TEST BANK FOR STATISTICS FOR
BUSINESS AND ECONOMICS GRADED A+
Question 1
A quality control inspector tests a batch of electronic components
until they find the first defective item. If the probability of finding a
defective item on any given trial is constant at 𝑝 = 0.05, which
distribution best models the number of items inspected up to and
including the first defective one, and what is its expected value?
A) Binomial distribution with 𝐸 (𝑋) = 0.05
B) Geometric distribution with 𝐸 (𝑋) = 20
C) Poisson distribution with 𝐸 (𝑋) = 20
D) Hypergeometric distribution with 𝐸 (𝑋) = 0.05
Correct Answer: B
Rationale: The Geometric distribution models the number of
Bernoulli trials needed until the first success (or defective item
1 1
occurs). The expected value is given by 𝐸 (𝑋) = = = 20.
𝑝 0.05
Question 2
Consider two events 𝐴 and 𝐵 in a sample space where 𝑃(𝐴) =
0.40, 𝑃(𝐵) = 0.30, and 𝑃(𝐴 ∪ 𝐵) = 0.58. What can be concluded
about the relationship between events 𝐴 and 𝐵?
A) 𝐴 and 𝐵 are mutually exclusive but not independent.
,B) 𝐴 and 𝐵 are independent.
C) 𝐴 and 𝐵 are neither mutually exclusive nor independent.
D) 𝐴 and 𝐵 are mutually exclusive and independent.
Correct Answer: B
Rationale: Using the addition rule, 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) −
𝑃(𝐴 ∪ 𝐵) = 0.40 + 0.30 − 0.58 = 0.12. Since 𝑃(𝐴) × 𝑃(𝐵) =
0.40 × 0.30 = 0.12 = 𝑃(𝐴 ∩ 𝐵), the two events are independent.
They are not mutually exclusive because 𝑃(𝐴 ∩ 𝐵) ≠ 0.
Question 3
An analyst wants to select a committee of 4 members from a
department consisting of 6 managers and 8 analysts. If selection
is random without replacement, what distribution determines the
probability of selecting exactly 3 analysts, and why?
A) Binomial, because each selection is independent with constant
probability.
B) Poisson, because selections occur continuously over time.
C) Hypergeometric, because sampling occurs without
replacement from a finite population with two distinct groups.
D) Negative Binomial, because the process continues until 3
analysts are chosen.
Correct Answer: C
Rationale: The Hypergeometric distribution applies when
sampling without replacement from a finite population divided into
,two distinct groups (managers and analysts), causing trial
outcomes to be dependent.
Question 4
Suppose 𝑋 is a continuous random variable representing call
duration with mean 𝜇 = 10 minutes and standard deviation 𝜎 = 2
minutes. According to Tchebysheff's Theorem, what is the
minimum probability that a call lasts between 4 and 16 minutes?
A) At least 0.68
B) At least 0.75
C) At least 0.89
D) Exactly 0.95
Correct Answer: C
Rationale: The interval 10 ± 6 corresponds to 𝜇 ± 𝑘𝜎 with 𝑘 = 3
standard deviations (3 × 2 = 6). By Tchebysheff's Theorem,
1 1 8
𝑃(|𝑋 − 𝜇| < 𝑘𝜎) ≥ 1 − = 1 − = ≈ 0.89.
𝑘2 9 9
Question 5
Which of the following functions 𝐹 (𝑥 ) represents a valid
Cumulative Distribution Function (CDF) for a continuous random
variable 𝑋?
A) 𝐹 (𝑥 ) decreases on the interval [𝑎, 𝑏] as 𝑥 increases.
B) 𝐹 (𝑥 ) approaches ∞ as 𝑥 → ∞.
, C) 𝐹 (𝑥 ) is continuous from above, non-decreasing, with
lim 𝐹 (𝑥 ) = 0 and lim 𝐹 (𝑥 ) = 1.
𝑥→−∞ 𝑥→∞
D) 𝐹 (𝑥 ) < 0 for small negative values of 𝑥.
Correct Answer: C
Rationale: A cumulative distribution function must be
monotonically non-decreasing, non-negative, continuous from
above, and bounded between 0 and 1, approaching 0 at −∞ and
1 at ∞.
Question 6
Two continuous random variables 𝑌1 and 𝑌2 have joint probability
density function 𝑓 (𝑦1 , 𝑦2 ). Which condition guarantees that 𝑌1 and
𝑌2 are independent?
A) Cov(𝑌1 , 𝑌2 ) = 0
B) 𝑓 (𝑦1 , 𝑦2 ) = 𝑓1 (𝑦1 )𝑓2 (𝑦2 ) for all pairs (𝑦1 , 𝑦2 )
C) 𝐸 (𝑌1 𝑌2 ) = 𝐸 (𝑌1 ) + 𝐸 (𝑌2 )
D) 𝑓(𝑦1 , 𝑦2 ) ≥ 0 over the region
Correct Answer: B
Rationale: Random variables 𝑌1 and 𝑌2 are independent if and
only if their joint density function factors into the product of their
marginal density functions for all values in their domain. Zero
covariance alone does not imply independence for general
continuous distributions.
BUSINESS AND ECONOMICS GRADED A+
Question 1
A quality control inspector tests a batch of electronic components
until they find the first defective item. If the probability of finding a
defective item on any given trial is constant at 𝑝 = 0.05, which
distribution best models the number of items inspected up to and
including the first defective one, and what is its expected value?
A) Binomial distribution with 𝐸 (𝑋) = 0.05
B) Geometric distribution with 𝐸 (𝑋) = 20
C) Poisson distribution with 𝐸 (𝑋) = 20
D) Hypergeometric distribution with 𝐸 (𝑋) = 0.05
Correct Answer: B
Rationale: The Geometric distribution models the number of
Bernoulli trials needed until the first success (or defective item
1 1
occurs). The expected value is given by 𝐸 (𝑋) = = = 20.
𝑝 0.05
Question 2
Consider two events 𝐴 and 𝐵 in a sample space where 𝑃(𝐴) =
0.40, 𝑃(𝐵) = 0.30, and 𝑃(𝐴 ∪ 𝐵) = 0.58. What can be concluded
about the relationship between events 𝐴 and 𝐵?
A) 𝐴 and 𝐵 are mutually exclusive but not independent.
,B) 𝐴 and 𝐵 are independent.
C) 𝐴 and 𝐵 are neither mutually exclusive nor independent.
D) 𝐴 and 𝐵 are mutually exclusive and independent.
Correct Answer: B
Rationale: Using the addition rule, 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) −
𝑃(𝐴 ∪ 𝐵) = 0.40 + 0.30 − 0.58 = 0.12. Since 𝑃(𝐴) × 𝑃(𝐵) =
0.40 × 0.30 = 0.12 = 𝑃(𝐴 ∩ 𝐵), the two events are independent.
They are not mutually exclusive because 𝑃(𝐴 ∩ 𝐵) ≠ 0.
Question 3
An analyst wants to select a committee of 4 members from a
department consisting of 6 managers and 8 analysts. If selection
is random without replacement, what distribution determines the
probability of selecting exactly 3 analysts, and why?
A) Binomial, because each selection is independent with constant
probability.
B) Poisson, because selections occur continuously over time.
C) Hypergeometric, because sampling occurs without
replacement from a finite population with two distinct groups.
D) Negative Binomial, because the process continues until 3
analysts are chosen.
Correct Answer: C
Rationale: The Hypergeometric distribution applies when
sampling without replacement from a finite population divided into
,two distinct groups (managers and analysts), causing trial
outcomes to be dependent.
Question 4
Suppose 𝑋 is a continuous random variable representing call
duration with mean 𝜇 = 10 minutes and standard deviation 𝜎 = 2
minutes. According to Tchebysheff's Theorem, what is the
minimum probability that a call lasts between 4 and 16 minutes?
A) At least 0.68
B) At least 0.75
C) At least 0.89
D) Exactly 0.95
Correct Answer: C
Rationale: The interval 10 ± 6 corresponds to 𝜇 ± 𝑘𝜎 with 𝑘 = 3
standard deviations (3 × 2 = 6). By Tchebysheff's Theorem,
1 1 8
𝑃(|𝑋 − 𝜇| < 𝑘𝜎) ≥ 1 − = 1 − = ≈ 0.89.
𝑘2 9 9
Question 5
Which of the following functions 𝐹 (𝑥 ) represents a valid
Cumulative Distribution Function (CDF) for a continuous random
variable 𝑋?
A) 𝐹 (𝑥 ) decreases on the interval [𝑎, 𝑏] as 𝑥 increases.
B) 𝐹 (𝑥 ) approaches ∞ as 𝑥 → ∞.
, C) 𝐹 (𝑥 ) is continuous from above, non-decreasing, with
lim 𝐹 (𝑥 ) = 0 and lim 𝐹 (𝑥 ) = 1.
𝑥→−∞ 𝑥→∞
D) 𝐹 (𝑥 ) < 0 for small negative values of 𝑥.
Correct Answer: C
Rationale: A cumulative distribution function must be
monotonically non-decreasing, non-negative, continuous from
above, and bounded between 0 and 1, approaching 0 at −∞ and
1 at ∞.
Question 6
Two continuous random variables 𝑌1 and 𝑌2 have joint probability
density function 𝑓 (𝑦1 , 𝑦2 ). Which condition guarantees that 𝑌1 and
𝑌2 are independent?
A) Cov(𝑌1 , 𝑌2 ) = 0
B) 𝑓 (𝑦1 , 𝑦2 ) = 𝑓1 (𝑦1 )𝑓2 (𝑦2 ) for all pairs (𝑦1 , 𝑦2 )
C) 𝐸 (𝑌1 𝑌2 ) = 𝐸 (𝑌1 ) + 𝐸 (𝑌2 )
D) 𝑓(𝑦1 , 𝑦2 ) ≥ 0 over the region
Correct Answer: B
Rationale: Random variables 𝑌1 and 𝑌2 are independent if and
only if their joint density function factors into the product of their
marginal density functions for all values in their domain. Zero
covariance alone does not imply independence for general
continuous distributions.
