Linear Regression Handbook Chapter Experimental Testing n n
Linear Regression Handbook Chapter
Experimental Testing n n 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.
Model Fitting Correlation = n 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. n The mathematical equation describes a functional relationship between the variables. n
Model Fitting It may be relatively simple to demonstrate the functional relationship between the dependent and independent variables if the relationship is direct. n Direct relationships (linear relationships) can be represented by a straight line. n
Linear Models A straight line is simplest statistically analyzable function. n 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. n
Slope –Intercept Formula y=mx+b y = dependent variable x = independent variable m = slope of the line b = y-intercept of the line
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.
y – Intercept b = y-intercept of the line • The y-intercept is the point at which a straight line intersects the y-axis.
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. n The points of the straight line have the coordinates corresponding to the (x, y) pairs. n
Linear Regression Since measurements inherently contain both systematic and random errors, data points will not fit a given equation perfectly. n Linear regression is a statistical method for finding the best fitting straight line to a set of (x, y) pairs. n
The expression giving the most probable slope of the fitted straight line is: N m= N N N S yixi – S yi S xi i=1 N N S i=1 2 xi – i=1 N [S xi i=1 2 ]
The expression giving the most probable y-intercept of the fitted straight line is: N S b= i=1 N 2 xi N N S yi – S xi S yixi i=1 N N S i=1 2 xi – i=1 N [S xi i=1 2 ]
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