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Linear Regression and Correlation

  • Understand the concept of correlation and regression.
  • Compute and interpret the correlation coefficient.
  • Learn the simple linear regression model.
  • Estimate regression coefficients using least squares.
  • Visualize regression lines and data correlation.

Definition of Correlation

Correlation measures the strength and direction of the relationship between two variables.

  • If two variables increase together, they have a positive correlation.
  • If one variable increases while the other decreases, they have a negative correlation.
  • If there is no systematic relationship, they are uncorrelated.

Correlation Coefficient

The **Pearson correlation coefficient** (\( r \)) measures the strength of a linear relationship:

\[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} \]

Interpretation of \( r \):

Correlation Coefficient (\( r \)) Interpretation
\( 1.0 \) Perfect positive correlation
\( 0.5 \) to \( 1.0 \) Strong positive correlation
\( 0.0 \) to \( 0.5 \) Weak positive correlation
\( -0.5 \) to \( 0.0 \) Weak negative correlation
\( -1.0 \) to \( -0.5 \) Strong negative correlation

Simple Linear Regression

Linear regression is a method for modeling the relationship between two variables using a straight-line equation:

\[ y = b_0 + b_1 x + \varepsilon \]

where:

  • \( y \) = dependent variable (response)
  • \( x \) = independent variable (predictor)
  • \( b_0 \) = intercept (value of \( y \) when \( x = 0 \))
  • \( b_1 \) = slope (change in \( y \) for a unit change in \( x \))
  • \( \varepsilon \) = error term

Least Squares Estimation

The regression coefficients (\( b_0 \) and \( b_1 \)) are estimated using the **least squares method**:

\[ b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \] \[ b_0 = \bar{y} - b_1 \bar{x} \]

Derivation

The sum of squared residuals is given by:

\[ S = \sum (y_i - (b_0 + b_1 x_i))^2 \]

Taking the derivative with respect to \( b_0 \) and \( b_1 \) and solving for zero gives:

\[ b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \] \[ b_0 = \bar{y} - b_1 \bar{x} \]

Visualization of Regression Line and Correlation

Examples

Example 1: A researcher collects the following data on study hours (\( x \)) and test scores (\( y \)):

Study Hours (\( x \)) Test Score (\( y \))
2 60
4 65
6 70
8 75
10 80

Compute the regression equation.

  • \( \bar{x} = 6 \), \( \bar{y} = 70 \)
  • \( b_1 = 2.5 \)
  • \( b_0 = 55 \)

Regression equation:

\[ y = 55 + 2.5x \]

Exercises

  • Question 1: Given the dataset \( (1,2), (2,4), (3,6), (4,8) \), compute \( b_0 \) and \( b_1 \).
  • Question 2: If the correlation coefficient between height and weight is \( r = 0.8 \), interpret its meaning.
  • Question 3: Find the regression line for \( (1,3), (2,5), (3,7) \).
  • Answer 1: \( b_0 = 0 \), \( b_1 = 2 \).
  • Answer 2: Strong positive correlation.
  • Answer 3: \( y = 1 + 2x \).

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