Statistics Case Studies & Practical Application 5 — Questions and Answers
Question 1: A sports analyst examines whether home-field advantage exists by testing if home teams win more than 50% of games across 500 games. 275 home wins are observed. What test is appropriate?
- Independent samples t-test comparing two team means
- One-sample z-test for a proportion against H₀: p=0.50 (Correct answer)
- Chi-square goodness-of-fit test comparing wins across seasons
- Paired t-test for home vs away scores
Correct answer: One-sample z-test for a proportion against H₀: p=0.50
Testing a single observed proportion against a known hypothesized value (0.50) calls for a one-sample z-test for proportions.
Question 2: A neuroscientist measures reaction times before and after caffeine intake for 15 subjects. Which test is most appropriate?
- Two-sample independent t-test
- Paired t-test (Correct answer)
- One-way ANOVA
- Mann-Whitney U test
Correct answer: Paired t-test
The same subjects are measured twice (before and after), creating dependent pairs, which is handled by the paired t-test to control for individual variability.
Question 3: A data scientist splits a dataset 80/20 into training and test sets, fits a model on training data, and evaluates on the test set. The test set accuracy is 78%. What does this metric measure?
- How well the model memorized the training data
- The model's ability to generalize to unseen data (Correct answer)
- The percentage of correct training predictions
- The statistical significance of the model's coefficients
Correct answer: The model's ability to generalize to unseen data
Evaluating on a held-out test set estimates how well the model generalizes to new, unseen observations not used during training.
Question 4: A researcher calculates Cohen's d = 0.82 when comparing two teaching methods. How should this effect size be interpreted?
- Small effect — the methods are barely different
- Medium effect — the methods differ by about half a standard deviation
- Large effect — the methods differ by nearly one standard deviation (Correct answer)
- The result is statistically significant at α=0.05
Correct answer: Large effect — the methods differ by nearly one standard deviation
Cohen's d = 0.82 falls in the 'large' range (d > 0.8), indicating the group means differ by approximately 0.82 standard deviations.
Question 5: A supply chain manager tracks weekly inventory levels over 2 years and notices a strong upward trend. She fits a simple linear regression with time as the predictor. What assumption violation is most likely?
- Homoscedasticity is violated because the sample is too small
- Autocorrelation: residuals from time-series data are often correlated across time (Correct answer)
- Normality: inventory values are never normally distributed
- Linearity: time cannot be used as a predictor variable
Correct answer: Autocorrelation: residuals from time-series data are often correlated across time
Time-series data often violates the independence assumption because consecutive residuals are correlated (autocorrelation), inflating the significance of regression estimates.
Question 6: A political scientist tests whether voter turnout differs by region using a 2×4 contingency table (two parties × four regions). Which test is most appropriate?
- Paired t-test
- One-way ANOVA
- Chi-square test of independence (Correct answer)
- Pearson correlation
Correct answer: Chi-square test of independence
A chi-square test of independence evaluates whether two categorical variables (party and region) are associated using frequency counts in a contingency table.
Question 7: An analyst fits a time-series forecasting model and plots the residuals. The residuals show a clear sinusoidal pattern. What does this indicate?
- The model is performing well because residuals are centered at zero
- The model has not captured seasonal or cyclical patterns in the data (Correct answer)
- Sinusoidal residuals indicate the model is overfitting
- The residuals should always follow a sinusoidal pattern
Correct answer: The model has not captured seasonal or cyclical patterns in the data
A systematic pattern in residuals (like a sine wave) indicates the model missed a periodic structure such as seasonality, violating the randomness assumption.
Statistics Case Studies & Practical Application 5
A sports analyst examines whether home-field advantage exists by testing if home teams win more than 50% of games across 500 games. 275 home wins are observed.
What test is appropriate?