 \
4 & 0.0209 \
5 & 0.0079
\end{array}$

Calculate the specified probability -Suppose that $W$ is a random variable. Given that $P ( W \leq 3 ) = 0.1$ , find $P ( W > 3 )$ .

Use random-variable notation to represent the event. -Suppose that two balanced dice are rolled. Let X denote the absolute value of the difference of the two numbers. Use random-variable notation to represent the event that the absolute value of the Difference of the two numbers is 2.

Find the standard deviation of the binomial random variable. -A company manufactures batteries in batches of 26 and there is a 3% rate of defects. Find the standard deviation for the random variable X, the number of defects per batch.

Provide an appropriate response. -For any discrete random variable, the possible values of the random variable form a finite set of numbers.

Use the Poisson Distribution to find the indicated probability. Round to three decimal places when necessary. -A computer salesman averages 1.9 sales per week. Use the Poisson distribution to find the probability that in a randomly selected week the number of computers sold is

Find the mean of the Poisson random variable. -The number of calls received by a car towing service in an hour has a Poisson distribution with parameter $\lambda = 1.33$ . Let $X$ denote the number of calls received by the service in a randomly selected hour. Find the mean of X.

Find the indicated binomial probability. Round to five decimal places when necessary. -A cat has a litter of 7 kittens. Find the probability that exactly 5 of the little furballs are female. Assume that male and female births are equally likely.

Determine the possible values of the random variable. -Suppose that two balanced dice are rolled. Let Y denote the product of the two numbers. What are the possible values of the random variable Y?

Find the mean of the binomial random variable. Round to two decimal places when necessary. -The probability that a person has immunity to a particular disease is 0.3. Find the mean for the random variable X, the number who have immunity in samples of size

Find the standard deviation of the Poisson random variable. Round to three decimal places. -Suppose X has a Poisson distribution with parameter ʎ = 1.300. Find the standard deviation of X.

The Nature of Statistics

Organizing Data

Descriptive Measures

Probability Concepts

The Normal Distribution

The Sampling Distribution of the Sample Mean

Confidence Intervals for One Population Mean

Hypothesis Tests for One Population Mean

Inferences for Two Population Means

Inferences for Population Standard Deviations

Inferences for Population Proportions

Chi-Square Procedures

Descriptive Methods in Regression and Correlation

Inferential Methods in Regression and Correlation

Analysis of Variance Anova

Filters

Exam 5: Discrete Random Variables

Calculate the specified probability -Suppose that WWW is a random variable. Given that P(W≤3)=0.1P ( W \leq 3 ) = 0.1P(W≤3)=0.1 , find P(W>3)P ( W > 3 )P(W>3) .