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| An equation or formula that simplifies and represents reality. |
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| A linear model is an equation of a line. To interpret a linear model, we need to know the variables (along with their W's) and their units. |
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| The value of y found for a given x-value in the data. A predicted value is found by substituting the x-value in the regression equation. A predicted value are the values on the fitted line; the points (x,y) all lie exactly on the fitted line. |
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Residuals are the differences between data values and the corresponding values predicted be the regression model-or, more generally, values predicted by any model.
Residual= observed value- predicted value= e=y-y |
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| The least square criterion specifics the unique line that minimizes the variance of the residuals or, equivalently, the sum of the squared residuals. |
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| Regression line (line of best fit) |
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| The particular linear equation (y=a+bx)that satisfies the least squares criterion is called the least squares regression line. Casually, we often just call it the regression line, or the line of best fit. |
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| The slope ,b, gives a value in "y-units per x-unit." Changes of one unit in x are associated with changes of b units in predicted values of y. |
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| The intercept,a, gives gives a starting value in y-units. It is the y-value when x is 0. |
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| Because the correlation is always less than 1.0 in magnitude each predicted y tends to be fewer standard deviations from its mean than its corresponding x was from its mean. |
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| Although linear models provide an easy way to predict values of y for a given value of x, it is unsafe to predict for values of x far from the ones used to find the linear models equation. Such extrapolation may pretend to see into the future, but the predictions should not be trusted. |
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| A variable that is not explicitly part of a model, but affects the way variables in the model appear to be related. Because we can never be certain that observational data are not hiding a lurking variable that influences both x and y, it is never safe to conclude that a linear model demonstrates a casual relationship, no matter how strong the linear association. |
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