17 Regression - Regression Predicts Y from X Linear...

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Regression ! Predicts Y from X ! Linear regression assumes that the relationship between X and Y can be described by a line Correlation vs. regression Regression assumes. .. ! Random sample ! Y is normally distributed with equal variance for all values of X The parameters of linear regression Y = ! + X
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Positive ! Negative ! ! = 0 Higher Lower Estimating a regression line Y = a + b X Nomenclature Residual: Y i ! ˆ Y i Finding the "least squares" regression line SS residual = Y i ! ˆ Y i ( ) 2 i = 1 n Minimize:
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Best estimate of the slope b = X i ! X ( ) Y i ! Y ( ) i = 1 n X i ! X ( ) 2 i = 1 n (= "Sum of cross products" over "Sum of squares of X") Remember the shortcuts: X i ! X ( ) Y i ! Y ( ) i = 1 n " = X i Y i " # $ % % & ( ( ! X i Y i " " n X i ! X ( ) 2 i = 1 n " = X i 2 ( ) " ! X i " # $ % % & ( ( 2 n Finding a Y = a + b X So. . a = Y ! b X Example: Predicting age based on radioactivity in teeth Many above ground nuclear bomb tests in the ‘50s and ’60s may have left a radioactive signal in developing teeth. Is it possible to predict a person’s age based on dental C 14 ?
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Teeth data: Estimated Actual 1982 1985 1985 1986 1987 1987 1990 1989 1991 1991 1991 1991 1992 1991 1954 1951 1956 1951 1954 1958 Estimated Actual 1963 1963 1963 1964 1965 1964 1967 1970 1969 1972 1970 1973 1973 1972 1974 1975 1980 1983 1982 1983 Teeth data: X = 39488, Y ! !
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17 Regression - Regression Predicts Y from X Linear...

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