,MATH 110 Module 3 Exam - Introduction to
Statistics Portage Learning - Latest (26/27)
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1. Which of the following correctly distinguishes between a discrete and a continuous random
variable?
A) A discrete variable has an infinite number of possible values; a continuous variable has a finite
number.
B) A discrete variable takes values that are countable; a continuous variable takes values that are
measurable on a continuum.
C) A discrete variable is always qualitative; a continuous variable is always quantitative.
D) A discrete variable can take any value in an interval; a continuous variable can only take whole
numbers.
Correct Answer: B) A discrete variable takes values that are countable; a continuous variable takes
values that are measurable on a continuum.
Rationale: Discrete variables have a countable number of distinct values (e.g., number of cars), while
continuous variables can take any value within a range (e.g., weight). Option A reverses the
definitions. Option C is false because both can be quantitative. Option D also reverses the definitions.
2. A researcher records the number of children in 200 families. This variable is best classified as:
A) Continuous quantitative
B) Discrete quantitative
C) Categorical nominal
D) Categorical ordinal
Correct Answer: B) Discrete quantitative
,Rationale: The number of children is a countable whole number (0, 1, 2, ...), making it a discrete
quantitative variable. Continuous variables take any value in an interval, and categorical variables
involve categories without numerical meaning.
3. A probability distribution for a discrete random variable must satisfy which two conditions?
A) Each probability is between 0 and 1, and the sum of all probabilities equals 1.
B) Each probability is between 0 and 1, and the sum of all probabilities equals the sample size.
C) Each probability is greater than 0, and the sum of all probabilities equals 100.
D) Each probability is less than 1, and the sum of all probabilities equals the number of outcomes.
Correct Answer: A) Each probability is between 0 and 1, and the sum of all probabilities equals 1.
Rationale: For any discrete probability distribution, all individual probabilities must lie between 0 and
1 inclusive, and their total must be exactly 1. The sum equals the sample size only for frequencies, not
probabilities.
4. The expected value of a discrete random variable is defined as:
A) The most frequently occurring value
B) The weighted average of all possible values, where the weights are the probabilities
C) The square root of the variance
D) The midpoint of the range of possible values
Correct Answer: B) The weighted average of all possible values, where the weights are the
probabilities
Rationale: The expected value (mean) of a discrete random variable is calculated as the sum of each
value multiplied by its probability. The mode is the most frequent value, the standard deviation is the
square root of the variance, and the range midpoint is unrelated.
5. A fair die is rolled once. Let X be the number showing on the die. What is the expected value of X?
A) 2.5
, B) 3.0
C) 3.5
D) 4.0
Correct Answer: C) 3.5
Rationale: For a fair six-sided die, each outcome 1 through 6 has probability 1/6. The expected value is
(1+2+3+4+5+6)/6 = 21/6 = 3.5. This is the average value over many rolls.
6. A random variable X has the following probability distribution: P(X=0)=0.2, P(X=1)=0.5, P(X=2)=0.3.
What is the expected value of X?
A) 0.8
B) 1.0
C) 1.1
D) 1.5
Correct Answer: C) 1.1
Rationale: The expected value is calculated as (0)(0.2) + (1)(0.5) + (2)(0.3) = 0 + 0.5 + 0.6 = 1.1. Options
A, B, and D result from incorrect arithmetic or misapplication of the formula.
7. For a discrete probability distribution, the variance measures:
A) The center of the distribution
B) The spread or variability of the distribution around the mean
C) The most likely outcome
D) The skewness of the distribution
Correct Answer: B) The spread or variability of the distribution around the mean
Statistics Portage Learning - Latest (26/27)
(PDF)
1. Which of the following correctly distinguishes between a discrete and a continuous random
variable?
A) A discrete variable has an infinite number of possible values; a continuous variable has a finite
number.
B) A discrete variable takes values that are countable; a continuous variable takes values that are
measurable on a continuum.
C) A discrete variable is always qualitative; a continuous variable is always quantitative.
D) A discrete variable can take any value in an interval; a continuous variable can only take whole
numbers.
Correct Answer: B) A discrete variable takes values that are countable; a continuous variable takes
values that are measurable on a continuum.
Rationale: Discrete variables have a countable number of distinct values (e.g., number of cars), while
continuous variables can take any value within a range (e.g., weight). Option A reverses the
definitions. Option C is false because both can be quantitative. Option D also reverses the definitions.
2. A researcher records the number of children in 200 families. This variable is best classified as:
A) Continuous quantitative
B) Discrete quantitative
C) Categorical nominal
D) Categorical ordinal
Correct Answer: B) Discrete quantitative
,Rationale: The number of children is a countable whole number (0, 1, 2, ...), making it a discrete
quantitative variable. Continuous variables take any value in an interval, and categorical variables
involve categories without numerical meaning.
3. A probability distribution for a discrete random variable must satisfy which two conditions?
A) Each probability is between 0 and 1, and the sum of all probabilities equals 1.
B) Each probability is between 0 and 1, and the sum of all probabilities equals the sample size.
C) Each probability is greater than 0, and the sum of all probabilities equals 100.
D) Each probability is less than 1, and the sum of all probabilities equals the number of outcomes.
Correct Answer: A) Each probability is between 0 and 1, and the sum of all probabilities equals 1.
Rationale: For any discrete probability distribution, all individual probabilities must lie between 0 and
1 inclusive, and their total must be exactly 1. The sum equals the sample size only for frequencies, not
probabilities.
4. The expected value of a discrete random variable is defined as:
A) The most frequently occurring value
B) The weighted average of all possible values, where the weights are the probabilities
C) The square root of the variance
D) The midpoint of the range of possible values
Correct Answer: B) The weighted average of all possible values, where the weights are the
probabilities
Rationale: The expected value (mean) of a discrete random variable is calculated as the sum of each
value multiplied by its probability. The mode is the most frequent value, the standard deviation is the
square root of the variance, and the range midpoint is unrelated.
5. A fair die is rolled once. Let X be the number showing on the die. What is the expected value of X?
A) 2.5
, B) 3.0
C) 3.5
D) 4.0
Correct Answer: C) 3.5
Rationale: For a fair six-sided die, each outcome 1 through 6 has probability 1/6. The expected value is
(1+2+3+4+5+6)/6 = 21/6 = 3.5. This is the average value over many rolls.
6. A random variable X has the following probability distribution: P(X=0)=0.2, P(X=1)=0.5, P(X=2)=0.3.
What is the expected value of X?
A) 0.8
B) 1.0
C) 1.1
D) 1.5
Correct Answer: C) 1.1
Rationale: The expected value is calculated as (0)(0.2) + (1)(0.5) + (2)(0.3) = 0 + 0.5 + 0.6 = 1.1. Options
A, B, and D result from incorrect arithmetic or misapplication of the formula.
7. For a discrete probability distribution, the variance measures:
A) The center of the distribution
B) The spread or variability of the distribution around the mean
C) The most likely outcome
D) The skewness of the distribution
Correct Answer: B) The spread or variability of the distribution around the mean
