Unit 6: Business Management Toolkit — BMT 15 Simple Linear Regression
What is Simple Linear Regression?
Simple Linear Regression is a statistical method used to model the relationship between two variables: a dependent variable (\(y\)) and an independent variable (\(x\)). It estimates how changes in \(x\) affect \(y\) with a straight line (the regression line), widely used in business forecasting, sales analysis, and performance prediction.
Purpose:
- Predict outcomes and trends
- Quantify relationships between variables
- Support data-driven management decisions
- Predict outcomes and trends
- Quantify relationships between variables
- Support data-driven management decisions
The Regression Line Equation
The general form of the simple linear regression equation is:
y = a + bx
where:
\(y\) = Dependent variable
\(x\) = Independent variable
\(a\) = Intercept (value of \(y\) when \(x=0\))
\(b\) = Slope (change in \(y\) for a unit change in \(x\))
where:
\(y\) = Dependent variable
\(x\) = Independent variable
\(a\) = Intercept (value of \(y\) when \(x=0\))
\(b\) = Slope (change in \(y\) for a unit change in \(x\))
Core Formulas
- Intercept (\(a\)): a = \overline{y} - b\overline{x}
- Slope (\(b\)): b = \frac{\sum (x_i - \overline{x})(y_i - \overline{y})}{\sum (x_i - \overline{x})^2}
- Predicted value of \(y\): y = a + bx
- Coefficient of Determination (\(R^2\)): R^2 = \frac{ \text{Explained Variation} }{ \text{Total Variation} } Indicates how well \(x\) explains changes in \(y\) (ranges from 0 to 1).
Key: Use a scatter plot for visualization and residuals (differences between predicted and actual values) to test model accuracy.
Worked Example
Scenario: A store wants to predict monthly sales (\(y\)) from advertising spend (\(x\)).
Collected data:
x:\ 5,\ 12,\ 18,\ 25,\ 32,\ 38 (thousand \$)
y:\ 14,\ 28,\ 32,\ 50,\ 62,\ 68 (thousand \$)
Collected data:
x:\ 5,\ 12,\ 18,\ 25,\ 32,\ 38 (thousand \$)
y:\ 14,\ 28,\ 32,\ 50,\ 62,\ 68 (thousand \$)
Regression calculation: (showing method, not full math)
Calculate means \(\overline{x},\ \overline{y}\), compute \(b\) using the formula above, then \(a\), and finally plug in values for prediction.
If \(b = 1.8,\ a = 4\), predicted sales for advertising \$20,000:
y = 4 + 1.8 \times 20 = 40 (thousand \$)
Calculate means \(\overline{x},\ \overline{y}\), compute \(b\) using the formula above, then \(a\), and finally plug in values for prediction.
If \(b = 1.8,\ a = 4\), predicted sales for advertising \$20,000:
y = 4 + 1.8 \times 20 = 40 (thousand \$)
Advantages & Limitations
| Advantages | Limitations |
|---|---|
|
- Easy to interpret - Fast forecasting method - Inspects direction and strength of relationships - Reveals simple patterns in data |
- Only models linear relationships - Sensitive to outliers - May oversimplify real-world scenarios - Requires variables to be quantified |
Conclusion
Simple linear regression is a powerful toolkit method for business managers—enabling robust predictions, evidence-based strategies, and clearer understanding of data-driven trends.