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2_3_Handout - 2.3 Least-Squares Regression correlation...

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2.3 Least-Squares Regression 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 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
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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 = a + bx In this equation, b is the slope, the amount by which y changes when x increases by one unit. The number a is the intercept, the value of y when x=0. Any straight line describing our data has the form: fat gain = a + (b * NEA increase) Figure 2.12 The computer generates the following regression line: fat gain = 3.505 - (0.00344 * NEA increase) slope b = -0.00344 tells us that fat gained goes down by 0.00344 kg for each calorie increase in NEA slope b tells us rate of change in response variable (y) as explanatory variable (x) changes intercept a = 3.505 kg is the estimated fat gain if NEA does not change when a person overeats Prediction #1) Suppose an individual’s NEA increase when overeating is 400 calories.
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