Finally, Gladly and Super Size run a model called m3 which has two predictor variables (temperature and mood) and the outcome variable meatballs as well as an interaction. The output is shown below. Coefficients: Estimate Std. Error t value Pr(|t|) (Intercept) 128.04578 2.43944 52.490

Question

Finally, Gladly and Super Size run a model called m3 which has two predictor variables (temperature and mood) and the outcome variable meatballs as well as an interaction. The output is shown below.

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)    
(Intercept)      128.04578    2.43944  52.490   <2e-16 ***
temperature        7.07231    0.16109  43.903   <2e-16 ***
mood              14.62052    1.62318   9.007   <2e-16 ***
temperature:mood   1.36555    0.09708  14.066   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 19.76 on 361 degrees of freedom
Multiple R-squared:  0.9416,    Adjusted R-squared:  0.9411 
F-statistic:  1941 on 3 and 361 DF,  p-value: < 2.2e-16

How would you interpret this model?

BlackTom AI · Accepted Answer

A positive interaction coefficient (1.36555) means the effect of temperature on meatballs increases as mood increases, and the effect of mood increases as temperature increases. This produces larger predicted meatball counts when temperature and mood have the same sign (both above zero or both below zero) and a reduction when they have opposite signs. 1) Relevant concepts - Linear regression with an interaction: outcome = intercept + β1*temp + β2*mood + β3*(temp*mood). - A positive interaction coefficient (β3 > 0) adds a positive contribution when temp*mood > 0 (same sign) and a negative contribution when temp*mood < 0 (opposite signs). - Statistical significance: very small p-values (<2e-16) indicate coefficients are significantly different from zero; high R-squared (0.9416) shows the model explains much of the variability. 2) Step-by-step reasoning - Model: meatballs = 128.04578 + 7.07231*temp + 14.62052*mood + 1.36555*(temp*mood). - Because β3 = 1.36555 > 0, the product temp*mood increases meatballs when it is positive (temp and mood both + or both -), and decreases meatballs when negative (one +, one -). 3) Sample calculations (intermediate steps) - Example A (both positive): temp = 10, mood = 5 - temp term: 7.07231*10 = 70.7231 - mood term: 14.62052*5 = 73.1026 - interaction: 1.36555*(10*5) = 1.36555*50 = 68.2775 - prediction = 128.04578 + 70.7231 + 73.1026 + 68.2775 = 340.14898 - Example B (opposite signs): temp = 10, mood = -5 - temp term: 70.7231 - mood term: 14.62052*(-5) = -73.1026 - interaction: 1.36555*(10*-5) = -68.2775 - prediction = 128.04578 + 70.7231 - 73.1026 - 68.2775 = 57.389 These show much higher predicted meatballs when temp and mood are both positive (same sign) versus opposite signs. 4) Why the correct answer is correct - The positive interaction means same-sign products add to the outcome, so "More meatballs are sold when mood & temperature have the same sign: both + or both -" correctly captures the interaction effect. 5) Why other options are incorrect - "Temperature and mood affect each other non-linearly": interaction modifies joint effect on meatballs, not that the predictors affect each other. - "Gladly and Super Size sell an average of 128.05 meatballs per day": 128.05 is the intercept (prediction when temp=0 and mood=0), not the overall average necessarily. - "The effects ... are independent": incorrect because a significant interaction shows dependence. - "People buy more meatballs when it is hot and the server is sad": hot (temp+) and sad (mood-) are opposite signs; interaction is negative then, so this statement is not supported. Conclusion: The correct interpretation is that more meatballs are sold when mood and temperature have the same sign (both + or both -), which matches the provided answer.

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