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Chapter 10 Regression

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Defining Regression Simple linear regression features one independent variable and one dependent variable, as in correlation the assumption is that the independent variable influences the variation in the dependant variable. Simple linear regression features one independent variable and one dependent variable, as in correlation the assumption is that the independent variable influences the variation in the dependant variable.

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Simple linear regression takes the form of a mathematical equation:

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Defining Regression Where: Where: a is the Y intercept. a is the Y intercept. It is the point at which the regression line crosses the Y- axis. It is the point at which the regression line crosses the Y- axis. It equals Y (the predicted value of Y) when X is 0. The value of the Y intercept can be positive or negative. It equals Y (the predicted value of Y) when X is 0. The value of the Y intercept can be positive or negative. b represents the slope of the regression line. b represents the slope of the regression line. It represents the change in Y per one unit change in X. As X increases by one, Y changes by the value of b. It represents the change in Y per one unit change in X. As X increases by one, Y changes by the value of b. The slope shows how much and in which direction Y will vary per one unit change in the value of X. The slope shows how much and in which direction Y will vary per one unit change in the value of X. A positive value for b indicates that a one unit increase in X leads to an increase in Y by the value of b. A positive value for b indicates that a one unit increase in X leads to an increase in Y by the value of b. A negative b value indicates that a one unit increase in X leads to a decrease in Y by the value of b. A negative b value indicates that a one unit increase in X leads to a decrease in Y by the value of b.

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Defining Regression Regression analysis uses the least squares technique to establish the equation. Regression analysis uses the least squares technique to establish the equation. It constructs the straight line that comes closest to all points in the scatter plot between X and Y. It constructs the straight line that comes closest to all points in the scatter plot between X and Y. The least squares procedure provides the best fit in that it lies at the point where the squared deviations from all the points on the scatter plot are minimized and thus the accuracy of the prediction is maximized. The least squares procedure provides the best fit in that it lies at the point where the squared deviations from all the points on the scatter plot are minimized and thus the accuracy of the prediction is maximized.

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Using Regression: Predicting Prison Population Size Regression analysis can provide information that is useful to legislators. Regression analysis can provide information that is useful to legislators. For instance legislators often want to know the expected size of the prison population. For instance legislators often want to know the expected size of the prison population. This is one of the major uses of regression, to attempt to anticipate future conditions. The assumption is that the past will be like the past. This is one of the major uses of regression, to attempt to anticipate future conditions. The assumption is that the past will be like the past. If present trends continue the regression equation gives us information as to what can be expected. If present trends continue the regression equation gives us information as to what can be expected.

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Using Regression: Predicting Prison Population Size In this example, we conduct a regression analysis on the murder rate per 100,000 for the United States during the period 1976-1998. Our goal is to predict the homicide rate in 2005. In this example, we conduct a regression analysis on the murder rate per 100,000 for the United States during the period 1976-1998. Our goal is to predict the homicide rate in 2005.

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Calculation Formulas To calculate regression coefficients by hand you must first construct a table. Table 10.3 consists of our murder rate data with one change. To calculate regression coefficients by hand you must first construct a table. Table 10.3 consists of our murder rate data with one change. Instead of entering the year as X you enter the number of the year in the data. This is done to simplify the calculations. Y is left as the murder rate per 100,000 population. Instead of entering the year as X you enter the number of the year in the data. This is done to simplify the calculations. Y is left as the murder rate per 100,000 population.

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Calculation Formulas Therefore we can see that the expected murder rate in the year 2005 will be 7.4/100,000. Therefore we can see that the expected murder rate in the year 2005 will be 7.4/100,000. This is limited to the idea the past pattern will continue into the future. This is limited to the idea the past pattern will continue into the future.

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Curvilinear Relationship Figure 10.2 shows us our best fit regression line the murder rates per 100,000 in the United States, 1976-1998 Figure 10.2 shows us our best fit regression line the murder rates per 100,000 in the United States, 1976-1998 You can see a curvilinear relationship, the murder rate follows a curved line for this time period You can see a curvilinear relationship, the murder rate follows a curved line for this time period We can also see that the murder rate is decreasing over time. We can also see that the murder rate is decreasing over time.

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Conclusion Linear regression – a statistical method based upon correlation that is used to make projections or predictions. Linear regression – a statistical method based upon correlation that is used to make projections or predictions. Assumes that the independent and dependent variables are correlated and that they share a linear relationship Assumes that the independent and dependent variables are correlated and that they share a linear relationship “Best fit” line equation is Y= a + bX “Best fit” line equation is Y= a + bX The value of a tells you the value of Y when X=0. The slope or b tells you how much Y increases or decreases for every one unit change in X The value of a tells you the value of Y when X=0. The slope or b tells you how much Y increases or decreases for every one unit change in X

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