Gladly wants to find out whether temperature or mood (or both) predicts how many meatballs are sold, and whether there is an interaction. He therefore decides to do a regression. Why is a regression the best analysis for this out of all of the tests we have learned this semester?

Question

BlackTom AI · Accepted Answer

Multiple linear regression estimates the simultaneous effects of several numeric predictors on a numeric outcome, so it naturally fits Gladly’s goal.

1) Relevant knowledge points and concepts
- Predictor vs outcome: temperature (numeric) and mood (numeric) are predictors; meatballs sold (numeric) is the continuous outcome.
- Multiple linear regression: models Y = β0 + β1X1 + β2X2 + β3(X1·X2) + ε, allowing estimation of unique effects and an interaction term.
- Interaction term: the product X1·X2 tests whether the effect of one predictor depends on the level of the other.

2) Step-by-step reasoning
- Identify variables: X1 = temperature, X2 = mood, Y = meatballs sold.
- Build model including interaction: Y = β0 + β1·Temp + β2·Mood + β3·(Temp×Mood) + ε.
- Estimate β coefficients from data (ordinary least squares). A significant β1 or β2 shows main effects; a significant β3 shows an interaction (moderation).

3) Intermediate/calculation steps (conceptual)
- Fit OLS to obtain β̂0, β̂1, β̂2, β̂3.
- Test H0: β1 = 0, H0: β2 = 0, H0: β3 = 0 using t-tests (t = β̂ / SE(β̂)).
- If β̂3 ≠ 0 and statistically significant, compute conditional slopes: ∂Y/∂Temp = β̂1 + β̂3·Mood (shows how Temp effect varies by Mood).

4) Why the correct answer is correct
- The chosen option states regression is the only test (among the listed choices) that handles multiple numeric predictors and a numeric outcome in one model. Regression uniquely allows simultaneous estimation of both predictors’ contributions to a continuous outcome and can include their product to test interaction, matching Gladly’s goals.

5) Why other options are incorrect
- "Only test that can capture an interaction": false—factorial ANOVA or other models can capture interactions too; regression is not unique in that sense.
- "Only test that can allow him to draw conclusions about causality": false—regression alone does not establish causality without experimental design or causal inference assumptions.
- "Only test that can handle a continuous outcome variable": false—t-tests, ANOVA and correlations also handle continuous outcomes, but they typically do not accommodate multiple numeric predictors simultaneously in one model.

6) Conclusion
Regression is the appropriate choice because it handles multiple numeric predictors and a numeric outcome in a single model; this matches the provided correct answer.

Gladly wants to find out whether temperature or mood (or both) predicts how many meatballs are sold, and whether there is an interaction. He therefore decides to do a regression. Why is a regression the best analysis for this out of all of the tests we have learned this semester?单项选择题

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