Essential Statistics Chapter 51 Least Squares Regression Line u Regression line equation: y = a + bx ^ –x is the value of the explanatory variable –“y-hat”

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Presentation transcript:

Essential Statistics Chapter 51 Least Squares Regression Line u Regression line equation: y = a + bx ^ –x is the value of the explanatory variable –“y-hat” is the predicted value for a x value –a and b are just the intercept and slope of a straight line

Essential Statistics Chapter 52 ^ u Regression equation: y = a + bx Regression Line Calculation where s x and s y are the standard deviations of the two variables, and r is their correlation

R-square (R 2 ) u To assess how well regression equation (a model) explain and predicts future outcomes u R-square value (0 ≤ r 2 ≤ 1) u In model analysis, measuring the accuracy of regression equation ◙ value of 1 indicates a reliable model for future prediction ◙ value of zero indicates the model fails. Essential Statistics Chapter 53

Essential Statistics Chapter 54 Coefficient of Determination (R 2 ) u In linear regression, R 2 is the square of the sample correlation coefficient u R 2 or r 2 is the fraction of the explained variation to the total variation in the values of the response variable (y) ◙ r=1: r 2 =1:regression line explains all (100%) of the variation in y ◙ r=.7: r 2 =.49: regression line explains almost half (50%) of the variation in y

Essential Statistics Chapter 55 Residuals u A residual is the difference between an observed value of the response variable and the value predicted by the regression line: residual = y  y ^

Essential Statistics Chapter 46 Measuring Strength & Direction of a Linear Relationship u The correlation coefficient r –measure of the strength of the relationship: the stronger the relationship, the larger the magnitude of r. –measure of the direction of the relationship: positive r indicates a positive relationship, negative r indicates a negative relationship.

Essential Statistics Chapter 17 Weight Data: Histogram Weight * Left endpoint is included in the group, right endpoint is not. Number of students

Essential Statistics Chapter 18 Weight Data: Stemplot (Stem & Leaf Plot) Key 20 | 3 means 203 pounds Stems = 10’s Leaves = 1’s

Essential Statistics Chapter 29 Five-Number Summary u minimum = 100 u Q 1 = u M = 165 u Q 3 = 185 u maximum = 260 The middle 50% of the data are between Q 1 and Q 3

Essential Statistics Chapter 210 M Weight Data: Boxplot Q1Q1 Q3Q3 minmax Weight