Question 1

A senior high school student is conducting a survey about transportation to school. They record the number of minutes it takes for each student to travel from home to school each day. Based on this situation, answer the question below.

Which of the following best defines a random variable?

Accepted Answer

A numerical value from a random process

Answer

A personal preference of the student

Answer

A constant value used in calculations

Answer

A description of daily travel experiences

Question 2

A senior high school student is conducting a survey about transportation to school. They record the number of minutes it takes for each student to travel from home to school each day. Based on this situation, answer the question below.

Which of the following best defines a random variable?

What type of random variable represents the number of minutes it takes for each student to travel from home to school each day?

Accepted Answer

Continuous random variable

Question 3

Which of the following is an example of a discrete random variable?

A. The distance traveled to school in kilometers.B. The amount of water consumed in a day.C. The number of absences in a school month. D.

Accepted Answer

The number of absences in a school

Answer

The amount of water consumed in a day

Answer

The distance traveled to school in kilometers

Answer

The speed of a vehicle during travel.

Question 4

Why is the number of minutes it takes for each student to travel from home to school each day considered a continuous random variable?

Accepted Answer

It can take any value within a given time interval.

Answer

It is chosen according to student preference.

Answer

It is based only on whole numbers of minutes.

Answer

It depends on a fixed and constant travel time.

Question 5

How does a discrete random variable differ from a continuous random variable?

Accepted Answer

A discrete random variable takes countable values, while a continuous random variable is measured.

Answer

A discrete random variable is measured, while a continuous random variable is counted.

Answer

A discrete random variable is theoretical, while a continuous random variable is practical.

Answer

A discrete random variable is always larger than a continuous random variable.

Question 6

A student records the number of classmates who submit their assignments on time each day. Which of the following correctly identifies the random variable and its type?

Accepted Answer

Random variable: Number of classmates; Type: Discrete

Answer

Random variable: Daily class duration; Type: Continuous

Answer

Random variable: Number of assignments; Type: Continuous

Answer

Random variable: Submission time; Type: Continuous

Question 7

A teacher measures the amount of time (in hours) students spend reviewing for a quiz. If the random variable is the time spent reviewing, which of the following could be a possible set of values?

Accepted Answer

{0.25, 1.5, 2.75, 3.25}

Answer

{5, 10, 15, 20}

Answer

{0, 1, 2, 3, 4}

Answer

{1, 2, 3, 4}

Question 8

Consider a random variable representing the number of school activities attended by a student in a month. Which option correctly identifies the variable and its possible values?

Accepted Answer

Random variable: Number of activities; Possible values: {0, 1, 2, 3, …}

Answer

Random variable: Number of activities; Possible values: {0.5, 1.0, 1.5, …}

Answer

Random variable: Time spent in activities; Possible values: {1.2, 2.5, 3.8, …}

Answer

Random variable: Number of activities; Possible values: {1.5, 2.5, 3.5, …}

Question 9

A student tracks the number of quizzes given each day for one week and records the following data: {1, 0, 2, 1, 3, 0, 2}. Which of the following represents the possible values of the random variable “number of quizzes given in a day”?

Accepted Answer

{0, 1, 2, 3}

Answer

{1&#44; 2&#44; 3}

Answer

{0, 1, 2, 3, 4}

Answer

{0&#44; 2&#44; 3}

Question 10

A student recorded the number of homework assignments submitted by classmates in a week. The recorded values were {1, 2, 0, 3, 2, 1, 0, 2, 4, 3}. Another student claims that the possible values of the random variable “number of homework submissions” are {0, 1, 2, 3, 4}.

How would you evaluate this claim?

Accepted Answer

The claim is correct because all unique values in the data are included.

Answer

The claim is correct because the values represent a continuous random variable.

Answer

The claim is incorrect because the possible values should include numbers beyond 4.

Answer

The claim is incorrect because the value 4 was observed only once.

Question 11

What is the sum of all probabilities in the probability distribution of a discrete random variable?

Accepted Answer

1.00

Question 12

For a spinner divided into 8 equal sections, what is the probability of landing on any specific section?

Accepted Answer

1/8

Answer

1/4

Answer

1/6

Answer

1/2

Question 13

Why is the outcome of spinning a spinner with 8 equal sections considered a discrete random variable?

Accepted Answer

Because the outcomes are whole numbers

Answer

Because the outcomes depend on the player’s choice

Answer

Because the outcomes can take any value in a range

Answer

Because the outcomes are uncountable

Question 14

Which statement best describes the probability distribution of spinning a fair spinner with 8 equal sections?

Accepted Answer

Each outcome has an equal probability of occurring

Answer

The highest-numbered section has the greatest probability

Answer

Each outcome has a different probability based on its position

Answer

The probabilities of the outcomes depend on previous spins

Question 15

A game involves spinning a fair spinner with 8 equal sections numbered 1 to 8. If the random variable X represents the outcome, what is the probability that X is an even number?

Accepted Answer

1/2

Answer

3/4

Answer

1/4

Answer

1/8

Question 16

A quality control officer records the number of customer complaints received in a day. The random variable X represents the number of complaints, and its probability distribution is shown below. Use the information to answer the question that follows.

x (Number of Complaints) 0 1 2 3

P(x) 0.25 0.35 0.25 0.15

Why is the given table considered a valid probability distribution?

x (Number of Complaints) 0 1 2 3

P(x) 0.25 0.35 0.25 0.15

Which formula is used to compute the mean (μ) of the discrete random variable X?

Accepted Answer

μ = ∑ X · P(X)

Answer

μ = ∑ P(X)

Answer

μ = ∑ X

Answer

μ = ∑ X² · P(X)

Question 17

In a probability distribution, what is the sum of all probabilities?

Accepted Answer

1.0

Question 18

A quality control officer records the number of customer complaints received in a day. The random variable X represents the number of complaints, and its probability distribution is shown below. Use the information to answer the question that follows.

x (Number of Complaints) 0 1 2 3

P(x) 0.25 0.35 0.25 0.15

Why is the given table considered a valid probability distribution?

x (Number of Complaints) 0 1 2 3

P(x) 0.25 0.35 0.25 0.15

Why is the given table considered a valid probability distribution?

Accepted Answer

Because the probabilities are within the range, [0, 1] and their sum equals 1.

Answer

Because all values of X are whole numbers and not negative numbers.

Answer

Because the probabilities are whole numbers and are equal for all values of X.

Answer

Because the values of X range from 0 to 3 and their sum equals 2.

Question 19

A quality control officer records the number of customer complaints received in a day. The random variable X represents the number of complaints, and its probability distribution is shown below. Use the information to answer the question that follows.

x (Number of Complaints) 0 1 2 3

P(x) 0.25 0.35 0.25 0.15

Why is the given table considered a valid probability distribution?

x (Number of Complaints) 0 1 2 3

P(x) 0.25 0.35 0.25 0.15

What is the mean (μ) of the given probability distribution? (Round off your answer up to 2 decimal places.)

Accepted Answer

1.3

Question 20

A quality control officer records the number of customer complaints received in a day. The random variable X represents the number of complaints, and its probability distribution is shown below. Use the information to answer the question that follows.

x (Number of Complaints) 0 1 2 3

P(x) 0.25 0.35 0.25 0.15

Why is the given table considered a valid probability distribution?

x (Number of Complaints) 0 1 2 3

P(x) 0.25 0.35 0.25 0.15

Compute the variance (σ2) of the given probability distribution. (Round off your answer up to 2 decimal places.)

Accepted Answer

1.01

Question 21

The following probability distribution represents the number of defective items produced by a machine during a shift:

x (Number of Defective Items) 0 1 2 3

𝑃(x) 0.2 0.5 0.2 0.1

Given that the mean of X is 1.1, what impact would increasing the probability of X=0 to 0.4 have on the variance?

Accepted Answer

It would decrease the variance because the values are closer to the mean.

Answer

The variance cannot be determined without recalculating all probabilities.

Answer

It would increase the variance because the distribution becomes more spread out.

Answer

The variance would remain unchanged because only X=0's probability increases.

Question 22

Consider the following probability distribution for a random variable X:

x (Number of Defective Items) 0 1 2 3

𝑃(x) 0.3 0.4 0.2 0.1

If the mean of X is 1.1, how would a decrease in the probability of X=1 affect the variance?

Accepted Answer

The variance would increase because the distribution becomes more spread out.

Answer

The variance cannot be determined without recalculating the mean.

Answer

The variance would decrease because the values become closer to the mean.

Answer

The variance would stay the same because the mean is unaffected.

Question 23

The number of customer complaints received daily by a service center is modeled by the probability distribution below:

x 0 1 2 3

P(x) 0.45 0.25 0.20 0.10

If the mean is 1.0 and the variance is 1.10, which conclusion is correct?

Accepted Answer

The distribution is spread out, indicating variability in outcomes.

Answer

The variance indicates no variability.

Answer

The data are highly concentrated near the mean.

Answer

The variance value is impossible for a discrete random variable.

Question 24

Given the following probability distribution for a discrete random variable X, where X represents the number of defective items:

X (Number of Defective Items) 0 1 2 3

𝑃(x) 0.49 0.49 0.01 0.01

The mean of X is1.0, and the variance is 0.02. Which of the following best describes this distribution?

Accepted Answer

The values of X are tightly clustered around the mean, with almost all values close to 1.

Answer

The distribution shows a high degree of variability, with large deviations from the mean.

Answer

The values of X are spread out evenly across all possible outcomes.

Answer

The mean of 1.0 is unreliable because the variance is very high.

Question 25

A restaurant manager records the number of customer complaints received per day. Based on 30 days of observation, the probability distribution of complaints per day is shown below. The manager claims that the restaurant rarely receives complaints, with only 1 complaint per day on average.

Number of Students (X) 0 1 2 3 4

Probability P(X) 0.33 0.13 0.23 0.15 0.16

Determine the mean of the probability distribution and evaluate if the manager’s claim is valid.

Accepted Answer

The mean is 1.68, so the manager’s claim is not valid.

Answer

The mean cannot be determined from the given data.

Answer

The mean is 1.0, so the manager’s claim is valid.

Answer

The mean is 2.5, so the manager’s claim is not valid.

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