Statistics Case Studies & Practical Application 4 — Questions and Answers
Question 1: A researcher reports: 'The 95% confidence interval for the mean recovery time is (12.4 days, 15.6 days).' What is the correct interpretation?
- There is a 95% probability the true mean falls in this interval
- If the study were repeated many times, 95% of such intervals would capture the true mean (Correct answer)
- 95% of individual patients recover between 12.4 and 15.6 days
- The sample mean is 95% accurate
Correct answer: If the study were repeated many times, 95% of such intervals would capture the true mean
A 95% CI means that the procedure, repeated across many samples, would produce intervals containing the true parameter 95% of the time; it is not a probability statement about one interval.
Question 2: An HR analyst runs a multiple regression predicting employee performance from age, tenure, and training hours. The VIF for tenure is 8.7. What does this indicate?
- Tenure is the most important predictor of performance
- Tenure is highly correlated with other predictors, indicating multicollinearity (Correct answer)
- The model explains 87% of the variance in performance
- Tenure should be removed because it has no effect
Correct answer: Tenure is highly correlated with other predictors, indicating multicollinearity
A VIF above 5-10 signals multicollinearity, meaning tenure is strongly correlated with other predictors, which inflates coefficient standard errors.
Question 3: A school wants to estimate the proportion of students who walk to school. They survey 200 students and find 80 walk. What is the 95% confidence interval for the true proportion?
- (0.33, 0.47) (Correct answer)
- (0.35, 0.45)
- (0.32, 0.48)
- (0.30, 0.50)
Correct answer: (0.33, 0.47)
With p̂=0.40 and n=200, the margin of error is 1.96×√(0.40×0.60/200)≈0.068, giving the interval (0.332, 0.468) ≈ (0.33, 0.47).
Question 4: A public health study finds that counties with more fast-food restaurants have higher obesity rates (r=0.71). A reporter headlines: 'Fast Food Causes Obesity.' Why is this misleading?
- The correlation is not strong enough to report
- Ecological correlations (county-level data) cannot support causal claims about individuals (Correct answer)
- Obesity rates should be measured at the national level
- The reporter should have used a regression model instead
Correct answer: Ecological correlations (county-level data) cannot support causal claims about individuals
Using aggregate data to make individual-level causal claims commits the ecological fallacy; county-level correlations don't establish causation for individuals.
Question 5: A company surveys employee satisfaction on a 1-5 Likert scale. Management wants to compare average satisfaction between departments using a t-test. What assumption concern is most relevant?
- The t-test requires a population size over 1,000
- Likert data is ordinal, and using means may be inappropriate; normality should be verified (Correct answer)
- The t-test cannot compare more than two groups
- Satisfaction scores must be normally distributed by law
Correct answer: Likert data is ordinal, and using means may be inappropriate; normality should be verified
Likert data is technically ordinal, and treating it as interval-level for t-tests is debated; checking for approximate normality or using nonparametric alternatives is advisable.
Question 6: A researcher tests whether a coin is fair by flipping it 100 times and getting 60 heads. He sets α=0.05 and calculates p=0.057. What conclusion does he draw?
- The coin is definitely fair because p > 0.05
- Fail to reject H₀; there is insufficient evidence to conclude the coin is unfair (Correct answer)
- The coin is biased because 60 heads is far from 50
- The test is inconclusive and must be repeated
Correct answer: Fail to reject H₀; there is insufficient evidence to conclude the coin is unfair
Since p=0.057 > α=0.05, we fail to reject the null hypothesis; this doesn't prove the coin is fair, only that the evidence is insufficient to conclude otherwise.
Question 7: An insurance company builds a logistic regression model to predict claim fraud. The model has 92% accuracy but the fraud rate is only 5%. Why might accuracy be misleading here?
- Logistic regression cannot achieve 92% accuracy
- A model predicting 'no fraud' for all cases would also be 95% accurate, hiding poor fraud detection (Correct answer)
- The model needs more predictor variables to be useful
- 92% accuracy is too high and indicates overfitting
Correct answer: A model predicting 'no fraud' for all cases would also be 95% accurate, hiding poor fraud detection
With a 5% base rate, predicting no fraud always yields 95% accuracy; metrics like precision, recall, and AUC better capture performance on imbalanced classes.
Statistics Case Studies & Practical Application 4
A researcher reports: 'The 95% confidence interval for the mean recovery time is (12.4 days, 15.6 days).' What is the correct interpretation?