1. Linear Regression Handbook Chapter

2. Experimental Testing Data are collected, in scientific experiments, to test the relationship between various measurable quantities that are predicted by a hypothesis, either to support or invalidate the hypothesis. Data are collected, in scientific experiments, to test the relationship between various measurable quantities that are predicted by a hypothesis, either to support or invalidate the hypothesis. The dependent variable is measured under varying, but preset, values of the independent variable. The dependent variable is measured under varying, but preset, values of the independent variable.

3. Model Fitting Correlation = Correlation = If the variables have a correlation, it is convenient to express the relationship in the form of a mathematical equation, known as a model or natural law. If the variables have a correlation, it is convenient to express the relationship in the form of a mathematical equation, known as a model or natural law. The mathematical equation describes a functional relationship between the variables. The mathematical equation describes a functional relationship between the variables.

4. Model Fitting It may be relatively simple to demonstrate the functional relationship between the dependent and independent variables if the relationship is direct. It may be relatively simple to demonstrate the functional relationship between the dependent and independent variables if the relationship is direct. Direct relationships (linear relationships) can be represented by a straight line. Direct relationships (linear relationships) can be represented by a straight line.

5. Linear Models A straight line is simplest statistically analyzable function. A straight line is simplest statistically analyzable function. One way to express the linear relationship between an independent variable (x) and a dependent variable (y) is the slope –intercept formula: y = m x + b. One way to express the linear relationship between an independent variable (x) and a dependent variable (y) is the slope –intercept formula: y = m x + b.

6. Slope –Intercept Formula y = m x + b y = dependent variable x = independent variable m = slope of the line b = y-intercept of the line

7. Slope m = slope of the line The slope represents the amount of tilt (slant) the straight line has relative to the x- and y- axes.The slope represents the amount of tilt (slant) the straight line has relative to the x- and y- axes.

8. y – Intercept b = y-intercept of the line The y-intercept is the point at which a straight line intersects the y-axis.The y-intercept is the point at which a straight line intersects the y-axis.

9. Slope –Intercept Formula If the slope and y –intercept of a straight line is given, then the y value can be calculated, with confidence, for every given value of x. If the slope and y –intercept of a straight line is given, then the y value can be calculated, with confidence, for every given value of x. The points of the straight line have the coordinates corresponding to the (x, y) pairs. The points of the straight line have the coordinates corresponding to the (x, y) pairs.

10. Linear Regression Since measurements inherently contain both systematic and random errors, data points will not fit a given equation perfectly. Since measurements inherently contain both systematic and random errors, data points will not fit a given equation perfectly. Linear regression is a statistical method for finding the best fitting straight line to a set of (x, y) pairs. Linear regression is a statistical method for finding the best fitting straight line to a set of (x, y) pairs.

11. The expression giving the most probable slope of the fitted straight line is: N N N N N  x i  –  [  x i ]  i=1 i=1 i=1 i=1 N N N N N N N  y i x i  –  y i  x i N  y i x i  –  y i  x i i=1 i=1 i=1 i=1 i=1 i=1 m =

12. The expression giving the most probable y-intercept of the fitted straight line is: N N N N N  x i  –  [  x i ]  i=1 i=1 i=1 i=1 N N N N N N N N  x i 2   y i  –   x i   y i x i i=1i=1 i=1 i=1 b =