Chapter 15 STATS.docx

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30 Mar 2012
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Chapter 15
Describing Relationships: Regression, Prediction, and Causation
Regression lines
A regression line is a straight line that describes how a response variable y changes as an
explanatory variable x changes. (predict the value of y for a given value of x)
Because we want to predict y from x, we want a line that is close to the poitsn in the vertical
direction
The least-squares regression line of y on x is the line that makes the sum of the squares of the
vertical distances of the data points from the line as small as possible
X stands for the explanatory variable and y for the response variable
Y = a + bx
The number b is the slope of the line (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)
To use the equation for prediction, just substitute your x-value into the equation and calculate the
resulting y-value
Understanding prediction
Prediction is based on fitting some “model” to set of data
Prediction works best when the model first the data closely
Prediction outside the range of the available data is risky
Beware of extraporlation (prediction outside the range of available data)
Correlation and regression
Correlation measures the direction and strength of a straight-line relationship
Regression draws a line to describe the relationship
Correlation does not require choosing an explanatory variable, regression is opposite
Both correlation and regression are strongly affected by outliers
The usefulness of the regression line for prediction depends on the strength of the association
(depends on the correlation between the variables)
The square of the correlation r^2 is the proportion of the variation in the values of y that is
explained by the least-square regression of y on x
The idea is that when there is a straight-line relationship, some of the variation in y is accounted
for by the fact that as x changes it pulls y along with it
In reporting a regression, it is usual to give r^2 as a measure of how successful the regression was
in explaining the response
Perfect correlation (r = 1 or r = -1) means the points lie exactly on a lie
The question of causation
A strong relationship between two variables does not always mean that change sin one variable
cause changes in the other
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