2.3
LeastSquares 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 (responsey, explanatoryx)
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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 Fall '08
 ABDUS,S.
 Regression Analysis, Standard Deviation, Variance, regression line, Nea

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