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Mathematics LibreTexts

11: Linear Models

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    • 11.1: Linear Equations
      Linear regression for two variables is based on a linear equation with one independent variable. The equation has the form: y=a+bx where a and b are constant numbers. The variable x is the independent variable, and y is the dependent variable. Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.
    • 11.2: Scatter Plots
      A scatter plot shows the direction of a relationship between the variables. A clear direction happens when there is either: High values of one variable occurring with high values of the other variable or low values of one variable occurring with low values of the other variable. High values of one variable occurring with low values of the other variable.
    • 11.3: The Regression Equation
      A regression line, or a line of best fit, can be drawn on a scatter plot and used to predict outcomes for the x and y variables in a given data set or sample data. There are several ways to find a regression line, but usually the least-squares regression line is used because it creates a uniform line. Residuals measure the distance from the actual value of y and the estimated value of y . The Sum of Squared Errors, when set to its minimum, calculates the points on the line of best fit.
    • 11.4: Testing the Significance of the Correlation Coefficient
      The correlation coefficient tells us about the strength and direction of the linear relationship between x and y. However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the value of the correlation coefficient r and the sample size n, and perform a hypothesis test of the "significance of the correlation coefficient" to decide whether the linear relationship in the sample data is strong enough to use to linear model.
    • 11.5: Prediction
      After determining the presence of a strong correlation coefficient and calculating the line of best fit, you can use the least squares regression line to make predictions about your data.  The process of predicting inside of the observed x values observed in the data is called interpolation. The process of predicting outside of the observed x-values observed in the data is called extrapolation.

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