AP Stats: 3.3 Least-Squares
Regression Line
• Regression Line is a straight line that
describes how a response variable (y)
changes as an explanatory variable (x)
changes. This is a MODEL for the data.
• We us the regression line to make
PREDICTIONS
Least-squares regression line
(LSRL)
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.
Equations of the LSRL
***SLOPE
br
sy
yˆ  a  bx
intercept
a  y  bx
sx
Calculator: LinReg(a+bx) L1, L2, Y1
Coefficient of Determination - r2
The fraction of the variation in the values
of y that is explained by least-squares
regression of y on x.
“we say ?% of the variation in y is
explained by least-squares regression
of y on x.”
Residuals
The difference between an observed
value of the response variable and the
value predicted by the regression line.
–
Residual = observed y – predicted y
Resid =
y  yˆ
Residual Plot
plots the residuals on the vertical axis
against the explanatory variable on the
horizontal axis.
:
The mean of the residuals is always 0
(if not we have rounding error)
Influential
An observation is Influential if
removing it would markedly change
the position of the regression line.
Points that are outliers in the x
direction are often influential.