multiple regresson

Correlation coefficient (r)

Measures strength and direction of relationship between two variables

Strong correlation

Value of r close to +1 or -1

Weak correlation

Value of r close to 0

Correlation between Courses and Wage (example)

0.765

Multicollinearity

High correlation between independent variables (X variables)

How to detect multicollinearity

Correlation between X variables is high (≈ 0.7 or higher)

VIF rule

VIF ≥ 5 indicates possible multicollinearity

Regression equation form

y = b0 + b1x1 + b2x2

Intercept (b0)

Predicted value of y when all x variables equal 0

Slope coefficient (b1 or b2)

Change in y for a one-unit increase in x while holding other variables constant

Selling price coefficient (example)

-0.00825

Ad expense coefficient (example)

0.00585

Interpretation of negative coefficient

As x increases

Interpretation of positive coefficient

As x increases

Coefficient interpretation rule

Always include "holding other variables constant"

R-squared (R²)

Percentage of variation in the dependent variable explained by independent variables

R² example

93.19%

Meaning of R²

Independent variables explain 93.19% of variation in the response variable

What R² does NOT explain

Does not explain variation in x variables

F-test (overall model)

Tests if at least one independent variable affects the dependent variable

t-test (individual variable)

Tests if a specific coefficient is significantly different from 0

Significance rule

If p-value < 0.05

Example p-values

0.000

Regression hypothesis (null)

H0: βi = 0

Regression hypothesis (alternative)

H1: βi ≠ 0

Confidence interval (CI)

Estimates the average value of the response variable

Prediction interval (PI)

Estimates the value for an individual observation

Difference between CI and PI

CI = average

Example CI interpretation

Average sales will be between 8.35 and 12.40 units

Correct CI wording

"average sales for all months"

Incorrect CI wording

"randomly selected month"

Extrapolation

Making predictions outside the range of observed data

Extrapolation rule

Predictions outside data range are unreliable

Example extrapolation case

Selling price = 2100 is outside observed range

Regression constant meaning

Point where regression line crosses y-axis

Coefficient meaning (selling price)

Sales decrease by 0.00825 units for each $1 increase in price

Coefficient meaning (ad expense)

Sales increase by 0.00585 units for each $1 increase in ad expense

Model significance

Determined using F-test p-value

When model is significant

When F-test p-value < 0.05

When variable is significant

When t-test p-value < 0.05

Standard error (S)

Measures typical prediction error

Good model indicator

High R² and low standard error

Key regression rule

Always interpret coefficients in context of holding other variables constant

Quick exam trick (R²)

Think "percent of variation explained"

Quick exam trick (CI vs PI)

CI = average

Quick exam trick (sign of coefficient)

Positive = increase