Deck 5: Probability Distributions and Data Modeling

Question 1

Use the data given below to answer the following questions).
On an average, the number of students that choose to study Arts subjects at Greyin Tide University is 17 each year. Hint: Use Poisson distribution formula).
Which of the following is true about probability density functions?

A) A graph of the density function must lie at or below the x-axis.
B) The total area under the density function below the x-axis is 2.0.
C) It calculates the probability of a random variable lying within a certain interval.
D) P (a) x P (b) is the area under the density function between two numbers a and b.

Answer 1

It calculates the probability of a random variable lying within a certain interval.

Question 2

Which of the following is true of the binomial distribution?

A) The expected value of the binomial distribution is np1 - p), where n is the number of experiments and p is the probability of success.
B) The binomial distribution can assume different shapes and amounts of skewness, depending on the parameters.
C) In Excel's binomial distribution function, setting cumulative to TRUE will provide the probability mass function for a specified value.
D) The expected value of the binomial distribution is λ, a constant.

Answer 2

The binomial distribution can assume different shapes and amounts of skewness, depending on the parameters.

Question 3

Which of the following characterizes a random variable having two possible outcomes, each with a constant probability of occurrence?

A) Beta distribution
B) Bernoulli distribution
C) Poisson distribution
D) Binominal distribution

Answer 3

Bernoulli distribution

Question 4

Which of the following is true about the classical definition of probability?

A) It is based on judgment and experience.
B) If the process that generates the outcomes is known, probabilities can be deduced from theoretical arguments.
C) The probability that an outcome will occur is simply the relative frequency associated with that outcome.
D) It is based on observed data.

Question 5

Use the data given below to answer the following questions).
15 students were asked to choose between the broad categories of Arts, Science, and Math as their preferred area of study. $\begin{array} { | c | c | c | } \hline \text { Respondent } & \text { Gender } & \text { Preference } \\\hline 1 & \text { Fernale } & \text { Arts } \\\hline 2 & \text { Male } & \text { Science } \\\hline 3 & \text { Male } & \text { Math } \\\hline 4 & \text { Fernale } & \text { Arts } \\\hline 5 & \text { Fernale } & \text { Math } \\\hline 6 & \text { Male } & \text { Science } \\\hline 7 & \text { Male } & \text { Math } \\\hline 8 & \text { Male } & \text { Math } \\\hline 9 & \text { Fernale } & \text { Arts } \\\hline 10 & \text { Male } & \text { Arts } \\\hline 11 & \text { Male } & \text { Science } \\\hline 12 & \text { Fernale } & \text { Science } \\\hline 13 & \text { Fernale } & \text { Math } \\\hline 14 & \text { Male } & \text { Arts } \\\hline 15 & \text { Fernale } & \text { Arts }\\\hline \end{array}$

-Identify the correct statement from the following.

A) The events that a respondent is female and chooses Math are not independent.
B) All events are independent of each other.
C) The events that a respondent is male and chooses Science are not dependent.
D) The events that a respondent is female and that she chooses Science are independent.

Question 6

Use the data given below to answer the following questions).
15 students were asked to choose between the broad categories of Arts, Science, and Math as their preferred area of study. $\begin{array} { | c | c | c | } \hline \text { Respondent } & \text { Gender } & \text { Preference } \\\hline 1 & \text { Fernale } & \text { Arts } \\\hline 2 & \text { Male } & \text { Science } \\\hline 3 & \text { Male } & \text { Math } \\\hline 4 & \text { Fernale } & \text { Arts } \\\hline 5 & \text { Fernale } & \text { Math } \\\hline 6 & \text { Male } & \text { Science } \\\hline 7 & \text { Male } & \text { Math } \\\hline 8 & \text { Male } & \text { Math } \\\hline 9 & \text { Fernale } & \text { Arts } \\\hline 10 & \text { Male } & \text { Arts } \\\hline 11 & \text { Male } & \text { Science } \\\hline 12 & \text { Fernale } & \text { Science } \\\hline 13 & \text { Fernale } & \text { Math } \\\hline 14 & \text { Male } & \text { Arts } \\\hline 15 & \text { Fernale } & \text { Arts }\\\hline \end{array}$

-Which of the following will hold true if events X and Y are independent?

A) PX) = PY) × PX and Y)
B) PY) = PX|Y)
C) PX and Y) = [PX) + PY)] - PX or Y)
D) PX) = PX|Y)

Question 7

The of a random variable corresponds to the notion of the mean, or average, for a sample.

A) mode
B) variance
C) expected value
D) standard deviation

Question 8

The binominal distribution:

A) is a discrete distribution used to model the number of occurrences in some unit of measure.
B) assumes that the average number of occurrences per unit is a constant and that occurrences are independent.
C) is symmetric irrespective of the value of the probability of success.
D) models n independent replications of a Bernoulli experiment, each with a probability p of success.

Question 9

Use the data given below to answer the following questions).
On an average, the number of students that choose to study Arts subjects at Greyin Tide University is 17 each year. Hint: Use Poisson distribution formula).
A probability density function:

A) is the probability distribution of discrete outcomes.
B) suggests that the probability that a random variable assumes a specific value must be positive.
C) characterizes outcomes of a continuous random variable.
D) can yield negative values depending on the values of the random variable, X.

Question 10

Use the data given below to answer the following questions).
15 students were asked to choose between the broad categories of Arts, Science, and Math as their preferred area of study. $\begin{array} { | c | c | c | } \hline \text { Respondent } & \text { Gender } & \text { Preference } \\\hline 1 & \text { Fernale } & \text { Arts } \\\hline 2 & \text { Male } & \text { Science } \\\hline 3 & \text { Male } & \text { Math } \\\hline 4 & \text { Fernale } & \text { Arts } \\\hline 5 & \text { Fernale } & \text { Math } \\\hline 6 & \text { Male } & \text { Science } \\\hline 7 & \text { Male } & \text { Math } \\\hline 8 & \text { Male } & \text { Math } \\\hline 9 & \text { Fernale } & \text { Arts } \\\hline 10 & \text { Male } & \text { Arts } \\\hline 11 & \text { Male } & \text { Science } \\\hline 12 & \text { Fernale } & \text { Science } \\\hline 13 & \text { Fernale } & \text { Math } \\\hline 14 & \text { Male } & \text { Arts } \\\hline 15 & \text { Fernale } & \text { Arts }\\\hline \end{array}$

-Use the multiplication law of probability to compute the probability that the respondent is male and prefers Math.

A) $\frac { 8 } { 75 }$
B) $\frac { 3 } { 8 }$
C) $\frac { 3 } { 225 }$
D) $\frac { 3 } { 15 }$

Question 11

Use the data given below to answer the following questions).
15 students were asked to choose between the broad categories of Arts, Science, and Math as their preferred area of study. $\begin{array} { | c | c | c | } \hline \text { Respondent } & \text { Gender } & \text { Preference } \\\hline 1 & \text { Fernale } & \text { Arts } \\\hline 2 & \text { Male } & \text { Science } \\\hline 3 & \text { Male } & \text { Math } \\\hline 4 & \text { Fernale } & \text { Arts } \\\hline 5 & \text { Fernale } & \text { Math } \\\hline 6 & \text { Male } & \text { Science } \\\hline 7 & \text { Male } & \text { Math } \\\hline 8 & \text { Male } & \text { Math } \\\hline 9 & \text { Fernale } & \text { Arts } \\\hline 10 & \text { Male } & \text { Arts } \\\hline 11 & \text { Male } & \text { Science } \\\hline 12 & \text { Fernale } & \text { Science } \\\hline 13 & \text { Fernale } & \text { Math } \\\hline 14 & \text { Male } & \text { Arts } \\\hline 15 & \text { Fernale } & \text { Arts }\\\hline \end{array}$

-What is the probability that a respondent prefers Science given that the respondent is female?

A) $\frac { 1 } { 15 }$
B) $\frac { 1 } { 7 }$
C) $\frac { 7 } { 15 }$
D) 4
7

Question 12

Which of the following is a weighted average of the squared deviations from the expected value?

A) skewness
B) mean difference
C) kurtosis
D) variance

Question 13

Use the data given below to answer the following questions).
On an average, the number of students that choose to study Arts subjects at Greyin Tide University is 17 each year. Hint: Use Poisson distribution formula).
What is the probability that exactly 13 students will take up Arts in the coming year?

A) 0.08795
B) 0.20087
C) 0.04798
D) 0.06585

Question 14

For a discrete random variable X, which of the following computes the expected value?

A) $\sum f \left( x _ { i } \right)$
B) $\sum _ { i = 1 } ^ { \infty } x _ { i } f \left( x _ { i } \right)$
C) $\sum _ { i = 1 } ^ { \infty } x _ { i }$
D) $\sum ^ { f ( x ) }$

Question 15

Use the data given below to answer the following questions).
On an average, the number of students that choose to study Arts subjects at Greyin Tide University is 17 each year. Hint: Use Poisson distribution formula).
Which of the following characterizes a continuous random variable for which all outcomes between some minimum and maximum value are equally likely?

A) exponential distribution
B) uniform distribution
C) normal distribution
D) lognormal distribution

Question 16

Use the data given below to answer the following questions).
On an average, the number of students that choose to study Arts subjects at Greyin Tide University is 17 each year. Hint: Use Poisson distribution formula).
What is the probability of 8 takers or less for Arts subjects in the coming year?

A) 0.00716
B) 0.01260
C) 0.03164
D) 0.04744

Question 17

Consider an event X comprised of three outcomes whose probabilities are $\frac { 9 } { 18 }$ , $\frac { 1 } { 18 }$ , and $\frac { 6 } { 18 }$ . Compute the probability of the complement of the event.

A) $\frac { 2 } { 18 }$
B) $\frac { 1 } { 2 }$
C) $\frac { 16 } { 3 }$
D) $\frac { 16 } { 18 }$

Question 18

Use the data given below to answer the following questions).
On an average, the number of students that choose to study Arts subjects at Greyin Tide University is 17 each year. Hint: Use Poisson distribution formula).
Which of the following is true of normal distributions?

A) The mean, median, and mode are all equal.
B) Its measure of skewness is always greater than 1.
C) The range of the random variable X is bounded.
D) Mathematical formulas make it easier to compute normal distributions.

Question 19

Use the data given below to answer the following questions).
15 students were asked to choose between the broad categories of Arts, Science, and Math as their preferred area of study. $\begin{array} { | c | c | c | } \hline \text { Respondent } & \text { Gender } & \text { Preference } \\\hline 1 & \text { Fernale } & \text { Arts } \\\hline 2 & \text { Male } & \text { Science } \\\hline 3 & \text { Male } & \text { Math } \\\hline 4 & \text { Fernale } & \text { Arts } \\\hline 5 & \text { Fernale } & \text { Math } \\\hline 6 & \text { Male } & \text { Science } \\\hline 7 & \text { Male } & \text { Math } \\\hline 8 & \text { Male } & \text { Math } \\\hline 9 & \text { Fernale } & \text { Arts } \\\hline 10 & \text { Male } & \text { Arts } \\\hline 11 & \text { Male } & \text { Science } \\\hline 12 & \text { Fernale } & \text { Science } \\\hline 13 & \text { Fernale } & \text { Math } \\\hline 14 & \text { Male } & \text { Arts } \\\hline 15 & \text { Fernale } & \text { Arts }\\\hline \end{array}$

-Which of the following is a continuous random variable?

A) the outcomes of rolling two dice
B) the time to complete a specific task
C) the number of new hires in a year
D) the number of hits on a Web site link

Question 20

Which of the following is true about variance?

A) It measures the uncertainty of a random variable.
B) Higher variance implies low uncertainty.
C) It is the square root of a random variable's standard deviation.
D) It is the weighted average of all possible outcomes.

Question 21

What are the properties of a probability mass function?

Question 22

The Excel function NORM.DIST finds probabilities for the standard normal distribution.

Question 23

How are probability, experiment, outcome, and sample space related to each other?

Question 24

Use the data given below to answer the following questions).
The profit from selling folding tables varies uniformly each quarter between $1,500 and $2,300.
What is the probability that profit will be between $2,000 and $2,200?

Question 25

Explain the similarity between the formulas for computing population mean and expected value of discrete random variable X.

Question 26

The expected value and variance of the Poisson distribution are equal to λ.

Question 27

Use the data given below to answer the following questions).
At a casino, a combination of two spinners is used to decide the winner based on the sum of scores from spinning. The spinners each have four colored spaces - red, yellow, blue, and green. Red = 1, Yellow = 2, Blue = 3, and Green = 4
What is the probability that the spinners land on colors summing up to exactly 3?

Question 28

If PX|Y) = PX), it implies that event Y is dependent on event X.

Question 29

Use the data given below to answer the following questions).
The profit from selling folding tables varies uniformly each quarter between $1,500 and $2,300.
What is the probability that profit will be less than $1,900?

Question 30

Use the data given below to answer the following questions).
On an average, the number of students that choose to study Arts subjects at Greyin Tide University is 17 each year. Hint: Use Poisson distribution formula).
The exponential distribution:

A) has the density function f x) = e-λx , for x ≥ 0.
B) models the time between randomly occurring events.
C) has an expected value λ.
D) is described by the familiar bell-shaped curve.

Question 31

Use the data given below to answer the following questions).
At a casino, a combination of two spinners is used to decide the winner based on the sum of scores from spinning. The spinners each have four colored spaces - red, yellow, blue, and green. Red = 1, Yellow = 2, Blue = 3, and Green = 4
Compute the expected value of the random variable that denotes the possible summed scores from the two spinners.

Question 32

How can normal probabilities be computed using Excel? Given the cumulative probability, how can the value of the random variable be found?

Question 33

Use the data given below to answer the following questions).
At a casino, a combination of two spinners is used to decide the winner based on the sum of scores from spinning. The spinners each have four colored spaces - red, yellow, blue, and green. Red = 1, Yellow = 2, Blue = 3, and Green = 4
Compute the variance of the random variable that denotes the possible summed scores from the two spinners.