Question 1

Find the standard deviation of the random variable.
-The random variable X is the number of people who have a college degree in a randomly selected group of four adults from a particular town. Its probability distribution is given in the table. Round
The answer to two decimal places. $\begin{array}{c|ccccc}
\mathrm{x} & 0 & 1 & 2 & 3 & 4 \
\hline \mathrm{P}(\mathrm{X}=\mathrm{x}) & 0.0081 & 0.0756 & 0.2646 & 0.4116 & 0.2401
\end{array}$
A)$2.95$
B) $1.06$
C) $0.92$
D) $0.84$

Accepted Answer

The answer of Find the standard deviation of the random...

Question 2

Find the expected value of the random variable. Round to the nearest cent unless stated otherwise.
-A contractor is considering a sale that promises a profit of $27,000 with a probability of 0.7 or a loss (due to bad weather, strikes, and such)of $2000 with a probability of 0.3. What is the expected
Profit? Round the answer to the nearest dollar. 
A)$18,300
B)$20,300
C)$25,000
D)$18,900

Accepted Answer

The expected profit is calculated as: (0.7 * $27,000) + (0.3 * -$2,000) = $18,900.

Question 3

Evaluate the expression.
-$\left( \begin{array} { c } 11 \ 3 \end{array} \right)$  
A)40,320
B) 330
C) 165
D) 4

Accepted Answer

The expression $\left( \begin{array} { c } 11 \ 3 \end{array} \right)$ represents a binomial coefficient, calculated as $\frac{11!}{3!(11-3)!}$. This simplifies to 165.

Question 4

Use random-variable notation to represent the event.
-Suppose that two balanced dice are rolled. Let Y denote the product of the two numbers. Use random-variable notation to represent the event that the product of the two numbers is greater
Than 4. 
A) $\{ X Y > 4 \}$
B) $\{ 5,6 \}$
C) $\mathrm { P } \{ \mathrm { Y } > 4 \}$
D) $\{ Y > 4 \}$

Accepted Answer

The correct notation for representing the event that the product of the two numbers is greater than 4 using a random variable (in this case, Y) is $\{ Y > 4 \}$. This notation correctly specifies the condition involving the random variable Y.

Question 5

Use random-variable notation to represent the event.
-Suppose a coin is tossed four times. Let X denote the total number of tails obtained in the four tosses. Use random-variable notation to represent the event that the total number of tails is three. 
A) HTTT, THTT, TTHT, TTTH
B) $\{ X \geq 3 \}$
C) $\mathrm { P } \{ \mathrm { X } = 3 \}$
D) $\{ X = 3 \}$

Accepted Answer

The correct notation for the event that the total number of tails is exactly three is $\{ X = 3 \}$, where X is the random variable representing the total number of tails.

Question 6

Determine the possible values of the random variable.
-Suppose that two balanced dice are rolled. Let X denote the absolute value of the difference of the two numbers. What are the possible values of the random variable X? 
A)0, 1, 2, 3, 4, 5, 6
B)-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5
C)0, 1, 2, 3, 4, 5
D)1, 2, 3, 4, 5

Accepted Answer

Since X is the absolute value of the difference between the numbers on two dice, the smallest difference is 0 (when both dice show the same number) and the largest possible difference is 5 (between 1 and 6). Negative differences are not possible because the absolute value converts them into positive numbers.

Question 7

Use the Poisson Distribution to find the indicated probability. Round to three decimal places when necessary.
-$\lambda = 4.5 ; \mathrm { P } ( \mathrm { X } < 3 )$  
A)$0.174$
B) $0.162$
C) $0.504$
D) $0.169$

Accepted Answer

The Poisson distribution formula for finding the probability of observing $x$ events in a fixed interval of time or space is given by $P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}$, where $\lambda$ is the average number of events in the given interval, and $x$ is the number of occurrences. To find $P(X < 3)$, we need to calculate $P(X = 0) + P(X = 1) + P(X = 2)$ with $\lambda = 4.5$. Calculating each and summing them gives a result close to $0.174$.

Question 8

Find the specified probability.
-A statistics professor has office hours from 9:00 am to 10:00 am each day. The number of students waiting to see the professor is a random variable, X, with the distribution shown in the table. $$\begin{array} { c | c | c | c | c | c | c } 
\mathrm { x } & 0 & 1 & 2 & 3 & 4 & 5 \
\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.05 & 0.10 & 0.40 & 0.25 & 0.15 & 0.05
\end{array}$$ The professor gives each student 10 minutes. Determine the probability that a student arriving just
After 9:00 am will have to wait at least 30 minutes to see the professor.
A) 0.25
B)0.85
C)0.15
D)0.45

Accepted Answer

For a student to wait at least 30 minutes, there must be at least 3 students ahead of them because each student takes 10 minutes. The probability for 3 or more students is P(X=3) + P(X=4) + P(X=5) = 0.25 + 0.15 + 0.05 = 0.45.

Question 9

A Poisson random variable has an infinite number of possible values?

Accepted Answer

A Poisson random variable can theoretically take on any non-negative integer value (0, 1, 2, 3, ...) ad infinitum, representing the number of events occurring in a fixed interval of time or space.

Question 10

Evaluate the expression.
-$\frac { 9 ! } { 7 ! }$  
A)2 !
B) 63,000
C) $\frac { 9 } { 7 }$
D) 72

Accepted Answer

The expression $\frac{9!}{7!}$ simplifies to $9 \times 8 = 72$ because the $7!$ terms cancel out.

Question 11

Find the standard deviation of the binomial random variable.
-The probability that a radish seed will germinate is 0.7. A gardener plants seeds in batches of 11. Find the standard deviation for the random variable X, the number of seeds germinating in each
Batch. 
A)1.52
B)1.502
C)1.432
D)1.449

Accepted Answer

The standard deviation of a binomial random variable is given by the formula σ = √(np(1-p)), where n is the number of trials and p is the probability of success. In this case, n = 11 and p = 0.7, so σ = √(11 * 0.7 * 0.3) = 1.502.

Question 12

Find the standard deviation of the binomial random variable.
-On a multiple choice test with 10 questions, each question has four possible answers, one of which is correct. For students who guess at all answers, find the standard deviation for the random
Variable X, the number of correct answers. 
A)1.299
B)1.342
C)1.369
D)1.273

Accepted Answer

The answer of Find the standard deviation of the binomial...

Question 13

Determine the possible values of the random variable.
-The following table displays a frequency distribution for the number of siblings for students in one middle school. For a randomly selected student in the school, let X denote the number of siblings of
The student. What are the possible values of the random variable X? $\begin{array}{r|rrrrrrrr}
\text { Number of siblings } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \
\hline \text { Frequency } & 189 & 245 & 102 & 42 & 24 & 13 & 5 & 2
\end{array}$
A)Brother, sister
B) $0,1,2,3,4,5,6,7$
C) 7
D) $189,245,102,42,24,13,5,2$

Accepted Answer

The possible values of the random variable X, which denotes the number of siblings, are directly listed in the table as 0, 1, 2, 3, 4, 5, 6, 7.

Question 14

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.
 
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)
$\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.167 \
1 & 0.278 \
2 & 0.222 \
3 & 0.167 \
4 & 0.111 \
5 & 0.056
\end{array}$

Accepted Answer

The probability distribution of $X$ (the absolute difference between the two dice) is calculated by counting the number of outcomes that result in each possible difference and dividing by the total number of outcomes (36). For $X=0$, there are 6 outcomes where the dice match. For $X=1$, there are 10 outcomes (e.g., (1,2), (2,1), etc.). This pattern continues, decreasing as the difference increases, leading to the probabilities listed in option D.

Question 15

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. 
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)
$\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 1 & 0.38 \
2 & 0.1444 \
3 & 0.0549 \
4 & 0.0209 \
5 & 0.0079
\end{array}$

Accepted Answer

The correct probabilities are calculated using the binomial probability formula $P(X=x) = \binom{n}{x}p^x(1-p)^{n-x}$, where $n=5$, $p=0.38$, and $x$ ranges from 0 to 5. The probabilities in choice C accurately reflect the outcomes of applying this formula to each possible value of $x$.

Question 16

Construct a probability histogram for the binomial random variable, X.
-A baseball player batting 0.300 comes to bat 4 times in a game. X is the number of hits. 
A)

B)

C)

D)

Accepted Answer

The answer of Construct a probability histogram for the binomial...

Question 17

Obtain the probability distribution of the random variable.
-When a coin is tossed four times, sixteen equally likely outcomes are possible as shown below: HHHH HHHT HHTH HHTT
HTHH HTHT HTTH HTTT
THHH THHT THTH THTT
TTHH TTHT TTTH TTTT
Let $\mathrm { X }$ denote the total number of tails obtained in the four tosses. Find the probability distribution of the random variable X. Leave your probabilities in fraction form. 
A)
$\begin{array}{r|r}
x & P(X=x) \
\hline 0 & 1 / 16 \
1 & 1 / 8 \
2 & 3 / 8 \
3 & 1 / 8 \
4 & 1 / 16
\end{array}$

B)
$\begin{array}{l|r}
x & P(X=x) \
\hline 0 & 1 / 16 \
1 & 3 / 16 \
2 & 1 / 2 \
3 & 3 / 16 \
4 & 1 / 16
\end{array}$

C)
$\begin{array}{r|r}
x & P(X=x) \
\hline 0 & 1 / 16 \
1 & 1 / 4 \
2 & 3 / 8 \
3 & 1 / 4 \
4 & 1 / 16
\end{array}$

D)
$\begin{array}{r|r}
x & P(X=x) \
\hline 1 & 1 / 4 \
2 & 7 / 16 \
3 & 1 / 4 \
4 & 1 / 16
\end{array}$

Accepted Answer

The probability of getting 0 tails is 1/16, the probability of getting 1 tail is 4/16, the probability of getting 2 tails is 6/16, the probability of getting 3 tails is 4/16, and the probability of getting 4 tails is 1/16.

Question 18

Use random-variable notation to represent the event.
-The following table displays a frequency distribution for the number of siblings for students in one middle school. For a randomly selected student in the school, let Y denote the number of siblings of
The student. $\begin{array}{r|rrrrrrrr}
\text { Number of siblings } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \
\hline \text { Frequency } & 189 & 245 & 102 & 42 & 24 & 13 & 5 & 2
\end{array}$

Use random-variable notation to represent the event that the student obtained has at least two but fewer than six siblings.  
A)$\{ 2 \leq Y \leq 6 \}$
B) $\{ 2,3,4,5 \}$
C) $\{ 2 < Y < 6 \}$
D) $\{ 2 \leq Y < 6 \}$

Accepted Answer

The correct representation for having at least two but fewer than six siblings is $\{ 2 \leq Y < 6 \}$, as it includes students with 2, 3, 4, and 5 siblings but not those with 6 or more.

Question 19

Determine the binomial probability formula given the number of trials and the success probability for Bernoulli trials.Let X denote the total number of successes. Round to three decimal places.
-$$\mathrm { n } = 5 , \mathrm { p } = 0.6 , \mathrm { P } ( \mathrm { X } = 2 )$$  
A)$0.360$
B) $0.230$
C) $0.300$
D) $0.346$

Accepted Answer

The answer of Determine the binomial probability formula given the...

Question 20

Determine the required probability by using the Poisson approximation to the binomial distribution. Round to threedecimal places.
-The probability that a car will have a flat tire while driving through a certain tunnel is 0.00004. Use the Poisson approximation to the binomial distribution to find the probability that among 16,000
Cars passing through this tunnel, exactly two will have a flat tire. 
A)0.151
B)0.173
C)0.108
D)0.130

Accepted Answer

The Poisson approximation is used when n is large and p is small. Here, λ = np = 16000 * 0.00004 = 0.64. The probability of exactly k events in a Poisson distribution is given by (e^(-λ) * λ^k) / k!. For k = 2, this is (e^(-0.64) * 0.64^2) / 2! = 0.108.

Quiz 5: Discrete Random Variables

Evaluate the expression. - $\left( \begin{array} { c } 11 \\ 3 \end{array} \right)$

Use random-variable notation to represent the event. -Suppose that two balanced dice are rolled. Let Y denote the product of the two numbers. Use random-variable notation to represent the event that the product of the two numbers is greater Than 4.

Use random-variable notation to represent the event. -Suppose a coin is tossed four times. Let X denote the total number of tails obtained in the four tosses. Use random-variable notation to represent the event that the total number of tails is three.

Determine the possible values of the random variable. -Suppose that two balanced dice are rolled. Let X denote the absolute value of the difference of the two numbers. What are the possible values of the random variable X?

Use the Poisson Distribution to find the indicated probability. Round to three decimal places when necessary. - $\lambda = 4.5 ; \mathrm { P } ( \mathrm { X } < 3 )$

A Poisson random variable has an infinite number of possible values?

Evaluate the expression. - $\frac { 9 ! } { 7 ! }$

Find the standard deviation of the binomial random variable. -The probability that a radish seed will germinate is 0.7. A gardener plants seeds in batches of 11. Find the standard deviation for the random variable X, the number of seeds germinating in each Batch.

Find the standard deviation of the binomial random variable. -On a multiple choice test with 10 questions, each question has four possible answers, one of which is correct. For students who guess at all answers, find the standard deviation for the random Variable X, the number of correct answers.

Construct a probability histogram for the binomial random variable, X. -A baseball player batting 0.300 comes to bat 4 times in a game. X is the number of hits.

Determine the binomial probability formula given the number of trials and the success probability for Bernoulli trials.Let X denote the total number of successes. Round to three decimal places. - $\mathrm { n } = 5 , \mathrm { p } = 0.6 , \mathrm { P } ( \mathrm { X } = 2 )$

Quiz 5: Discrete Random Variables

Evaluate the expression. - (113) \left( \begin{array} { c } 11 \\ 3 \end{array} \right) (113​)

Use random-variable notation to represent the event. -Suppose that two balanced dice are rolled. Let Y denote the product of the two numbers. Use random-variable notation to represent the event that the product of the two numbers is greater Than 4.

Use random-variable notation to represent the event. -Suppose a coin is tossed four times. Let X denote the total number of tails obtained in the four tosses. Use random-variable notation to represent the event that the total number of tails is three.

Determine the possible values of the random variable. -Suppose that two balanced dice are rolled. Let X denote the absolute value of the difference of the two numbers. What are the possible values of the random variable X?

Use the Poisson Distribution to find the indicated probability. Round to three decimal places when necessary. - λ=4.5;P(X<3) \lambda = 4.5 ; \mathrm { P } ( \mathrm { X } < 3 ) λ=4.5;P(X<3)

A Poisson random variable has an infinite number of possible values?

Evaluate the expression. - 9!7!\frac { 9 ! } { 7 ! }7!9!​

Find the standard deviation of the binomial random variable. -The probability that a radish seed will germinate is 0.7. A gardener plants seeds in batches of 11. Find the standard deviation for the random variable X, the number of seeds germinating in each Batch.

Find the standard deviation of the binomial random variable. -On a multiple choice test with 10 questions, each question has four possible answers, one of which is correct. For students who guess at all answers, find the standard deviation for the random Variable X, the number of correct answers.

Construct a probability histogram for the binomial random variable, X. -A baseball player batting 0.300 comes to bat 4 times in a game. X is the number of hits.

Evaluate the expression. - $\left( \begin{array} { c } 11 \\ 3 \end{array} \right)$

Use the Poisson Distribution to find the indicated probability. Round to three decimal places when necessary. - $\lambda = 4.5 ; \mathrm { P } ( \mathrm { X } < 3 )$

Evaluate the expression. - $\frac { 9 ! } { 7 ! }$