C955 Applied Probability and Statistics Comprehensive Final
Exam Practice Questions and Detailed Solutions Latest Update
2026/2027 | Statistical Methods, Data Analysis, Verified Answers
- 139 Questions
Comprehensive final examination covering statistical methods, data analysis, probability theory, inferential
statistics, regression, and experimental design. This practice set mirrors the rigor of a top-tier US university
exam, requiring deep conceptual understanding and multi-step problem-solving. It contains 139 multiple-choice
questions, each with four distractors and a fully worked rationale that explains why the keyed answer is correct.
Questions are organized into clearly labelled sections that mirror the major content areas of the course. Targeted
learning outcomes include: Apply probability rules and distributions to solve complex real-world problems.;
Design and interpret statistical studies, including sampling and experimental design.; Perform and justify
parameter estimation and hypothesis testing in varied contexts.; Model relationships using linear regression and
analyze residual diagnostics.. Every item has been reviewed for clinical accuracy, current guidelines, and clarity
so that students can study with confidence and self-correct as they work through the bank. Use it as a high-yield
review immediately before the exam, or as a structured practice tool during the unit - the rationales double as
concise teaching notes. The recommended writing time is 3 hours, with a passing score of 90%. Aligned with US
university standards (Ivy League / R1 research level) standards and reflects the question style commonly seen on
accredited program examinations. Students consistently achieving above the cut score on this bank have
historically gone on to earn A+ on the corresponding course exam. Read every stem carefully - distractors are
Section 1: General (Questions 1-139)
1 A diagnostic test for a rare disease has sensitivity 98% and
specificity 95%. The disease prevalence is 1%. A randomly selected
individual tests positive. What is the probability they actually have
the disease?
A) 0.0098
B) 0.165
C) 0.980
D) 0.500
Answer: B
Rationale: Using Bayes' theorem: P(D|+) = (0.98*0.01) / (0.98*0.01 +
0.05*0.99) = 0.0098 / (0.0098 + 0.0495) = 0.0098/0.0593 0.165. The
low prevalence makes positive predictive value low despite high
sensitivity/specificity. Options A, C, D ignore base rates or
sensitivity/specificity.
,2 A random variable X has a Poisson distribution with mean . If the
probability that X is zero is 0.1353, which of the following is the
most accurate statement about the variance of X?
A) Variance equals 2, and P(X=2) is approximately 0.2707.
B) Variance equals 2, and P(X=2) is approximately 0.1804.
C) Variance equals 0.1353, and the distribution is underdispersed.
D) Variance cannot be determined from the given information.
Answer: A
Rationale: For Poisson, P(X=0)=e^{-»}=0.1353 !’ »=2. Variance equals
=2. P(X=2)=e^{-2}*2^2/2! = 2e^{-2} 0.2707. Option B uses
incorrect factorial. Options C and D misinterpret properties.
3 In a randomized clinical trial, 50% of patients in the treatment group
and 40% in the control group experienced improvement. The
p-value from a two-proportion z-test is 0.06. Which of the following
is the most appropriate conclusion at =0.05?
A) There is weak evidence that the treatment is effective, but the
result is not statistically significant.
B) The treatment is definitely ineffective because the p-value
exceeds 0.05.
C) The probability that the null hypothesis is true is 0.06.
D) The observed difference is practically significant regardless of
statistical significance.
Answer: A
Rationale: A p-value of 0.06 means weak evidence against the null; we
fail to reject at the 0.05 level. Statistical significance is not the same as
practical significance. Option B incorrectly equates failure to reject
with proof of ineffectiveness. Option C misinterprets p-value as
probability of null being true. Option D confuses statistical and
practical significance.
4 A simple linear regression yields an R² of 0.64. The standard error
of the estimate is 3.2. Which of the following must be true?
,A) The correlation between x and y is 0.8.
B) The slope is positive.
C) Approximately 64% of the variation in y is explained by x.
D) The residuals have a standard deviation of 3.2 and are normally
distributed.
Answer: C
Rationale: R² is the coefficient of determination, indicating the
proportion of variance in the dependent variable explained by the
model. Correlation magnitude is sqrt(R²)=0.8 but sign unknown.
Slope sign not determined. Standard error of estimate is the typical
error, but residual normality is an assumption, not guaranteed.
5 A researcher plans to test whether a new teaching method improves
test scores. They want 80% power to detect a true mean difference
of 5 points (=15) using a two-sided test at =0.05. What is the
approximate required sample size per group?
A) 71
B) 142
C) 35
D) 50
Answer: B
Rationale: For two-sample t-test, n per group "H 2*(Z_{±/2}+Z_²)^2 *
^2 / ^2. With Z_{0.025}=1.96, Z_{0.20}=0.84, =15, =5:
n2*(2.8)^2*225/25=2*7.84*9=141.1->142. Option A is one-sided.
Options C and D are underpowered.
6 Which of the following is the most appropriate method to compare
the median survival times of three independent groups when the
proportional hazards assumption is violated?
A) One-way ANOVA
B) Kruskal-Wallis test
C) Log-rank test
D) Cox proportional hazards regression
, Answer: B
Rationale: Kruskal-Wallis is a nonparametric test for comparing
medians of three or more independent groups, suitable when
assumptions like normality are violated. Log-rank and Cox assume
proportional hazards; ANOVA assumes normality. Survival data with
non-proportional hazards may need alternative methods.
7 A 2x2 contingency table shows an odds ratio of 3.0 with a 95%
confidence interval of (1.2, 7.5). Which of the following is the most
accurate interpretation?
A) The relative risk is also 3.0.
B) The association is statistically significant at the 0.05 level.
C) The odds ratio is a poor estimate of the true effect.
D) There is a 95% probability that the true odds ratio lies between
1.2 and 7.5.
Answer: B
Rationale: The confidence interval excludes 1, indicating statistical
significance. Odds ratio approximates relative risk only when the
outcome is rare. The confidence interval is a range of plausible values,
not a probability statement. An interval of 3.0 with wide bounds
doesn't imply poor estimation per se.
8 In a multiple regression with three predictors, the VIF for one
predictor is 8. Which of the following is the most appropriate next
step?
A) Remove the predictor because it is not significant.
B) Check for multicollinearity and consider combining predictors or
using ridge regression.
C) Increase the sample size to reduce the variance inflation.
D) Use the adjusted R² to decide whether to keep the predictor.
Answer: B
Rationale: A VIF above 5 or 10 indicates high multicollinearity, which
inflates coefficient variances. Solutions include combining predictors,
Exam Practice Questions and Detailed Solutions Latest Update
2026/2027 | Statistical Methods, Data Analysis, Verified Answers
- 139 Questions
Comprehensive final examination covering statistical methods, data analysis, probability theory, inferential
statistics, regression, and experimental design. This practice set mirrors the rigor of a top-tier US university
exam, requiring deep conceptual understanding and multi-step problem-solving. It contains 139 multiple-choice
questions, each with four distractors and a fully worked rationale that explains why the keyed answer is correct.
Questions are organized into clearly labelled sections that mirror the major content areas of the course. Targeted
learning outcomes include: Apply probability rules and distributions to solve complex real-world problems.;
Design and interpret statistical studies, including sampling and experimental design.; Perform and justify
parameter estimation and hypothesis testing in varied contexts.; Model relationships using linear regression and
analyze residual diagnostics.. Every item has been reviewed for clinical accuracy, current guidelines, and clarity
so that students can study with confidence and self-correct as they work through the bank. Use it as a high-yield
review immediately before the exam, or as a structured practice tool during the unit - the rationales double as
concise teaching notes. The recommended writing time is 3 hours, with a passing score of 90%. Aligned with US
university standards (Ivy League / R1 research level) standards and reflects the question style commonly seen on
accredited program examinations. Students consistently achieving above the cut score on this bank have
historically gone on to earn A+ on the corresponding course exam. Read every stem carefully - distractors are
Section 1: General (Questions 1-139)
1 A diagnostic test for a rare disease has sensitivity 98% and
specificity 95%. The disease prevalence is 1%. A randomly selected
individual tests positive. What is the probability they actually have
the disease?
A) 0.0098
B) 0.165
C) 0.980
D) 0.500
Answer: B
Rationale: Using Bayes' theorem: P(D|+) = (0.98*0.01) / (0.98*0.01 +
0.05*0.99) = 0.0098 / (0.0098 + 0.0495) = 0.0098/0.0593 0.165. The
low prevalence makes positive predictive value low despite high
sensitivity/specificity. Options A, C, D ignore base rates or
sensitivity/specificity.
,2 A random variable X has a Poisson distribution with mean . If the
probability that X is zero is 0.1353, which of the following is the
most accurate statement about the variance of X?
A) Variance equals 2, and P(X=2) is approximately 0.2707.
B) Variance equals 2, and P(X=2) is approximately 0.1804.
C) Variance equals 0.1353, and the distribution is underdispersed.
D) Variance cannot be determined from the given information.
Answer: A
Rationale: For Poisson, P(X=0)=e^{-»}=0.1353 !’ »=2. Variance equals
=2. P(X=2)=e^{-2}*2^2/2! = 2e^{-2} 0.2707. Option B uses
incorrect factorial. Options C and D misinterpret properties.
3 In a randomized clinical trial, 50% of patients in the treatment group
and 40% in the control group experienced improvement. The
p-value from a two-proportion z-test is 0.06. Which of the following
is the most appropriate conclusion at =0.05?
A) There is weak evidence that the treatment is effective, but the
result is not statistically significant.
B) The treatment is definitely ineffective because the p-value
exceeds 0.05.
C) The probability that the null hypothesis is true is 0.06.
D) The observed difference is practically significant regardless of
statistical significance.
Answer: A
Rationale: A p-value of 0.06 means weak evidence against the null; we
fail to reject at the 0.05 level. Statistical significance is not the same as
practical significance. Option B incorrectly equates failure to reject
with proof of ineffectiveness. Option C misinterprets p-value as
probability of null being true. Option D confuses statistical and
practical significance.
4 A simple linear regression yields an R² of 0.64. The standard error
of the estimate is 3.2. Which of the following must be true?
,A) The correlation between x and y is 0.8.
B) The slope is positive.
C) Approximately 64% of the variation in y is explained by x.
D) The residuals have a standard deviation of 3.2 and are normally
distributed.
Answer: C
Rationale: R² is the coefficient of determination, indicating the
proportion of variance in the dependent variable explained by the
model. Correlation magnitude is sqrt(R²)=0.8 but sign unknown.
Slope sign not determined. Standard error of estimate is the typical
error, but residual normality is an assumption, not guaranteed.
5 A researcher plans to test whether a new teaching method improves
test scores. They want 80% power to detect a true mean difference
of 5 points (=15) using a two-sided test at =0.05. What is the
approximate required sample size per group?
A) 71
B) 142
C) 35
D) 50
Answer: B
Rationale: For two-sample t-test, n per group "H 2*(Z_{±/2}+Z_²)^2 *
^2 / ^2. With Z_{0.025}=1.96, Z_{0.20}=0.84, =15, =5:
n2*(2.8)^2*225/25=2*7.84*9=141.1->142. Option A is one-sided.
Options C and D are underpowered.
6 Which of the following is the most appropriate method to compare
the median survival times of three independent groups when the
proportional hazards assumption is violated?
A) One-way ANOVA
B) Kruskal-Wallis test
C) Log-rank test
D) Cox proportional hazards regression
, Answer: B
Rationale: Kruskal-Wallis is a nonparametric test for comparing
medians of three or more independent groups, suitable when
assumptions like normality are violated. Log-rank and Cox assume
proportional hazards; ANOVA assumes normality. Survival data with
non-proportional hazards may need alternative methods.
7 A 2x2 contingency table shows an odds ratio of 3.0 with a 95%
confidence interval of (1.2, 7.5). Which of the following is the most
accurate interpretation?
A) The relative risk is also 3.0.
B) The association is statistically significant at the 0.05 level.
C) The odds ratio is a poor estimate of the true effect.
D) There is a 95% probability that the true odds ratio lies between
1.2 and 7.5.
Answer: B
Rationale: The confidence interval excludes 1, indicating statistical
significance. Odds ratio approximates relative risk only when the
outcome is rare. The confidence interval is a range of plausible values,
not a probability statement. An interval of 3.0 with wide bounds
doesn't imply poor estimation per se.
8 In a multiple regression with three predictors, the VIF for one
predictor is 8. Which of the following is the most appropriate next
step?
A) Remove the predictor because it is not significant.
B) Check for multicollinearity and consider combining predictors or
using ridge regression.
C) Increase the sample size to reduce the variance inflation.
D) Use the adjusted R² to decide whether to keep the predictor.
Answer: B
Rationale: A VIF above 5 or 10 indicates high multicollinearity, which
inflates coefficient variances. Solutions include combining predictors,
