Math 3 Quiz: Comparing Groups Via Simulation
16 questions · exam conditions
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Comparing Groups Via SimulationQuestion 1 of 16

A researcher compares customer wait times at two bank branches. Branch A (n=60): mean = 4.2 minutes, Branch B (n=55): mean = 5.8 minutes. She runs 1200 simulations assuming equal wait times and finds that 84 simulations show differences of 1.6 minutes or greater (in either direction). She concludes the branches have significantly different wait times. What is the primary concern with this analysis?

The sample sizes should be exactly equal for simulation-based comparisons, making the statistical test invalid with n=60 vs n=55.
The researcher should have used 1000 simulations instead of 1200 to align with standard statistical software defaults and ensure valid results.
Wait times are likely right-skewed rather than normally distributed, making simulation-based methods inappropriate for this type of data.
The p-value of 0.07 exceeds the conventional 0.05 significance level, so the conclusion of significant difference is not statistically justified.
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Math 3 Quiz

Math 3 Quiz: Comparing Groups Via Simulation

Practice Comparing Groups Via Simulation in Math 3 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 Comparing Groups Via Simulation, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 3.

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 researcher compares customer wait times at two bank branches. Branch A (n=60): mean = 4.2 minutes, Branch B (n=55): mean = 5.8 minutes. She runs 1200 simulations assuming equal wait times and finds that 84 simulations show differences of 1.6 minutes or greater (in either direction). She concludes the branches have significantly different wait times. What is the primary concern with this analysis?

  1. The sample sizes should be exactly equal for simulation-based comparisons, making the statistical test invalid with n=60 vs n=55.
  2. The researcher should have used 1000 simulations instead of 1200 to align with standard statistical software defaults and ensure valid results.
  3. Wait times are likely right-skewed rather than normally distributed, making simulation-based methods inappropriate for this type of data.
  4. The p-value of 0.07 exceeds the conventional 0.05 significance level, so the conclusion of significant difference is not statistically justified. (correct answer)
Explanation: When you encounter a hypothesis testing question involving simulations, focus on interpreting the p-value correctly and checking whether the conclusion matches the statistical evidence. Let's work through this step-by-step. The researcher found that 84 out of 1200 simulations showed differences of 1.6 minutes or greater. This gives us a p-value of 841200=0.07\frac{84}{1200} = 0.07. Since this p-value (0.07) exceeds the conventional significance level of 0.05, we cannot reject the null hypothesis of equal wait times. The researcher's conclusion of "significantly different wait times" contradicts this statistical evidence, making option D correct. Now let's examine why the other options miss the mark. Option A incorrectly suggests that unequal sample sizes invalidate simulation-based tests—while equal sizes are ideal, the difference between n=60 and n=55 doesn't invalidate the analysis. Option B focuses on an irrelevant detail about simulation count; 1200 simulations is actually more robust than 1000, and the specific number doesn't affect validity as long as it's sufficiently large. Option C wrongly claims that right-skewed distributions make simulation methods inappropriate—in fact, simulation-based methods are particularly valuable because they don't require normality assumptions like traditional t-tests do. The key takeaway: Always calculate the p-value and compare it to your significance level before accepting any conclusion about statistical significance. A p-value above 0.05 means the evidence isn't strong enough to claim a significant difference, regardless of what the researcher concludes.

Question 2

A researcher conducts a simulation to compare the effectiveness of two teaching methods. She randomly assigns 40 students to Method A and 40 students to Method B, then measures test scores. The observed difference in mean scores is 6.2 points (Method A higher). She then runs 1000 simulations assuming no real difference between methods, and finds that 78 out of 1000 simulated differences are 6.2 points or greater. What is the most appropriate conclusion?

  1. The difference is statistically significant at the 0.05 level, providing strong evidence that Method A is more effective than Method B.
  2. The difference is statistically significant at the 0.10 level but not at the 0.05 level, providing moderate evidence for Method A's superiority.
  3. The difference is not statistically significant at conventional levels, suggesting the observed difference could reasonably be due to random variation alone. (correct answer)
  4. The simulation is invalid because it assumes no difference exists, which contradicts the observed data showing Method A performed better.
Explanation: The p-value is 78/1000 = 0.078. This means there's a 7.8% chance of observing a difference of 6.2 points or greater purely by random chance if there's no real difference between methods. Since 0.078 > 0.05, the result is not statistically significant at the conventional 0.05 level, so we cannot conclude the difference is meaningful. Choice A is wrong because 0.078 > 0.05. Choice B is wrong because while 0.078 < 0.10, we typically use 0.05 as the standard cutoff. Choice D misunderstands that the null hypothesis simulation is the correct approach.

Question 3

Two manufacturing processes are compared for defect rates. Process A produces 8 defects in 200 items (4%), Process B produces 18 defects in 250 items (7.2%). A simulation of 1000 trials assuming equal defect rates shows 127 trials where the difference in percentages was 3.2% or greater. What additional information is needed to properly interpret these results?

  1. The exact number of trials where Process B exceeded Process A by 3.2% or more, to calculate the correct two-tailed p-value. (correct answer)
  2. The standard deviation of defect rates within each process, to verify that the simulation assumptions about variability are valid.
  3. The confidence interval for the difference in proportions, to determine whether the observed difference is practically significant.
  4. The power analysis for this study design, to ensure sufficient sample size for detecting meaningful differences between processes.
Explanation: The simulation result gives only one tail of the distribution (differences of 3.2% or greater favoring Process A). For proper hypothesis testing, we need to know how many trials showed Process B performing better by 3.2% or more, to calculate the complete two-tailed p-value. Choice B is wrong because simulations for proportions don't require knowing within-group standard deviations. Choice C addresses practical significance but isn't needed for statistical interpretation. Choice D is about study planning, not interpretation of results.

Question 4

An educational researcher tests whether small class sizes improve student performance. Large classes (n=80) averaged 76.2% on a standardized test, while small classes (n=70) averaged 79.8%. A simulation assuming no class size effect shows that only 3.2% of trials produced differences this large or larger. The researcher concludes that small classes cause improved performance. What is the main limitation of this conclusion?

  1. The sample sizes are too different (80 vs 70) to make valid statistical comparisons between the groups using simulation methods.
  2. A 3.2% probability is above the standard 1% threshold required for educational research, so the evidence is insufficient for causal claims.
  3. The study design cannot establish causation because other factors besides class size might explain the observed performance difference. (correct answer)
  4. The simulation methodology is inappropriate for educational data because test scores are not normally distributed in most populations.
Explanation: While the statistical test suggests a significant difference (3.2% < 5%), establishing causation requires controlling for confounding variables. Small classes might be in different schools, have different teachers, or serve different student populations. The study shows association, not causation. Choice A is wrong because moderate differences in sample size don't invalidate simulations. Choice B incorrectly states that educational research requires 1% significance (it typically uses 5%). Choice D is wrong because simulation methods are robust and don't require normal distributions.

Question 5

A company compares customer satisfaction scores between two store locations. Store A: mean = 7.8 (n=50), Store B: mean = 7.2 (n=45). A simulation of 2000 trials assuming no real difference shows that 312 trials produced differences of 0.6 or greater in favor of either store. What is the correct interpretation?

  1. The p-value is 0.156, so the difference is not statistically significant at the 0.05 level, indicating no meaningful difference between stores. (correct answer)
  2. The p-value is 0.312, so the difference is not statistically significant at any conventional level, indicating no meaningful difference between stores.
  3. The p-value is 0.078, so the difference is not statistically significant at the 0.05 level but significant at the 0.10 level.
  4. The simulation tested differences in both directions, so we need additional information about one-tailed versus two-tailed tests to interpret the results.
Explanation: The simulation found 312 out of 2000 trials with differences of 0.6 or greater 'in favor of either store,' meaning this is already a two-tailed test result. The p-value is 312/2000 = 0.156. Since 0.156 > 0.05, the difference is not statistically significant at the conventional level. Choice B incorrectly uses 312 as if it were out of 1000. Choice C would be correct if the p-value were 0.078, but that's not the calculation here. Choice D misunderstands that the simulation already accounted for both directions.

Question 6

A study compares weight loss between two diet plans over 12 weeks. Diet A (n=60): mean loss = 5.2 kg, Diet B (n=55): mean loss = 3.8 kg. A researcher runs 1500 simulations assuming both diets are equally effective and finds that 67 simulations show Diet A performing better by 1.4 kg or more. However, 89 simulations show Diet B performing better by 1.4 kg or more. What is the most appropriate statistical conclusion?

  1. The p-value is approximately 0.045, providing statistically significant evidence that Diet A is more effective than Diet B at the 0.05 level.
  2. The p-value is approximately 0.059, so there is insufficient evidence to conclude a significant difference between the diets at the 0.05 level.
  3. The asymmetric simulation results (67 vs 89) indicate a flaw in the randomization process, invalidating the statistical test completely.
  4. The p-value is approximately 0.104, providing insufficient evidence for a significant difference between diets at conventional significance levels. (correct answer)
Explanation: For a two-tailed test, we consider extreme differences in either direction. The total number of simulations with differences as extreme as 1.4 kg (in either direction) is 67 + 89 = 156. The p-value is 156/1500 = 0.104. Since 0.104 > 0.05, there's insufficient evidence for statistical significance. Choice A uses only one tail (67/1500 ≈ 0.045). Choice B incorrectly calculates the p-value. Choice C misunderstands that some asymmetry in simulation results is normal due to random variation.

Question 7

A fitness study compares two exercise programs. Program A (n=42): mean weight loss = 6.8 kg, Program B (n=38): mean weight loss = 4.9 kg. A simulation of 1500 trials assuming no program difference shows that 73 trials produced differences of 1.9 kg or greater favoring Program A, while 68 trials produced differences of 1.9 kg or greater favoring Program B. What is the most appropriate interpretation?

  1. Since 73 > 68, there is systematic bias in the simulation favoring Program A, invalidating the statistical test results completely.
  2. The p-value is approximately 0.094, indicating insufficient evidence for a statistically significant difference between programs at the 0.05 level. (correct answer)
  3. The p-value is approximately 0.049, providing statistically significant evidence that Program A is more effective than Program B.
  4. The asymmetric results (73 vs 68) suggest the null hypothesis assumption of equal effectiveness is violated, requiring a different statistical approach.
Explanation: For a two-tailed test, we sum the extreme results in both directions: (73 + 68)/1500 = 141/1500 = 0.094. Since 0.094 > 0.05, the difference is not statistically significant. The slight asymmetry (73 vs 68) is normal random variation in simulations. Choice A incorrectly interprets normal variation as bias. Choice C uses only one tail (73/1500 ≈ 0.049). Choice D misunderstands that small asymmetries are expected in random simulations.

Question 8

A medical study compares pain relief scores between two medications. Drug A (n=65): mean improvement = 4.8 points, Drug B (n=70): mean improvement = 3.1 points. Researchers conduct 2500 simulations assuming equal drug effectiveness and find that 183 simulations show differences of 1.7 points or greater in either direction. However, they realize that 12 patients in the Drug A group had more severe initial pain scores. How does this affect the interpretation?

  1. The p-value of 0.073 remains valid because simulation accounts for expected variation in randomized trials.
  2. The p-value of 0.073 is still meaningful for statistical significance, but the confounding limits causal interpretation about drug effectiveness. (correct answer)
  3. The confounding variable invalidates the statistical test completely, requiring matched-pair analysis or covariate adjustment before any conclusions can be drawn.
  4. The unbalanced initial pain scores actually strengthen the evidence for Drug A, since it performed better despite treating more severe cases.
Explanation: When you encounter a study with both statistical results and potential confounding variables, you need to distinguish between statistical validity and causal interpretation—two separate but related concepts. The simulation gives a p-value of 1832500=0.073\frac{183}{2500} = 0.073, which is statistically valid. The researchers properly tested whether the observed 1.7-point difference could reasonably occur by chance if the drugs were equally effective. This statistical calculation remains mathematically sound regardless of confounding variables. However, the 12 patients with more severe initial pain in Drug A's group create a confounding variable that limits causal interpretation. Since initial pain severity likely affects improvement scores, we can't definitively attribute Drug A's better performance to the medication itself versus the different baseline conditions. Choice A is wrong because while the p-value calculation is valid, this doesn't mean the study design adequately controls for all relevant variables. Choice C overstates the problem—confounding doesn't "invalidate" the statistical test, but rather limits what we can conclude from it. The test still tells us something meaningful about the probability of observing this difference by chance. Choice D makes a logical error by assuming Drug A's performance is "better despite" treating severe cases, when severe initial pain might actually make larger improvements more likely. Study tip: In research interpretation questions, always separate statistical validity (was the test done correctly?) from causal validity (can we attribute the results to the intervention?). Confounding variables typically affect the latter more than the former.

Question 9

Two basketball coaches compare their teams' free-throw percentages. Team X shoots 68% and Team Y shoots 72%. A simulation assuming equal ability shows that in 2000 trials, differences of 4 percentage points or more occurred 380 times. However, Team X had 25 players while Team Y had 45 players. What is the most appropriate interpretation?

  1. The 4% difference is not meaningful since 19% of simulations showed such differences could occur by chance with equal teams (correct answer)
  2. The simulation results are invalid because the teams have unequal sample sizes, making direct comparison inappropriate
  3. The 4% difference is meaningful because Team Y's larger sample size makes their percentage more reliable than Team X's percentage
  4. The simulation demonstrates that Team Y is significantly better since larger samples reduce the likelihood of chance differences
Explanation: The p-value is 380/2000 = 0.19 = 19%. Since this exceeds typical significance levels (5% or 10%), the difference could reasonably be due to chance. Choice B incorrectly suggests unequal sample sizes invalidate simulation comparison. Choice C confuses sample size reliability with statistical significance. Choice D misinterprets how sample size affects the simulation results.

Question 10

Researchers compare reaction times between two age groups: teenagers (mean = 0.31 seconds) and adults (mean = 0.37 seconds). Using 800 simulations assuming equal reaction times, they find 47 cases where the difference equals or exceeds 0.06 seconds. They plan to publish results claiming teenagers react faster. What is the primary concern with their conclusion?

  1. The 0.06-second difference is too small to have practical significance in real-world applications requiring quick reactions
  2. The sample sizes for each age group are not specified, preventing proper evaluation of the statistical power
  3. The simulation method cannot account for individual variation within age groups, making group comparisons statistically invalid
  4. The p-value of 5.9% is borderline significant, requiring additional studies to confirm the difference before publication (correct answer)
Explanation: When you encounter a simulation-based statistical test, you're looking at a method to calculate p-values by seeing how often random chance could produce results as extreme as what was observed. In this study, researchers found a 0.06-second difference in reaction times and used 800 simulations assuming no real difference between groups. They found 47 cases where random variation alone produced differences of 0.06 seconds or greater. This gives a p-value of 47/800 = 0.0588, or about 5.9%. The correct answer is D because a p-value of 5.9% sits right at the boundary of statistical significance. Most scientific fields use 5% (0.05) as the cutoff, meaning this result is marginally non-significant. Publishing results that barely miss the significance threshold as definitive evidence of faster teenage reactions is problematic—the evidence isn't strong enough to support such a claim. Answer A incorrectly focuses on practical significance rather than statistical validity. Answer B suggests sample size is the issue, but the simulation method already accounts for sampling variation. Answer C misunderstands simulation methodology—simulations can and do account for within-group variation by modeling the expected distribution under the null hypothesis. The key takeaway: When p-values hover around 0.05, be extremely cautious about making strong claims. A p-value of 5.9% suggests the observed difference could easily be due to chance, making it inappropriate to conclude that teenagers definitively react faster without additional evidence.

Question 11

A pharmaceutical company compares two pain medications by measuring pain reduction on a 10-point scale. Drug X shows 6.4-point reduction (n=50) while Drug Y shows 5.9-point reduction (n=55). Simulations assuming equal effectiveness produce differences of 0.5 points or more in 89 out of 1200 trials. The company must decide which drug to manufacture. What statistical guidance should inform their decision?

  1. Conduct additional trials because the 7.4% probability suggests possible superiority but requires confirmation before major investment (correct answer)
  2. Manufacture Drug X because the 7.4% simulation probability indicates strong evidence of superior effectiveness over Drug Y
  3. Choose Drug Y because its larger sample size provides more reliable evidence despite the lower mean reduction
  4. The 7.4% probability confirms both drugs are essentially equivalent, so manufacturing choice should depend on cost factors
Explanation: When you encounter pharmaceutical trial comparisons, focus on interpreting simulation results within the context of statistical significance and practical decision-making under uncertainty. The simulation shows that when two equally effective drugs are tested, differences of 0.5 points or greater occur in 89 out of 1200 trials, giving a 7.4% probability. This represents a p-value of 0.074, which falls just above the conventional 0.05 significance threshold. This suggests possible but not statistically confirmed superiority of Drug X. Choice A correctly recognizes that 7.4% indicates suggestive evidence requiring additional confirmation before major manufacturing investments. The probability is low enough to warrant interest but high enough to demand caution. Choice B incorrectly interprets 7.4% as "strong evidence." In statistical terms, this probability is actually considered weak to moderate evidence, falling short of the standard significance level that would justify confident conclusions about superiority. Choice C misunderstands sample size interpretation. While Drug Y has a larger sample (55 vs 50), this small difference doesn't override the observed effect size difference, and both sample sizes are reasonably adequate for initial comparison. Choice D wrongly concludes the drugs are "essentially equivalent." A 7.4% probability doesn't confirm equivalence—it indicates insufficient evidence to conclude superiority, which is different from proving equality. Remember: p-values near but above 0.05 signal "promising but inconclusive" results. In high-stakes decisions like drug manufacturing, such borderline results typically warrant additional data collection rather than immediate action.

Question 12

Two online learning platforms are compared for course completion rates. Platform A shows 73% completion while Platform B shows 68% completion. Researchers run simulations assuming equal platform effectiveness and find that 5% differences or larger occur in 112 out of 800 trials when sample sizes are 200 students each. They then repeat the simulation with sample sizes of 50 students each, finding such differences in 156 out of 800 trials. What does this comparison reveal?

  1. Platform A is significantly better with large samples (14% probability) but not with small samples (19.5% probability)
  2. Larger sample sizes make it easier to detect meaningful differences, as shown by the decreased probability from 19.5% to 14%
  3. The platform difference is not significant in either case since both probabilities exceed 10%, indicating chance variation
  4. Sample size affects the likelihood of observing differences by chance, with larger samples producing more reliable comparisons (correct answer)
Explanation: The key insight is that larger samples reduce random variation, making large differences less likely to occur by chance (14% vs 19.5%). This demonstrates how sample size affects simulation results. Choice A incorrectly applies different significance thresholds. Choice B reverses the relationship. Choice C applies an arbitrary 10% threshold and misses the sample size lesson.

Question 13

A researcher claims that a new teaching method increases student test scores. To test this claim, she randomly assigns 30 students to the new method (Group A) and 30 students to the traditional method (Group B). Group A has a mean score of 78.5, and Group B has a mean score of 75.2. She then runs 1000 simulations assuming no difference between methods, finding that 156 simulations show a difference of 3.3 points or greater. What conclusion should she draw at a 5% significance level?

  1. The difference is statistically significant because the observed difference (3.3) is greater than the typical simulation differences
  2. The difference is not statistically significant because 15.6% of simulations showed differences this large or larger by chance alone (correct answer)
  3. The difference is statistically significant because only 156 out of 1000 simulations exceeded this difference under random variation
  4. The difference is not statistically significant because the sample sizes are too small to detect meaningful differences between groups
Explanation: With 156 out of 1000 simulations showing differences of 3.3 or greater, the p-value is 156/1000 = 0.156 = 15.6%. Since 15.6% > 5%, the difference is not statistically significant at the 5% level. Choice A incorrectly focuses on the magnitude rather than probability. Choice C misinterprets fewer simulations as evidence of significance. Choice D incorrectly assumes sample size is the issue.

Question 14

A fitness study compares weight loss between Diet Plan A (average 8.3 lbs lost) and Diet Plan B (average 6.8 lbs lost) over 12 weeks. Simulations assuming equal effectiveness show that differences of 1.5 lbs or greater occur in 67 out of 2000 trials. However, Diet Plan A participants exercised an average of 3.2 hours weekly while Diet Plan B participants exercised 3.0 hours weekly. What is the most appropriate interpretation?

  1. Diet Plan A is significantly more effective since only 3.35% of simulations showed such large differences by chance alone
  2. Both factors likely contribute to the difference, but the 3.35% probability suggests the diet difference is statistically meaningful
  3. The exercise difference invalidates the comparison since the diet plans were not tested under equivalent conditions for fair evaluation (correct answer)
  4. The 0.2-hour exercise difference is negligible, so the 3.35% probability confirms Diet Plan A's superior effectiveness for weight loss
Explanation: When evaluating experimental results, you must always check whether the comparison groups were truly equivalent except for the variable being tested. This principle of controlled experimentation is fundamental to drawing valid conclusions. The key issue here is that Diet Plan A and Diet Plan B participants didn't just differ in their diets—they also had different exercise patterns (3.2 vs 3.0 hours weekly). This creates what researchers call a confounding variable. Even though the simulation shows only a 3.35% chance of seeing such a large weight loss difference by chance, we cannot attribute the observed difference solely to the diet plans because exercise also affects weight loss. Choice C correctly identifies that the exercise difference "invalidates the comparison" because the diet plans weren't tested under equivalent conditions. When multiple variables differ between groups, you cannot isolate which factor caused the observed effect. Choice A incorrectly assumes the diet difference is the only explanation for the weight loss difference, ignoring the exercise confound. Choice B acknowledges both factors but wrongly suggests the statistical probability still makes the diet difference "meaningful"—but statistical significance becomes meaningless when confounding variables exist. Choice D dismisses the 0.2-hour exercise difference as "negligible," but any systematic difference between groups can affect results, especially in studies measuring small effects. Remember: Before interpreting statistical results, always verify that the comparison groups were truly equivalent except for the variable being studied. Confounding variables invalidate conclusions regardless of statistical significance.

Question 15

A company tests two website designs by measuring time spent on site. Design A averages 4.2 minutes (n=40) and Design B averages 3.7 minutes (n=35). Simulations assuming no difference between designs show that 73 out of 500 trials produce differences of 0.5 minutes or greater. The company wants to implement the better design. What should they conclude?

  1. Implement Design A because the simulation shows only 14.6% probability that such differences occur by chance, indicating real improvement
  2. Collect more data because 14.6% probability suggests the difference might be due to chance, making the decision uncertain (correct answer)
  3. Implement Design B because the smaller sample size makes its average more representative of typical user behavior patterns
  4. The simulation confirms no meaningful difference exists since 14.6% of trials showed similar differences under equal conditions
Explanation: With p-value = 73/500 = 0.146 = 14.6%, the result falls in a gray area - not clearly significant at 5% but somewhat low. More data would help clarify whether this represents a real difference. Choice A incorrectly treats 14.6% as strong evidence. Choice C illogically favors smaller sample size. Choice D incorrectly interprets 14.6% as confirming no difference.

Question 16

Educational researchers compare reading comprehension scores between two teaching methods: Method A (mean = 84.2) and Method B (mean = 81.7). They run 1500 simulations assuming both methods are equally effective. The simulation shows 284 trials where Method A exceeds Method B by 2.5 points or more, and 298 trials where Method B exceeds Method A by 2.5 points or more. What does this suggest about the observed 2.5-point difference?

  1. The difference favoring Method A is significant since Method A exceeded Method B by this amount in only 18.9% of simulations
  2. The difference is significant since the combined 38.8% of extreme differences demonstrates high variability between methods
  3. No meaningful difference exists since both directions of 2.5-point differences occurred in similar frequencies during simulations (correct answer)
  4. Method A is superior since it exceeded Method B more frequently (284 times) than the reverse (298 times) in simulations
Explanation: When you encounter a simulation study testing statistical significance, focus on what the simulation reveals about whether an observed difference could reasonably occur by chance alone. This simulation assumes both teaching methods are equally effective (null hypothesis) and tests whether a 2.5-point difference is unusual. The key insight is that extreme differences in both directions occurred at similar rates: Method A exceeded Method B by 2.5+ points in 284 trials, while Method B exceeded Method A by 2.5+ points in 298 trials. These frequencies are nearly identical (284 vs. 298), indicating that 2.5-point differences in either direction happen regularly when methods are actually equal. This suggests the observed difference is likely due to random variation, not a meaningful distinction between methods. Answer A misinterprets the 18.9% (284/1500) as evidence of significance, but this percentage alone doesn't determine significance - you need the full picture. Answer B incorrectly treats the combined 38.8% of extreme differences as evidence the methods differ, when this actually shows such differences are common under the null hypothesis. Answer D makes a logical error by claiming Method A is superior when Method B actually exceeded Method A more frequently (298 > 284), and regardless, the similar frequencies support the null hypothesis. The correct answer is C because the similar frequencies of extreme differences in both directions demonstrate that 2.5-point differences occur regularly by chance alone. Study tip: In simulation studies, look for whether extreme results occur frequently under the null hypothesis - high frequency suggests the observed difference isn't meaningful.