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Linear Regression
Regressions of y on x, regressions of x on y, the correlation coefficient r, the rank correlation coefficient rs, extrapolation and interpolation of data.
Regressions of y on x, regressions of x on y, the correlation coefficient r, the rank correlation coefficient rs, extrapolation and interpolation of data.
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Best fit lines can also be drawn approximately by eye. We start by finding the average x and y, giving the point (xˉ,yˉ). We then take a ruler and place it on this point, and adjust the slope until we find a reasonable best fit line.
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Linear regression is a statistical method used to model the relationship between two variables when data is given as pairs of points (x,y). We fit a straight line (called the regression line) that minimizes the average vertical distance from the points:
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The general equation of the regression line is:
where a is the slope and b is the y-intercept.
The values of a and b can be found using a calculator:
Use Stat>Edit to fill in x- and y-values into L1 and L2.
Then, press Stat, right arrow to the CALC menu, and select 4:LinReg(ax+b).
Pearson's product-moment correlation coefficient, denoted by r, measures the strength and direction of a linear relationship between two numerical variables x and y. Its value always lies between −1 and +1:
r=+1: perfect positive linear relationship
r=−1: perfect negative linear relationship
r=0: no linear relationship
A positive value means y generally increases as x increases; a negative value means y generally decreases as x increases. The closer r is to ±1, the stronger the linear relationship.
If you clickmode, scroll to STAT DIAGNOSTICS , hover over ON, and click ENTER, then any time you perform a linear regression, the calculator will provide Pearson's coefficient in addition to the regression line.
Once we have a regression line y=ax+b, we can use it to predict y by plugging in a value of x.
When using a regression line to predict y from x, we need to be aware of the danger of extrapolation. This occurs when we try to predict y for a value of x far outside the range of x values in our data. For such an x, we cannot trust that the relationship is the same.
While it is possible to use a regression line y=ax+b to predict x with
this is not a reliable process. The best fit line is determined to minimize the difference between the real y’s and the predicted y’s,so the difference between real and predicted values for x may be much larger.