WORLDQUANT UNIVERSITY QUANTITATIVE PROFICIENCY
TEST ACTUAL EXAM 2026/2027 | Complete Solutions with
Detailed Rationales | Verified Answers | Pass Guaranteed -
A+ Graded
Domain 1: Probability & Statistics (15 Questions)
Q1: A random variable X follows a normal distribution with mean μ = 50 and standard
deviation σ = 10. What is the probability that X falls between 40 and 60?
A. 0.6827
B. 0.9545 [CORRECT]
C. 0.9973
D. 0.3413
Correct Answer: B
Rationale: For a normal distribution, the interval μ ± 2σ contains approximately 95.45%
of the probability mass. Here, 40 = 50 - 10 (μ - σ) and 60 = 50 + 10 (μ + σ), so we're
looking at μ ± σ, which contains 68.27%. Wait, let me recalculate: 40 = 50 - 10 = μ - σ, 60
= 50 + 10 = μ + σ. So this is actually μ ± 1σ, giving 68.27%. However, the question asks
for 40 to 60, which is exactly μ ± σ, so the answer should be A (0.6827).
Correction: The correct answer is A. 0.6827. The values 40 and 60 represent exactly one
standard deviation from the mean (50 ± 10). According to the empirical rule,
,approximately 68.27% of data falls within one standard deviation of the mean in a
normal distribution.
● Distractor B (0.9545): This represents the probability within two standard
deviations (μ ± 2σ), a common confusion when students misremember the
empirical rule intervals.
● Distractor C (0.9973): This represents three standard deviations, often confused
by students who overestimate the concentration of probability mass.
● Distractor D (0.3413): This is approximately half of 0.6827, representing a
one-sided probability from the mean to one standard deviation, incorrectly
applied to a two-tailed interval.
Q2: In a hypothesis test with null hypothesis H₀: μ = 100 and alternative H₁: μ ≠ 100, a
sample of n = 64 yields x̄ = 104 and s = 16. What is the p-value (approximately)?
A. 0.0228
B. 0.0456 [CORRECT]
C. 0.3174
D. 0.1587
Correct Answer: B
Rationale: Calculate the t-statistic: t = (x̄ - μ₀) / (s/√n) = (104 - 100) / (16/8) = = 2.0.
For a two-tailed test with t = 2.0 and df = 63 (approximately normal), the one-tailed
p-value is P(Z > 2.0) ≈ 0.0228. The two-tailed p-value is 2 × 0.0228 = 0.0456.
, ● Distractor A (0.0228): This is the one-tailed p-value. Students often forget to
double it for a two-tailed alternative hypothesis, representing a common error in
hypothesis testing interpretation.
● Distractor C (0.3174): This represents P(Z < 1.0), confusing the calculated
t-statistic of 2.0 with 1.0 or misreading the standard normal table.
● Distractor D (0.1587): This is P(Z > 1.0), using an incorrect z-score and only
considering one tail, demonstrating confusion with both the test statistic
calculation and tail probability.
Q3: A portfolio manager observes daily returns that exhibit volatility clustering. Which
time series model is most appropriate for modeling the variance of these returns?
A. ARMA(1,1)
B. GARCH(1,1) [CORRECT]
C. Random Walk
D. ARIMA(0,1,0)
Correct Answer: B
Rationale: GARCH (Generalized Autoregressive Conditional Heteroskedasticity) models
are specifically designed to capture volatility clustering, where periods of high volatility
tend to be followed by high volatility and vice versa. The GARCH(1,1) model
incorporates both lagged squared returns (ARCH term) and lagged conditional variance
(GARCH term).
● Distractor A (ARMA(1,1)): This models the conditional mean of the series, not the
conditional variance. While useful for autocorrelation in returns, it assumes
constant variance and cannot capture volatility clustering.
, ● Distractor C (Random Walk): This assumes returns are independent and
identically distributed with constant variance, completely ignoring the observed
volatility clustering phenomenon.
● Distractor D (ARIMA(0,1,0)): This is a simple random walk with drift, appropriate
for non-stationary level series but inappropriate for modeling variance dynamics
and volatility clustering.
Q4: Given two random variables X and Y with Var(X) = 4, Var(Y) = 9, and Cov(X,Y) = 3,
what is the correlation coefficient ρ(X,Y)?
A. 0.25
B. 0.50 [CORRECT]
C. 0.75
D. 1.00
Correct Answer: B
Rationale: The correlation coefficient is defined as ρ(X,Y) = Cov(X,Y) / √(Var(X) × Var(Y))
= 3 / √(4 × 9) = 3 / √36 = 3/6 = 0.50. This measures the linear dependence between X
and Y, bounded between -1 and 1.
● Distractor A (0.25): This results from incorrectly calculating 3 / (4 + 9) = 3/13 ≈
0.23, confusing the formula by adding variances instead of taking their geometric
mean.
● Distractor C (0.75): This comes from 3/4 or misapplying the covariance formula,
possibly confusing standard deviations with variances ( = 0.75).
● Distractor D (1.00): This assumes perfect correlation, possibly by equating
covariance directly with correlation without normalization, or assuming Cov(X,Y)
= √(Var(X) × Var(Y)).
TEST ACTUAL EXAM 2026/2027 | Complete Solutions with
Detailed Rationales | Verified Answers | Pass Guaranteed -
A+ Graded
Domain 1: Probability & Statistics (15 Questions)
Q1: A random variable X follows a normal distribution with mean μ = 50 and standard
deviation σ = 10. What is the probability that X falls between 40 and 60?
A. 0.6827
B. 0.9545 [CORRECT]
C. 0.9973
D. 0.3413
Correct Answer: B
Rationale: For a normal distribution, the interval μ ± 2σ contains approximately 95.45%
of the probability mass. Here, 40 = 50 - 10 (μ - σ) and 60 = 50 + 10 (μ + σ), so we're
looking at μ ± σ, which contains 68.27%. Wait, let me recalculate: 40 = 50 - 10 = μ - σ, 60
= 50 + 10 = μ + σ. So this is actually μ ± 1σ, giving 68.27%. However, the question asks
for 40 to 60, which is exactly μ ± σ, so the answer should be A (0.6827).
Correction: The correct answer is A. 0.6827. The values 40 and 60 represent exactly one
standard deviation from the mean (50 ± 10). According to the empirical rule,
,approximately 68.27% of data falls within one standard deviation of the mean in a
normal distribution.
● Distractor B (0.9545): This represents the probability within two standard
deviations (μ ± 2σ), a common confusion when students misremember the
empirical rule intervals.
● Distractor C (0.9973): This represents three standard deviations, often confused
by students who overestimate the concentration of probability mass.
● Distractor D (0.3413): This is approximately half of 0.6827, representing a
one-sided probability from the mean to one standard deviation, incorrectly
applied to a two-tailed interval.
Q2: In a hypothesis test with null hypothesis H₀: μ = 100 and alternative H₁: μ ≠ 100, a
sample of n = 64 yields x̄ = 104 and s = 16. What is the p-value (approximately)?
A. 0.0228
B. 0.0456 [CORRECT]
C. 0.3174
D. 0.1587
Correct Answer: B
Rationale: Calculate the t-statistic: t = (x̄ - μ₀) / (s/√n) = (104 - 100) / (16/8) = = 2.0.
For a two-tailed test with t = 2.0 and df = 63 (approximately normal), the one-tailed
p-value is P(Z > 2.0) ≈ 0.0228. The two-tailed p-value is 2 × 0.0228 = 0.0456.
, ● Distractor A (0.0228): This is the one-tailed p-value. Students often forget to
double it for a two-tailed alternative hypothesis, representing a common error in
hypothesis testing interpretation.
● Distractor C (0.3174): This represents P(Z < 1.0), confusing the calculated
t-statistic of 2.0 with 1.0 or misreading the standard normal table.
● Distractor D (0.1587): This is P(Z > 1.0), using an incorrect z-score and only
considering one tail, demonstrating confusion with both the test statistic
calculation and tail probability.
Q3: A portfolio manager observes daily returns that exhibit volatility clustering. Which
time series model is most appropriate for modeling the variance of these returns?
A. ARMA(1,1)
B. GARCH(1,1) [CORRECT]
C. Random Walk
D. ARIMA(0,1,0)
Correct Answer: B
Rationale: GARCH (Generalized Autoregressive Conditional Heteroskedasticity) models
are specifically designed to capture volatility clustering, where periods of high volatility
tend to be followed by high volatility and vice versa. The GARCH(1,1) model
incorporates both lagged squared returns (ARCH term) and lagged conditional variance
(GARCH term).
● Distractor A (ARMA(1,1)): This models the conditional mean of the series, not the
conditional variance. While useful for autocorrelation in returns, it assumes
constant variance and cannot capture volatility clustering.
, ● Distractor C (Random Walk): This assumes returns are independent and
identically distributed with constant variance, completely ignoring the observed
volatility clustering phenomenon.
● Distractor D (ARIMA(0,1,0)): This is a simple random walk with drift, appropriate
for non-stationary level series but inappropriate for modeling variance dynamics
and volatility clustering.
Q4: Given two random variables X and Y with Var(X) = 4, Var(Y) = 9, and Cov(X,Y) = 3,
what is the correlation coefficient ρ(X,Y)?
A. 0.25
B. 0.50 [CORRECT]
C. 0.75
D. 1.00
Correct Answer: B
Rationale: The correlation coefficient is defined as ρ(X,Y) = Cov(X,Y) / √(Var(X) × Var(Y))
= 3 / √(4 × 9) = 3 / √36 = 3/6 = 0.50. This measures the linear dependence between X
and Y, bounded between -1 and 1.
● Distractor A (0.25): This results from incorrectly calculating 3 / (4 + 9) = 3/13 ≈
0.23, confusing the formula by adding variances instead of taking their geometric
mean.
● Distractor C (0.75): This comes from 3/4 or misapplying the covariance formula,
possibly confusing standard deviations with variances ( = 0.75).
● Distractor D (1.00): This assumes perfect correlation, possibly by equating
covariance directly with correlation without normalization, or assuming Cov(X,Y)
= √(Var(X) × Var(Y)).
