Biostatistics Quiz: Interpreting Statistical Output
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Interpreting Statistical OutputQuestion 1 of 20

A chi-square test of independence examines the relationship between smoking status (Yes/No) and lung disease (Yes/No). The output shows: χ² = 8.41, df = 1, p = 0.004, Cramer's V = 0.31. What does the Cramer's V value indicate about this relationship?

31% of the variation in lung disease is explained by smoking status
There is a moderate strength of association between smoking and lung disease
The probability of lung disease increases by 31% among smokers
31% of smokers in the sample have lung disease
The odds ratio between smoking and lung disease is approximately 0.31
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Biostatistics Quiz

Biostatistics Quiz: Interpreting Statistical Output

Practice Interpreting Statistical Output in Biostatistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Interpreting Statistical Output, giving you a quick way to practice the rules, question types, and explanations that matter most for Biostatistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A chi-square test of independence examines the relationship between smoking status (Yes/No) and lung disease (Yes/No). The output shows: χ² = 8.41, df = 1, p = 0.004, Cramer's V = 0.31. What does the Cramer's V value indicate about this relationship?

  1. 31% of the variation in lung disease is explained by smoking status
  2. There is a moderate strength of association between smoking and lung disease (correct answer)
  3. The probability of lung disease increases by 31% among smokers
  4. 31% of smokers in the sample have lung disease
  5. The odds ratio between smoking and lung disease is approximately 0.31
Explanation: When you encounter a chi-square test of independence with effect size measures, you're analyzing both whether variables are related and how strong that relationship is. Cramer's V is specifically designed to measure the strength of association between categorical variables, with values ranging from 0 (no association) to 1 (perfect association). Cramer's V = 0.31 indicates a moderate strength of association between smoking status and lung disease. The conventional interpretation is that values around 0.10 represent small effects, 0.30 moderate effects, and 0.50 large effects. This value tells you that while smoking and lung disease are significantly related (p = 0.004), the association is moderately strong rather than weak or very strong. Answer A incorrectly interprets Cramer's V as explained variance (like R²). While both range from 0 to 1, Cramer's V measures association strength, not the percentage of variance explained. Answer C confuses the effect size with a probability or risk ratio – Cramer's V doesn't tell you how much risk increases. Answer D treats the statistic as a proportion of the sample, but 0.31 has nothing to do with what percentage of smokers have lung disease. Remember that effect size measures like Cramer's V are crucial for interpreting practical significance beyond statistical significance. A significant p-value tells you there's likely a real relationship, but the effect size tells you whether that relationship matters in practical terms. Always look for these complementary statistics together.

Question 2

A correlation analysis between BMI and blood pressure yields r = 0.34, n = 150, p = 0.002, 95% CI for r = (0.18, 0.48). A colleague claims this proves that higher BMI causes higher blood pressure. What does the statistical output actually support?

  1. The output confirms causation since the correlation is statistically significant with p = 0.002
  2. The output suggests causation since 34% of blood pressure variation is explained by BMI
  3. The output shows a moderate positive association but cannot establish causal direction (correct answer)
  4. The output proves reverse causation since the confidence interval excludes zero
  5. The output shows no meaningful relationship since r < 0.5 indicates weak correlation
Explanation: When you encounter correlation analysis results, remember that correlation never establishes causation, regardless of statistical significance or effect size. This is a fundamental principle in biostatistics that researchers must always keep in mind. The statistical output shows r = 0.34, indicating a moderate positive linear relationship between BMI and blood pressure. With p = 0.002 and a confidence interval of (0.18, 0.48) that excludes zero, you can confidently conclude this association is statistically significant and likely represents a real relationship in the population. However, correlation analysis only measures the strength and direction of linear association—it cannot determine which variable influences the other, or whether both are influenced by external factors. Answer C correctly identifies that while the data demonstrates a moderate positive association, it cannot establish causal direction. The correlation could mean BMI affects blood pressure, blood pressure affects BMI, or both are influenced by confounding variables like genetics, diet, or lifestyle factors. Answer A incorrectly conflates statistical significance with causation. A p-value only tells you the association is unlikely due to chance. Answer B misinterprets r² (coefficient of determination) as r—the actual explained variance is r² = (0.34)² = 11.6%, not 34%. Answer D incorrectly suggests that excluding zero in the confidence interval indicates reverse causation, when it simply confirms the association isn't due to random variation. Remember: correlation coefficients measure association strength, never causation. To establish causality, you need experimental designs, temporal relationships, or sophisticated causal inference methods.

Question 3

A study examines the effectiveness of three pain medications using a one-way ANOVA. The output shows significant differences between groups (F = 6.84, p = 0.003). Post-hoc Tukey tests were then performed.

The Tukey post-hoc results show: Drug A vs Drug B: p = 0.041, Drug A vs Drug C: p = 0.002, Drug B vs Drug C: p = 0.167. Using α = 0.05, which statement correctly summarizes the pairwise comparisons?

  1. All three drugs differ significantly from each other since the overall ANOVA was significant
  2. Drug A differs significantly from both Drug B and Drug C, but Drugs B and C do not differ (correct answer)
  3. Only the Drug A vs Drug C comparison is significant after adjusting for multiple comparisons
  4. Drug C is significantly different from both other drugs since it appears in the two significant comparisons
  5. No pairwise comparisons are significant since the Tukey test is more conservative than the overall F-test
Explanation: When you encounter ANOVA followed by post-hoc tests, remember that a significant overall F-test only tells you that some groups differ—it doesn't specify which pairs are significantly different. That's why post-hoc tests like Tukey are essential for pairwise comparisons. Looking at the Tukey results with α = 0.05: Drug A vs Drug B (p = 0.041) is significant since 0.041 < 0.05. Drug A vs Drug C (p = 0.002) is also significant since 0.002 < 0.05. However, Drug B vs Drug C (p = 0.167) is not significant since 0.167 > 0.05. This means Drug A differs from both other medications, but Drugs B and C don't significantly differ from each other. Choice A incorrectly assumes that a significant overall ANOVA means all pairwise comparisons are significant—this is a common misconception. The F-test only indicates that at least one pair differs significantly. Choice C is wrong because two comparisons (A vs B and A vs C) are significant, not just one. The Tukey test already adjusts for multiple comparisons, so you simply compare each p-value to your α level. Choice D misinterprets which drug is driving the differences. While Drug C appears in two comparisons, Drug A is actually the one that differs significantly from the others—Drug C vs Drug B wasn't significant. Study tip: In post-hoc analysis, examine each pairwise p-value independently against your α level. The drug that appears in multiple significant comparisons is typically the outlier driving the overall ANOVA result.

Question 4

A multiple regression analysis predicting weight loss (kg) includes exercise hours per week (β₁ = 1.8, p = 0.003) and dietary restriction score (β₂ = 0.4, p = 0.156). The R² = 0.32, adjusted R² = 0.28. What can be concluded about the dietary restriction variable?

  1. Dietary restriction significantly contributes to the model since R² = 0.32 is significant
  2. Dietary restriction has no relationship with weight loss since p > 0.05
  3. Dietary restriction is not significantly associated with weight loss when controlling for exercise (correct answer)
  4. Dietary restriction should be removed from the model since it reduces the adjusted R²
  5. Dietary restriction explains 40% of the variance in weight loss based on β₂ = 0.4
Explanation: When interpreting multiple regression results, you need to understand what each p-value tells you about individual predictors while controlling for other variables in the model. The dietary restriction variable has β2=0.4\beta_2 = 0.4 with p=0.156p = 0.156. Since this p-value exceeds the conventional significance threshold of 0.05, we cannot conclude that dietary restriction is significantly associated with weight loss when exercise hours are held constant. This is the key insight: in multiple regression, each coefficient's p-value tests whether that specific variable adds significant predictive value beyond what the other variables already provide. Option A incorrectly conflates the overall model's explanatory power (R² = 0.32) with the significance of individual predictors. A model can have meaningful overall fit while containing non-significant individual variables. Option B oversimplifies by suggesting no relationship exists - the p-value only indicates we cannot detect a statistically significant relationship in this sample with these controls, not that no relationship exists at all. Option D misinterprets the adjusted R². The comparison between R² (0.32) and adjusted R² (0.28) reflects the penalty for including additional variables, but this alone doesn't determine whether to remove a variable - you'd need to compare models with and without the variable. Remember this pattern: in multiple regression, each coefficient's p-value tests that variable's unique contribution while controlling for all other predictors in the model. Don't confuse overall model significance with individual predictor significance - they answer different questions.

Question 5

A two-sample t-test comparing mean recovery time between Treatment A (n₁ = 30, mean = 12.4 days) and Treatment B (n₂ = 35, mean = 15.1 days) yields: t = -2.18, df = 63, p = 0.033, 95% CI for difference = (-5.15, -0.25). What does the confidence interval indicate?

  1. Treatment A patients recover 2.7 days faster on average with 95% confidence
  2. The true difference in population means is between -5.15 and -0.25 days with 95% confidence (correct answer)
  3. Treatment A is significantly better since the entire confidence interval is negative
  4. 95% of Treatment A patients recover between 0.25 and 5.15 days faster than Treatment B patients
  5. The difference could be as small as 0.25 days or as large as 5.15 days in favor of Treatment A
Explanation: When interpreting confidence intervals for the difference between two means, you're looking at a range of plausible values for the true population parameter. This particular 95% CI tells you where the real difference between Treatment A and Treatment B means likely falls. The confidence interval (-5.15, -0.25) represents the difference calculated as Treatment A minus Treatment B (12.4 - 15.1 = -2.7). Since both bounds are negative, Treatment A patients recover faster on average. However, the key insight is that this interval captures our uncertainty about the true population difference. Answer B correctly states that we can be 95% confident the true difference in population means falls between -5.15 and -0.25 days. This is the precise definition of a confidence interval. Answer A is wrong because it claims we know the exact difference (2.7 days) with 95% confidence. The 2.7 days is just our sample estimate—the confidence interval shows this estimate could be off by quite a bit. Answer C makes a valid statistical observation (the negative interval does indicate significance), but it doesn't answer what the confidence interval "indicates" in terms of interpreting the range of values. Answer D fundamentally misunderstands confidence intervals. It incorrectly suggests the interval describes individual patient outcomes rather than the population parameter. Confidence intervals are about population means, not individual predictions. Study tip: Remember that confidence intervals quantify uncertainty about population parameters, not individual outcomes. The interval gives you a range of plausible values for the true difference, not a guarantee about specific patients.

Question 6

A clinical trial uses a chi-square test to compare cure rates between a new drug (48/60 cured) and standard therapy (35/55 cured). The output shows: χ² = 2.89, df = 1, p = 0.089, OR = 1.85, 95% CI for OR = (0.91, 3.77). How should these results be interpreted?

  1. The new drug is significantly better since OR = 1.85 > 1.0 indicates higher cure rates
  2. The new drug shows promise but the difference is not statistically significant at α = 0.05 (correct answer)
  3. The results are inconclusive since the confidence interval includes both beneficial and harmful effects
  4. The new drug is not effective since p = 0.089 > 0.05 indicates no difference in cure rates
  5. The new drug is 85% more effective than standard therapy based on the odds ratio
Explanation: When interpreting clinical trial results, you need to consider both statistical significance and clinical meaningfulness together. Statistical significance tells you whether the observed difference is likely due to chance, while effect size and confidence intervals reveal the magnitude and precision of that difference. The correct interpretation is B because the data shows a promising trend that falls short of statistical significance. The odds ratio of 1.85 indicates the new drug has 85% higher odds of cure compared to standard therapy - a clinically meaningful difference. However, with p = 0.089 > 0.05, this difference doesn't reach conventional statistical significance, meaning we cannot rule out chance as an explanation. Choice A incorrectly conflates effect size with statistical significance. While OR = 1.85 does suggest higher cure rates, the p-value shows this difference isn't statistically significant. Choice C misinterprets the confidence interval. Though the 95% CI (0.91, 3.77) includes 1.0 (indicating potential for no difference), it doesn't include values below 1.0 that would suggest harm - the lower bound is 0.91. Choice D is overly pessimistic, incorrectly concluding "no effect" when the p-value simply indicates insufficient evidence for statistical significance. Study tip: Remember that p > 0.05 doesn't prove "no difference" - it means "insufficient evidence for a difference." Look for answer choices that acknowledge promising trends while respecting statistical significance thresholds. The phrase "shows promise but not statistically significant" is often correct in borderline results like p = 0.089.

Question 7

A linear regression of systolic blood pressure (Y) on age (X) produces: Ŷ = 95.2 + 0.8X, R² = 0.41, SE of slope = 0.15, t = 5.33, p < 0.001, n = 85. What is the predicted systolic blood pressure for a 60-year-old person, and what does R² indicate?

  1. 143.2 mmHg; 41% of individuals have blood pressure explained by this model
  2. 143.2 mmHg; age explains 41% of the variance in blood pressure in this sample (correct answer)
  3. 135.2 mmHg; there is a 41% correlation between age and blood pressure
  4. 143.2 mmHg; the model correctly predicts blood pressure within 41% accuracy
  5. 135.2 mmHg; 41% of the variance in blood pressure is explained by age
Explanation: When you encounter linear regression questions, focus on two key tasks: making predictions using the regression equation and interpreting the coefficient of determination (R²). To predict systolic blood pressure for a 60-year-old, substitute X = 60 into the equation: Ŷ = 95.2 + 0.8(60) = 95.2 + 48 = 143.2 mmHg. This straightforward calculation gives you the predicted value. The R² = 0.41 tells you that age explains 41% of the variance in systolic blood pressure within this sample. Think of R² as the proportion of "spread" or variability in blood pressure that can be attributed to differences in age. Answer A incorrectly interprets R² as applying to individuals rather than variance. R² doesn't tell you what percentage of people are "explained" by the model—it quantifies how much of the overall variability is accounted for. Answer C confuses R² with correlation. While related, R² (0.41) represents explained variance, not the correlation coefficient itself. The correlation would be 0.41=0.64\sqrt{0.41} = 0.64. Answer D misinterprets R² as prediction accuracy. R² doesn't indicate that 41% of predictions are within some error range—it measures how well the model explains variability compared to simply using the mean. Study tip: Remember that R² always represents "proportion of variance explained." When you see R² = 0.41, immediately think "age accounts for 41% of the variability in blood pressure." This interpretation applies universally across regression problems.

Question 8

A Mann-Whitney U test comparing pain scores between two treatments yields: U = 89.5, n₁ = 15, n₂ = 18, p = 0.027, median₁ = 4.2, median₂ = 6.1. The normal approximation z-score is -2.21. What conclusion is most appropriate?

  1. Treatment 1 produces significantly lower mean pain scores than Treatment 2
  2. There is a significant difference in the distribution of pain scores between treatments
  3. Treatment 1 has a significantly lower median pain score with 97.3% confidence
  4. The pain score distributions differ significantly, with Treatment 1 tending toward lower scores (correct answer)
  5. Treatment 1 reduces pain scores by an average of 1.9 points compared to Treatment 2
Explanation: When you encounter a Mann-Whitney U test question, remember this test compares the distributions of two independent groups without assuming normality—it's about ranking data, not just central tendencies. The significant p-value (0.027 < 0.05) tells us there's a statistically significant difference between the two treatment groups. The negative z-score (-2.21) combined with the medians (4.2 vs 6.1) indicates Treatment 1 consistently ranks lower than Treatment 2, meaning Treatment 1 tends toward lower pain scores. This supports answer D perfectly. Let's examine why the other options miss the mark: Answer A incorrectly focuses on "mean" pain scores. The Mann-Whitney U test doesn't compare means—it's a nonparametric test that compares distributions based on ranks, making no assumptions about means. Answer B is technically correct that distributions differ significantly, but it's incomplete. It fails to specify the direction of the difference, which is crucial information provided by both the medians and the negative z-score. Answer C misinterprets the p-value as a confidence level. A p-value of 0.027 doesn't mean "97.3% confidence"—it means there's a 2.7% chance of observing this difference if no true difference existed. Also, while we can infer Treatment 1 has lower scores, the Mann-Whitney test primarily compares entire distributions, not just medians. Study tip: For Mann-Whitney questions, always remember you're comparing distributions and their tendencies, not specific parameters like means. The direction of the z-score combined with descriptive statistics tells you which group tends higher or lower.

Question 9

An independent samples t-test output shows: Group 1 (n = 25): mean = 78.4, SD = 12.2; Group 2 (n = 30): mean = 82.7, SD = 15.8; t = -1.15, df = 53, p = 0.255, 95% CI = (-11.9, 3.3). What does the 95% confidence interval suggest about the population means?

  1. Group 2's mean could be up to 11.9 points higher or Group 1's could be up to 3.3 points higher
  2. The true difference between population means is likely between -11.9 and 3.3 with 95% confidence (correct answer)
  3. Since the CI includes zero, we can conclude the population means are exactly equal
  4. 95% of individuals in Group 1 score between 11.9 points lower and 3.3 points higher than Group 2
  5. The difference is not significant since the CI width of 15.2 points indicates high variability
Explanation: When interpreting confidence intervals from independent samples t-tests, you're looking at the plausible range of values for the true difference between population means. The CI of (-11.9, 3.3) represents the difference calculated as Group 1 mean minus Group 2 mean. Answer B correctly states that we can be 95% confident the true difference between population means falls between -11.9 and 3.3 points. This means Group 1's population mean could be anywhere from 11.9 points lower to 3.3 points higher than Group 2's population mean. Answer A misinterprets the interval by treating the bounds as separate, independent statements rather than endpoints of a single range for the difference. While the numerical values seem similar, this phrasing incorrectly suggests two separate possibilities rather than a continuous range. Answer C commits a classic error: concluding that because the CI includes zero, the means are definitely equal. The CI including zero simply means we cannot rule out equality—it's one possible value among many in the range, not a definitive conclusion. Answer D confuses confidence intervals about mean differences with prediction intervals about individual scores. This completely misapplies what the CI represents—it's about population parameters, not individual observations. Study tip: Remember that confidence intervals for mean differences tell you the plausible range for how far apart the true population means could be. If the interval includes zero, you cannot conclude the groups differ significantly, but you also cannot conclude they're identical.

Question 10

A repeated measures ANOVA comparing cholesterol levels at 4 time points (baseline, 1 month, 3 months, 6 months) shows: F(3,72) = 12.8, p < 0.001, η² = 0.35. Sphericity was violated (Mauchly's p = 0.032), so Greenhouse-Geisser correction was applied: F(2.1, 50.4) = 12.8, p < 0.001. What should be concluded?

  1. The sphericity violation invalidates the results, so no conclusion can be drawn about time effects
  2. Cholesterol levels change significantly over time, and the effect explains 35% of the variance (correct answer)
  3. The Greenhouse-Geisser correction shows the time effect is no longer significant after adjustment
  4. Only 2.1 of the 4 time points show significant differences based on the corrected degrees of freedom
  5. The violation of sphericity requires using a different statistical test for valid results
Explanation: When you encounter repeated measures ANOVA with sphericity violations, the key is understanding that corrections like Greenhouse-Geisser adjust the degrees of freedom to provide valid results, rather than invalidating the analysis. The results show a significant main effect of time both before and after correction. The original F(3,72) = 12.8, p < 0.001 becomes F(2.1, 50.4) = 12.8, p < 0.001 after Greenhouse-Geisser adjustment. Notice that the F-statistic remains the same (12.8) and the p-value stays highly significant (p < 0.001). The η² = 0.35 indicates that time explains 35% of the variance in cholesterol levels, representing a large effect size. This leads us to conclude that cholesterol levels change significantly over time with substantial practical importance. Answer A is wrong because sphericity violations don't invalidate results when proper corrections are applied. The Greenhouse-Geisser correction specifically addresses this issue. Answer C misinterprets the correction—the p-value remains < 0.001, so significance is maintained, not lost. Answer D fundamentally misunderstands what the corrected degrees of freedom mean. The 2.1 doesn't indicate that only 2.1 time points differ; it's a mathematical adjustment to account for violated sphericity assumptions while testing differences across all four time points. Study tip: Remember that sphericity corrections (Greenhouse-Geisser, Huynh-Feldt) fix the problem rather than ruin your analysis. If the corrected p-value remains significant, you can confidently interpret your results using the original effect size measures.

Question 11

A survival analysis using Kaplan-Meier curves compares two treatments. The log-rank test yields: χ² = 4.67, df = 1, p = 0.031. At 24 months, Treatment A has 68% survival (95% CI: 54%-78%) and Treatment B has 45% survival (95% CI: 31%-59%). What is the most accurate interpretation?

  1. Treatment A patients live 23% longer than Treatment B patients on average
  2. There is a significant difference in survival curves, with Treatment A showing better survival (correct answer)
  3. Treatment A reduces the risk of death by 68% compared to Treatment B
  4. The 24-month survival rates are significantly different since their confidence intervals don't overlap
  5. Treatment A has a 23 percentage point survival advantage that is statistically significant
Explanation: When interpreting survival analysis results, you need to distinguish between what the statistical test tells you versus what individual survival estimates mean. The log-rank test evaluates whether the entire survival curves differ significantly over time, not just at specific time points. The log-rank test result (χ² = 4.67, p = 0.031) indicates a statistically significant difference between the two survival curves since p < 0.05. Combined with the 24-month survival rates (68% vs 45%), this shows Treatment A has superior survival throughout the follow-up period. Answer B correctly captures both the statistical significance and the direction of the difference. Answer A incorrectly treats survival percentages as if they represent average lifespan increases. The 23% difference (68% - 45%) between survival rates doesn't translate to "23% longer life" - survival analysis measures the probability of surviving to a specific time point, not average survival duration. Answer C misinterprets the 68% survival rate as a risk reduction measure. A 68% survival rate means 68% of patients survived to 24 months, not that Treatment A reduces death risk by 68% compared to Treatment B. Answer D focuses on confidence interval overlap, which isn't how you interpret log-rank test results. While non-overlapping confidence intervals can suggest differences, the definitive test for comparing survival curves is the log-rank test itself, not confidence interval comparison. Remember: In survival analysis, always interpret the log-rank test results first for overall curve comparison, then use individual time-point estimates to understand the magnitude and direction of differences.

Question 12

A two-way ANOVA examines the effects of drug type (A vs B) and dose level (low vs high) on blood pressure reduction. Results show: Drug main effect F(1,76) = 8.4, p = 0.005; Dose main effect F(1,76) = 2.1, p = 0.151; Interaction F(1,76) = 6.8, p = 0.011. How should these results be interpreted?

  1. Both drug type and dose level significantly affect blood pressure, with a significant interaction
  2. Only drug type significantly affects blood pressure; dose level and interaction are non-significant
  3. Drug type has a significant effect, but the significant interaction indicates this effect depends on dose level (correct answer)
  4. The significant interaction overrides the main effects, so only combined drug-dose effects matter
  5. Drug type is significant and dose approaches significance, with an interaction requiring post-hoc tests
Explanation: When interpreting two-way ANOVA results, you must evaluate main effects and interactions systematically, with special attention to significant interactions since they change how you interpret main effects. Let's examine each statistical result: The drug main effect is significant (p = 0.005), indicating drug type matters. The dose main effect is non-significant (p = 0.151), suggesting dose level alone doesn't significantly impact blood pressure. However, the interaction effect is significant (p = 0.011), which is crucial for interpretation. A significant interaction means the effect of one factor depends on the level of the other factor. Here, while drug type shows an overall main effect, this effect varies depending on the dose level. You cannot simply say "drug A is better than drug B" without specifying the dose, because the drug effectiveness might differ dramatically between low and high doses. Answer A is incorrect because dose level alone is not significant (p = 0.151). Answer B misses the critical significant interaction effect, oversimplifying the findings. Answer D goes too far by claiming interactions "override" main effects entirely—significant main effects still provide valuable information, but must be interpreted within the context of the interaction. Answer C correctly captures that drug type has a significant effect, but this effect is qualified by the significant interaction, meaning the drug's effectiveness depends on the dose level. Study tip: When you see a significant interaction in ANOVA, always interpret main effects cautiously—the interaction tells you the story is more complex than main effects alone suggest.

Question 13

A McNemar's test compares the effectiveness of two diagnostic tests on the same 120 patients. The results show: discordant pairs = 28, χ² = 6.12, df = 1, p = 0.013. Test A was positive in 15 cases where Test B was negative, while Test B was positive in 13 cases where Test A was negative. What conclusion is appropriate?

  1. Test A is significantly more sensitive than Test B since it detected 15 vs 13 additional cases
  2. There is a significant difference in the diagnostic performance of the two tests (correct answer)
  3. Test A and Test B have significantly different specificities based on the discordant pairs
  4. The tests disagree in 28 cases, but this difference is not statistically meaningful
  5. Test A correctly identifies 15 more cases than Test B with statistical significance
Explanation: When you encounter McNemar's test questions, focus on what this test actually measures: whether two paired tests perform significantly differently on the same subjects, not which test is "better" or what type of performance differs. The key evidence here is the significant p-value (0.013 < 0.05), which tells you there's a statistically significant difference between the two tests' performance. McNemar's test uses only the discordant pairs (cases where the tests disagree) to determine if one test systematically performs differently than the other. With χ2=6.12χ² = 6.12 and p=0.013p = 0.013, you can conclude the tests perform significantly differently. Option A incorrectly assumes that Test A is "better" because it detected 2 more positive cases (15 vs 13). However, without knowing the true disease status, you can't determine which test is more accurate or sensitive. Those extra positives could be true positives or false positives. Option C makes an unsupported claim about specificity. McNemar's test tells you the tests differ significantly, but it doesn't specify whether the difference is in sensitivity, specificity, or both. The discordant pairs alone don't reveal which aspect of performance differs. Option D contradicts the statistical results. The p-value of 0.013 clearly indicates the difference IS statistically meaningful, making this option factually incorrect. Option B correctly interprets the results: there's a significant difference in diagnostic performance without making assumptions about which test is superior or which performance metric differs. Study tip: Remember that McNemar's test detects significant differences between paired tests, but it doesn't tell you which test is better or what aspect of performance differs.

Question 14

A Cox proportional hazards model analyzing time to hospital readmission includes age as a covariate: β = 0.032, SE = 0.014, HR = 1.033, 95% CI = (1.005, 1.061), p = 0.021. What does the hazard ratio indicate about the effect of age?

  1. Each year of age increases the probability of readmission by 3.3%
  2. Each year of age increases the instantaneous risk of readmission by 3.3%
  3. Patients who are 33 years old have hazard ratio of 1.033 compared to newborns
  4. Each year of age multiplies the baseline hazard of readmission by 1.033 (correct answer)
  5. Each year of age increases the median time to readmission by 3.3%
Explanation: When you encounter Cox proportional hazards models, focus on understanding what the hazard ratio represents: it's a multiplicative effect on the baseline hazard function, not an additive probability increase. The hazard ratio of 1.033 means that for each one-year increase in age, a patient's hazard (instantaneous risk) of readmission is multiplied by 1.033. This is fundamentally different from adding 3.3% to a probability. The Cox model assumes that covariates have multiplicative effects on the baseline hazard function, so if the baseline hazard at time t is h0(t)h_0(t), then a patient one year older has hazard 1.033×h0(t)1.033 \times h_0(t). Option A incorrectly interprets the hazard ratio as a probability increase. Hazard ratios don't directly translate to probability changes—that would require converting to risk differences. Option B misunderstands what "increases by 3.3%" means; the hazard doesn't increase by 3.3%, it increases by a factor of 1.033 (which is actually a 3.3% relative increase). Option C makes an error about the reference point—hazard ratios in regression represent the effect per unit change in the covariate, not a comparison to age zero. Study tip: Remember that hazard ratios are always multiplicative factors. An HR > 1 means increased risk, HR < 1 means decreased risk, and the baseline hazard gets multiplied by this factor for each unit increase in the covariate. Don't confuse hazard ratios with probability differences or percentage point increases.

Question 15

A paired t-test comparing pre- and post-treatment blood glucose levels in 25 patients yields: Mean difference = -15.2 mg/dL, SE = 4.8, t = -3.17, df = 24, p = 0.004, 95% CI = (-25.1, -5.3). If the study were repeated with 100 patients and the same effect size, what would happen to the standard error?

  1. It would remain approximately the same since the effect size is unchanged
  2. It would become approximately half as large due to the increased sample size (correct answer)
  3. It would become approximately four times smaller since n increased four-fold
  4. It would become approximately twice as large due to increased variability
  5. It cannot be determined without knowing the population variance
Explanation: When you encounter questions about how sample size affects statistical measures, focus on the fundamental relationship between sample size and standard error. The standard error measures the precision of your sample estimate, and it has a predictable mathematical relationship with sample size. The standard error formula includes sn\frac{s}{\sqrt{n}} in its structure, where s is the sample standard deviation and n is the sample size. This means standard error is inversely proportional to the square root of sample size. When sample size increases from 25 to 100 patients (a 4-fold increase), the denominator n\sqrt{n} increases from 25=5\sqrt{25} = 5 to 100=10\sqrt{100} = 10. This doubles the denominator, making the standard error approximately half as large. Choice A incorrectly assumes effect size and standard error are directly related. While effect size remains constant, standard error still decreases with larger samples because you're measuring the same effect with greater precision. Choice C falls into the common trap of thinking standard error decreases proportionally to sample size increase. Since standard error involves the square root of n, a 4-fold increase in sample size only halves (doesn't quarter) the standard error. Choice D contradicts the fundamental principle that larger samples provide more precise estimates, not less precise ones. Remember this key pattern: Standard error always decreases as sample size increases, following the square root relationship. Doubling sample size doesn't halve standard error—you need to quadruple sample size to halve standard error.

Question 16

A logistic regression predicting hospital readmission (1 = readmitted, 0 = not readmitted) produces the following for age: Coefficient = 0.045, SE = 0.018, Odds Ratio = 1.046, 95% CI for OR = (1.009, 1.084), p = 0.013. What is the correct interpretation of the odds ratio?

  1. For each year increase in age, the probability of readmission increases by 4.6%
  2. For each year increase in age, the odds of readmission increase by 46%
  3. For each year increase in age, the odds of readmission are multiplied by 1.046 (correct answer)
  4. Patients who are 46 years old have odds ratio of 1.046 compared to newborns
  5. For each year increase in age, the risk of readmission increases by 1.046%
Explanation: When interpreting logistic regression results, you need to understand what an odds ratio represents and how it functions as a multiplicative factor. The odds ratio tells you how the odds change when the predictor variable increases by one unit. The coefficient of 0.045 represents the natural log of the odds ratio. When exponentiated, this gives us the odds ratio of 1.046, which means the odds are multiplied by this factor for each one-unit increase in age. Answer C is correct because it accurately describes this multiplicative relationship. For every additional year of age, you multiply the current odds of readmission by 1.046. This is the fundamental interpretation of odds ratios in logistic regression. Answer A confuses odds ratios with probability. An odds ratio of 1.046 does not mean a 4.6% increase in probability – probability changes depend on the baseline probability and follow a more complex relationship. Answer B misinterprets the magnitude. An odds ratio of 1.046 represents a 4.6% increase in odds (1.046 - 1 = 0.046 = 4.6%), not a 46% increase. A 46% increase would correspond to an odds ratio of 1.46. Answer D incorrectly treats this as a comparison between specific age groups. The odds ratio of 1.046 applies to each single-year increase in age, not to a comparison between 46-year-olds and newborns. Remember: odds ratios are multiplicative factors that apply to each unit change in the predictor. Always distinguish between odds, probability, and relative comparisons when interpreting logistic regression output.

Question 17

A Pearson correlation between study hours and exam scores yields r = 0.52, n = 75, p < 0.001, 95% CI = (0.33, 0.67). A student argues that since r² = 0.27, study hours are not very important because they only explain 27% of exam score variance. How should this interpretation be evaluated?

  1. The student is correct; variables explaining less than 50% of variance are not practically significant
  2. The student is incorrect; r = 0.52 indicates a strong positive correlation regardless of r²
  3. The student misunderstands r²; 27% explained variance can still represent a substantial relationship (correct answer)
  4. The student is correct; the confidence interval shows too much uncertainty for practical conclusions
  5. The student is incorrect; statistical significance (p < 0.001) proves practical importance
Explanation: When interpreting correlation coefficients and their practical significance, you need to understand both the correlation strength and what the coefficient of determination (r²) actually represents in context. The correct interpretation recognizes that r² = 0.27 means study hours account for 27% of the variance in exam scores, which can indeed represent a substantial and practically meaningful relationship. In educational and behavioral research, explaining over a quarter of the variance in an outcome is often considered quite meaningful, especially given the many factors that influence academic performance (prior knowledge, test anxiety, sleep, nutrition, etc.). Option A is wrong because there's no universal threshold where variables below 50% explained variance become unimportant. Practical significance depends on the research context, not arbitrary cutoffs. Option B misses the point entirely—while r = 0.52 does indicate a moderate-to-strong positive correlation, the student's concern was specifically about the practical meaning of the explained variance, not correlation strength. Option D incorrectly focuses on the confidence interval; the CI (0.33, 0.67) actually shows reasonable precision and doesn't include zero, confirming statistical significance. The key insight is that "only 27%" reflects a common misunderstanding. In complex real-world phenomena, single variables rarely explain most of the variance. A correlation of 0.52 with 27% explained variance represents a practically significant relationship that could meaningfully inform study strategies. Study tip: In biostatistics, avoid arbitrary thresholds for practical significance. Instead, evaluate effect sizes within the context of the field and consider what percentage of variance explanation is realistic given the complexity of the phenomenon being studied.

Question 18

An ANOVA comparing mean cholesterol levels across three diet groups produces F = 4.82, df = (2, 57), p = 0.012. The means are: Group A = 180 mg/dL, Group B = 195 mg/dL, Group C = 205 mg/dL. What can be concluded from this output?

  1. Group C has significantly higher cholesterol than both Groups A and B individually
  2. There is sufficient evidence that at least one group mean differs from the others (correct answer)
  3. The difference between Groups A and C is statistically significant at α = 0.05
  4. All three groups have significantly different means from each other at α = 0.05
  5. The linear trend across groups A to C is statistically significant at α = 0.05
Explanation: When you encounter ANOVA results, remember that this test has a specific and limited scope: it only tells you whether there's evidence that at least one group differs from the others, not which specific groups differ. The key evidence here is the significant F-statistic (F = 4.82, p = 0.012 < 0.05). This p-value means we can reject the null hypothesis that all three group means are equal. Since we're rejecting "all means are equal," we're accepting that "at least one mean differs from the others." This directly supports answer B. However, ANOVA alone cannot tell us which specific pairs of groups differ significantly. Looking at the wrong answers: Answer A assumes Group C differs from both A and B individually, but we haven't done post-hoc tests to confirm these specific comparisons. Answer C makes the same error, claiming A and C specifically differ without proper pairwise testing. Answer D goes even further, assuming all three groups differ from each other pairwise – again, something ANOVA cannot establish on its own. The means do show a pattern (180 < 195 < 205), but statistical significance of individual comparisons requires additional testing, such as Tukey's HSD or Bonferroni corrections. These post-hoc tests control for multiple comparisons and determine which specific pairs are significantly different. Remember this ANOVA principle: a significant F-test is like a smoke detector – it tells you there's a "fire" (difference) somewhere, but you need additional investigation (post-hoc tests) to locate exactly where the fire is burning.

Question 19

A regression analysis examining the relationship between exercise hours per week (X) and systolic blood pressure (Y) yields the following output: Coefficient = -2.3, Standard Error = 0.8, p-value = 0.006, 95% CI = (-3.87, -0.73). If a patient increases their exercise by 3 hours per week, what is the predicted change in their systolic blood pressure?

  1. A decrease of 6.9 mmHg with 95% confidence
  2. A decrease of 6.9 mmHg based on the point estimate (correct answer)
  3. A decrease of 2.3 mmHg with statistical significance
  4. A change between -11.61 and -2.19 mmHg with 95% confidence
  5. A decrease of 2.3 mmHg based on the point estimate
Explanation: When interpreting regression coefficients, you need to understand what each piece of output tells you and how to apply it to predict changes in the outcome variable. The coefficient of -2.3 represents the predicted change in systolic blood pressure (in mmHg) for each one-hour increase in weekly exercise. To find the predicted change for a 3-hour increase, you multiply the coefficient by 3: −2.3×3=−6.9-2.3 \times 3 = -6.9 mmHg. This is a point estimate based on the regression equation. Answer B correctly identifies this as a decrease of 6.9 mmHg based on the point estimate. Answer A is incorrect because while the magnitude (-6.9 mmHg) is right, you cannot claim "95% confidence" for this specific prediction - the confidence interval given applies only to the coefficient itself, not to individual predictions. Answer C uses the wrong multiplier, giving you the change for just 1 hour of additional exercise rather than 3 hours. Answer D attempts to multiply the entire confidence interval by 3, but this misapplies the concept - the CI describes uncertainty around the coefficient, not around a specific patient's predicted change. The key distinction is between the coefficient's confidence interval (which reflects uncertainty in estimating the true population relationship) and a point prediction (which uses the estimated coefficient as if it were the true value). For exam questions asking about predicted changes, use the point estimate calculation unless specifically asked about prediction intervals or confidence bounds for individual predictions.

Question 20

A multiple regression model predicting hospital length of stay includes: age (β = 0.08, p = 0.045), severity score (β = 1.2, p < 0.001), and insurance type (β = -2.1, p = 0.067). The overall model: F(3,156) = 18.7, p < 0.001, R² = 0.26, adjusted R² = 0.25. Which statement about model interpretation is most accurate?

  1. All three predictors significantly contribute to the model since the overall F-test is significant
  2. Age and severity score are significant predictors, while insurance type is marginally significant
  3. The model explains 26% of length of stay variance, with severity score being the strongest predictor (correct answer)
  4. Insurance type should be removed from the model since it's not significant and barely affects R²
  5. The adjusted R² of 0.25 indicates the model is not practically useful for prediction
Explanation: When interpreting multiple regression output, you need to distinguish between overall model performance and individual predictor significance. The key metrics here are the regression coefficients (β), their p-values, R², and the overall F-test. Looking at the model performance, the significant F-test (p < 0.001) confirms the model as a whole is meaningful, and R² = 0.26 indicates it explains 26% of the variance in hospital length of stay. Among the predictors, severity score has the largest absolute coefficient (β = 1.2) and highest significance (p < 0.001), making it the strongest predictor. This reasoning confirms answer C is correct. Answer A is wrong because overall model significance doesn't guarantee all individual predictors are significant. The F-test evaluates whether the model explains more variance than chance, but individual coefficients must be evaluated separately using their own p-values. Answer B incorrectly labels insurance type as "marginally significant." With p = 0.067, it fails to reach the conventional α = 0.05 threshold and is technically non-significant, not marginal. Answer D makes a premature recommendation to remove insurance type. While it's not statistically significant, the decision to remove predictors should consider clinical importance, effect size, and model goals—not just statistical significance. The minimal difference between R² (0.26) and adjusted R² (0.25) actually suggests the model isn't overly complex. Study tip: In regression interpretation questions, always separate overall model performance from individual predictor effects, and remember that coefficient magnitude indicates practical importance while p-values indicate statistical significance.