Multiple Regression Exercise

Linear Regression Analysis: Key Formulas and Calculations

Given Model:

$\text{visitors}={\beta }_0+{\beta }_1\cdot \text{prices}+{\beta }_2\cdot \text{artists}$

From Output:

• ${\beta }_0=700.8493$
• ${\beta }_1=-3.7173$
• ${\beta }_2=4.0851$

Call:
lm(formula = visitors ~ prices + artists)

Residuals:
     Min       1Q   Median       3Q      Max 
-155.458  -52.355   -9.802   50.293  192.999 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 700.8493    30.4836  22.991  < 2e-16 ***
prices       -3.7173     0.3046 -12.205  < 2e-16 ***
artists       4.0851     0.6796   6.011 8.48e-08 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Statements and Solutions

1. Correlation between number of visitors and entrance fee:

   • Negative correlation: If ${\beta }_1<0$.

2. Impact of Entrance Fee Difference:

   • Calculate change in visitors when price difference is 30 EUR:
   $$\Delta \text{visitors}={\beta }_1\cdot \Delta \text{prices}=-3.7173\cdot 30=-111.519\approx -112$$

3. Variability in Visitors Explained by the Model:

   • Given $R^2$:
   $$R^2=0.7449$$
   • $74.49\%$ of the variability in visitors is explained by the model.

4. Significance of the Coefficient for Artists:

   • Check p-value for ${\beta }_2$:
   $$\text{p-value}=8.48\times 10^{-8}<0.05$$
   • The coefficient for artists is significantly different from 0 at the 5% significance level.

5. Expected Number of Visitors with Given Prices and Artists:

   • Calculate expected visitors for 30 EUR fee and 22 artists:
   $$\text{visitors}={\beta }_0+{\beta }_1\cdot \text{prices}+{\beta }_2\cdot \text{artists}$$ $$\text{visitors}=700.8493+(-3.7173\cdot 30)+(4.0851\cdot 22)$$ $$\text{visitors}=700.8493-111.519+89.8722=679.2025$$
   • Expected visitors: $679$ (less than 700).