Statistics Case Studies & Practical Application 3 — Questions and Answers
Question 1: A food company conducts an A/B test on two package designs. Design A (n=500) gets 120 purchases; Design B (n=500) gets 145 purchases. A chi-square test yields p=0.04. What is the correct business decision?
- Switch to Design B because it is statistically significantly better (Correct answer)
- Keep Design A because the absolute difference is small
- Run another test before deciding because p is close to 0.05
- Neither design is better because the effect size is unknown
Correct answer: Switch to Design B because it is statistically significantly better
With p=0.04 < 0.05, the difference is statistically significant, supporting adoption of Design B which outperformed Design A.
Question 2: A meteorologist models monthly rainfall using a Poisson distribution. Last month had 14 rain events; the historical mean is 10. What does this data suggest?
- The month was a statistical impossibility
- The month's rainfall was above average but within plausible Poisson variation (Correct answer)
- The Poisson model is no longer valid
- The standard deviation equals 14 for this month
Correct answer: The month's rainfall was above average but within plausible Poisson variation
For a Poisson distribution with λ=10, values of 14 are uncommon but not impossible, as the standard deviation is √10 ≈ 3.16.
Question 3: A sociologist uses stratified random sampling to study income across 4 age groups. She samples proportionally: 60 from ages 18-30, 90 from 31-50, 80 from 51-65, and 20 from 65+. What is the primary advantage of this approach over simple random sampling?
- It eliminates all sampling bias
- It ensures each age group is adequately represented for group-level comparisons (Correct answer)
- It is faster and cheaper than random sampling
- It avoids the need for a sampling frame
Correct answer: It ensures each age group is adequately represented for group-level comparisons
Stratified sampling ensures representation from all subgroups, enabling more precise estimates within and comparisons across strata.
Question 4: A factory monitors defect rates daily. Over 30 days, the p-chart shows all points within control limits but exhibiting a steady downward trend. What does this pattern suggest?
- The process is in control and no action is needed
- A systematic non-random pattern signals a potential special cause (Correct answer)
- The control limits should be widened
- The defect rate trend is due to common-cause variation
Correct answer: A systematic non-random pattern signals a potential special cause
A consistent trend, even within control limits, is a non-random pattern (run rule violation) suggesting an assignable cause is present.
Question 5: A clinical trial uses a crossover design where each patient receives both the drug and placebo in randomized order. What statistical test is most appropriate for comparing outcomes?
- Independent samples t-test
- Paired samples t-test (Correct answer)
- One-sample z-test
- Chi-square test of independence
Correct answer: Paired samples t-test
A crossover design produces paired observations from the same subjects, so a paired t-test accounts for within-subject correlation and reduces variability.
Question 6: A market researcher finds a correlation of r=0.85 between advertising spend and product sales. A colleague concludes that increasing ad spend causes higher sales. What is wrong with this conclusion?
- The correlation is too high to be meaningful
- Correlation does not establish causation; confounders may explain the relationship (Correct answer)
- A correlation above 0.80 always implies causation
- The sample size must be reported before interpreting r
Correct answer: Correlation does not establish causation; confounders may explain the relationship
Correlation measures the strength of association, not causation; other variables (e.g., season, competitor actions) could drive both variables.
Question 7: A city planner wants to compare average commute times across 5 neighborhoods. She collects random samples from each. Which statistical test should she use?
- Paired t-test
- One-way ANOVA (Correct answer)
- Two-sample z-test
- Pearson correlation
Correct answer: One-way ANOVA
One-way ANOVA is designed to compare means across three or more independent groups simultaneously while controlling the Type I error rate.
Statistics Case Studies & Practical Application 3
A food company conducts an A/B test on two package designs.
Design A (n=500) gets 120 purchases; Design B (n=500) gets 145 purchases.
A chi-square test yields p=0.04.
What is the correct business decision?