Review
estimate the correlation
• quadrant: a quadrant is any of the four regions into which a
plane is divided by a horizontal line and a vertical line.
Make Predictions
• Regression Line: an equation for predicting the response
outcome.
The regression line predicts the value for the response
variable y as a straight-line function of the value x of the
explanatory variable.
Let ŷ denote the predicted value of y . The equation for the
regression line has the form
ŷ = a + bx
In the above formula, a denotes the y-intercept and b
denotes the slope.
• The y-intercept is the predicted value of y when x = 0.
• The slope b is the amount that ŷ changes when x changes by
one unit.
• residual: the prediction error for an observation, which is
the differency ŷ − y between the actual value and the
predicted value of the response variable, is called a residual.
Graphically in the scatterplot, the residual for an observation is
the vertical distance between the point and the regression line.
• Residual sum of squares:
Residual sum of squares =
X
(residual)2 =
X
(y − ŷ )2
• Least Squares Method
Among the possible lines that can go through data points in a
scatterplot, this method gives the regression line that has the
smallest value for the residual sum of squares in using
ŷ = a + bx to predict y .
• having some positive residuals and some negative
residuals, but the sum (and the mean) of the residuals
equals 0.
• passing through the point (x̄, ȳ ).
• The slope:
sy
b=r
sx
The y-intercept:
a = ȳ − bx̄
• r-Squared (r 2 )
Interpretation: the proportion of the variation in the y-values
that is accounted for by the linear relationship of y with x.
Some Cautions in Analyzing Associations
• Extrapolation is dangerous. Extrapolation refers to using a
regression line to predict y values for x values outside the
observed range of data.
• Be cautious of influential outliers
• Correlation does not imply causation.