Linear Correlation and Regression Classrooms

90.1 Introduction to Linear Correlation

Linear correlation measures the strength and direction of a linear relationship between two quantitative variables. It describes how closely data points cluster around a straight line.

• Positive Correlation: As one variable increases, the other tends to increase.
• Negative Correlation: As one variable increases, the other tends to decrease.
• No Correlation: No apparent linear relationship between the variables.
• Scatter Plot: A graphical representation of the relationship between two variables, where each point represents a pair of values.

90.2 Product-Moment Formula

The Pearson Product-Moment Correlation Coefficient ($r$) quantifies the linear relationship between two variables. It ranges from -1 to +1.

• Formula: $$r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}$$ where $n$ is the number of data pairs, $x$ and $y$ are the individual data points.
• Interpretation:
  • $r = 1$: Perfect positive linear correlation.
  • $r = -1$: Perfect negative linear correlation.
  • $r = 0$: No linear correlation.
  • Values closer to $\pm 1$ indicate stronger linear relationships.

90.3 Significance of Correlation Coefficient

Determining the statistical significance of the correlation coefficient helps us decide if the observed linear relationship in a sample is likely to exist in the population.

• Hypothesis Testing:
  • Null Hypothesis ($H_0$): There is no linear correlation in the population ($\rho = 0$).
  • Alternative Hypothesis ($H_1$): There is a linear correlation in the population ($\rho \neq 0$, or $\rho > 0$, or $\rho < 0$).
• T-test for Correlation: $$t = r \sqrt{\frac{n-2}{1-r^2}}$$ with $n-2$ degrees of freedom. Compare calculated $t$ to critical $t$ from a t-distribution table.
• P-value: The probability of observing a sample correlation as extreme as, or more extreme than, the one observed, assuming the null hypothesis is true. A small p-value (typically < 0.05) indicates statistical significance.

90.4 Correlation Problems

Practice applying the concepts of linear correlation to various datasets and interpret the results.

91.1 Introduction to Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable (response variable) and one or more independent variables (predictor variables) by fitting a linear equation to observed data.

• Purpose:
  1. To predict the value of the dependent variable based on the independent variable.
  2. To understand the strength and direction of the relationship between variables.
• Regression Line: The straight line that best describes the linear relationship between the variables on a scatter plot. It is also known as the line of best fit.
• Dependent Variable (Y): The variable being predicted or explained.
• Independent Variable (X): The variable used to predict or explain the dependent variable.

91.2 Least-Squares Regression Lines

The least-squares regression line is the line that minimizes the sum of the squared vertical distances (residuals) from the data points to the line.

• Equation of the Line: $$\hat{y} = a + bx$$ where $\hat{y}$ is the predicted value of the dependent variable, $x$ is the independent variable, $a$ is the y-intercept, and $b$ is the slope.
• Slope ($b$): $$b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$$
• Y-intercept ($a$): $$a = \bar{y} - b\bar{x}$$ where $\bar{y}$ is the mean of the dependent variable and $\bar{x}$ is the mean of the independent variable.

91.3 Regression Problems

Practice calculating and interpreting least-squares regression lines to make predictions and understand relationships between variables.