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60 62 64 66 68 70 110 120 130 140 150 160 170 Scatterplot of Weight vs. Height r = 0.74 Height Weight 8 / 34
Today’s Goals The Intuition The Mechanics Some Caveats Simplification Summary There is a linear association between x and y (height and weight) The correlation coefficient tells us about the strength of the relationship We want to know more–describing y (weights), based on x (heights) This technique–using information about one variable to predict the value of another, based on a linear relationship–is known as regression Best way to do this–line through averages of groups But which groups? How do we generalize this? 9 / 34
Today’s Goals The Intuition The Mechanics Some Caveats Simplification The Formula Problem: We want to find the line that smooths the averages of groups. Solution: Match up the SDs, and account for the correlation. Associated with a one SD increase in x is an r SD increase in y , on average: y - ¯ y SD y = r x - ¯ x SD x This is the regression equation 10 / 34
Today’s Goals The Intuition The Mechanics Some Caveats Simplification Example: Averages Data is collected on the heights and weights of adults. It is found that the average height is 65 inches, with an SD of 2 inches, while the average weight is 140 pounds, with an SD of 10 pounds. Height and weight have a correlation coefficient of 0.74. Find the average weight of people with with height 62 inches. 1 Decide which variable is x (explanatory/independent) and which variable is y (response/dependent): Since we are given height, it is the x , while weight, which we want to predict, is the y variable 11 / 34
Today’s Goals The Intuition The Mechanics Some Caveats Simplification Example: Averages (cont.) 2 Calculate y in standard units: Since we are given the x means and SDs, plug them into the regression equation. y - ¯ y SD y = r x - ¯ x SD x y - 140 10 = 0 . 74 62 - 65 2 = - 1 . 11 12 / 34
Today’s Goals The Intuition The Mechanics Some Caveats Simplification Example: Averages (cont.) 3 Solve for y , in its desired units: Since we desire y in its original units, we convert the standard units into pounds y - 140 10 = - 1 . 11 y - 140 = - 11 . 10 y = - 11 . 10 + 140 = 128 . 9 4 Interpret the answer: People with height 62 inches weigh, on average, 128.9 pounds 13 / 34
Today’s Goals The Intuition The Mechanics Some Caveats Simplification Example: Individuals Data is collected on the heights and weights of adults. It is found that the average height is 65 inches, with an SD of 2 inches, while the average weight is 140 pounds, with an SD of 10 pounds.
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