They decide to begin simply by running a model called mP which has just one predictor variable (price) and the outcome variable attendance. The output of that model is shown below. Coefficients: Estimate Std. Error t value Pr(|t|) (Intercept) 1223.3395 22.2201 55.06

Question

They decide to begin simply by running a model called mP which has just one predictor variable (price) and the outcome variable attendance. The output of that model is shown below.

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 1223.3395    22.2201   55.06   <2e-16 ***
price        -18.5397     0.4864  -38.11   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 45.38 on 132 degrees of freedom
Multiple R-squared:  0.9167,    Adjusted R-squared:  0.9161 
F-statistic:  1453 on 1 and 132 DF,  p-value: < 2.2e-16

Which of the following best describes the relationship between attendance and price?

BlackTom AI · Accepted Answer

The slope coefficient of -18.5397 means that for each one-unit increase in price (one dollar), attendance is expected to change by -18.5397 people — i.e., to decrease by about 18.54 people.

1) Relevant knowledge points and concepts
- In a simple linear regression Attendance = β0 + β1*Price + ε, β1 (the coefficient on price) gives the expected change in attendance for a one-unit (here $1) increase in price.
- The sign of β1 indicates direction (negative → decrease).
- Statistical significance (very small p-values) tells us this relationship is unlikely due to chance.

2) Step-by-step reasoning
- From the output: price Estimate = -18.5397, Std. Error = 0.4864, t = -38.11, Pr(<|t|) < 2e-16.
- The coefficient is negative, so attendance falls as price increases.
- The magnitude -18.5397 gives the change per $1 increase.

3) Intermediate/calculation steps
- Round the coefficient to two decimal places: -18.5397 ≈ -18.54.
- Interpretation: ΔAttendance ≈ -18.54 × ΔPrice(in $). For ΔPrice = +1, ΔAttendance ≈ -18.54.

4) Why the correct answer is correct
- Because regression coefficients are in units of outcome per one unit of predictor, the negative value directly means ~18.54 fewer attendees per additional dollar.
- The very small p-value (<2e-16) and large t-statistic show this estimate is highly statistically significant.

5) Why other options are incorrect
- Options claiming “18.54 more” reverse the sign (wrong direction).
- Options that say “Every increase of $18.54 is associated with 1 person” invert the units (they confuse per-$1 vs per-$18.54).
- “There is no significant relationship” is wrong because p-values are extremely small and R-squared is high (0.9167), indicating a strong, significant relationship.

6) Conclusion
The correct description is: “Every increase in price of $1 is associated with 18.54 fewer people in attendance.” (option 4).

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