handout_2_3

handout_2_3 - 2.3 Least-Squares Regression (OLS)...

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2.3 Least-Squares Regression (OLS) •correlation measures the direction and strength of the linear relationship between two quantitative variables •we would like to summarize the overall pattern by drawing a line on the scatterplot A regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes. We often use a regression line to predict the value of y for a given value of x . Regression, unlike correlation, requires that we have an explanatory variable and a response variable. Example Does fidgeting keep you slim? Some people don’t gain weight even when they overeat. Perhaps fidgeting and other “nonexercise activity” (NEA) explains why- the body might spontaneously increase NEA when fed more. Researchers deliberately overfed 16 healthy adults for 8 weeks. The measured fat gain (response variable) and the increase in NEA (explanatory variable). Here are the data: NEA increase (calories)- x Fat gain (kg)- y -94 4.2 -57 3.0 -29 3.7 135 2.7 143 3.2 151 3.6 245 2.4 355 1.3 392 3.8 473 1.7 486 1.6 535 2.2 571 1.0 580 0.4 620 2.3 690 1.1
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Figure 2.11 displays the scatterplot (response- y , explanatory- x ) we can describe the overall pattern by drawing a straight line through the points no straight line passes exactly through all the points fitting a line to data means drawing a line that comes as close as possible to the points 2
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Straight Lines Suppose that y is a response variable (vertical axis) and x is an explanatory variable (horizontal axis). A straight line relating y to x has an equation of the form: y = b 0 + b 1 x In this equation, b 1 is the slope, the amount by which y changes when x increases by one unit. The number b 0 is the intercept, the value of y when x =0. Any straight line describing our data has the form: fat gain = b 0 + ( b 1 × NEA increase) The computer generates the following regression line: fat gain = 3.505 - (0.00344 × NEA increase) slope b 1 = -0.00344 tells us that fat gained goes down by 0.00344 kg for each cal increase in NEA slope b 1 tells us rate of change in response variable ( y ) as explanatory variable ( x ) changes intercept b 0 = 3.505 kg is the estimated fat gain if NEA does not change when a person overeats
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handout_2_3 - 2.3 Least-Squares Regression (OLS)...

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