1. Least-Squares Regression
Section 3.3

2. Regression line
A straight line that describes how a response variable y changes as an explanatory variable x changes
Used to predict y for a value of x

3. Data must have explanatory and response variables
Used for linear trends

4. Least-squares regression line
The line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible
y hat = a + bx
where y hat is the predicted value, a is the y-intercept, and b is the slope of the line

5. LSRL
The squares are made from the vertical distance from each point to the line.

6. y hat = a + bx b= r(sy/sx) a = ybar – b*xbar
The slope, b, is equal to the standard deviation of y (sy) divided by the standard deviation of x (sx) all multiplied by the correlation, r
a = ybar – b*xbar
The y-intercept, a, is equaly to the slope, b, multiplied by the mean for the x values, xbar, subtracted from the mean for the y values, ybar

7. Calculators
You can calculate a, b, and r by using the LinReg(a+bx) tool on your calculator
Go to Stat—Calc--#8(LinReg(a+bx))
There is a #4 under Calc that says LinReg(ax+b), which will give you the same a and b values for the data set, but the slope is a and y-intercept is b in this function, so use #8

8. Coefficient of determination
The fraction of the variation in the values of y that is explained by the least- squares regression of y on x r2
Sometimes represented as a percentage
% of the variation in y is explained by the least-squares regression of y on x

9. Facts about LSR:
The distinction between explanatory and response variables is essential
A change of one standard deviation in x corresponds to a change of r standard deviations in y
The LSRL always passes through the point (x bar, y bar)

10. Residual
The difference between an observed value of the response variable and the value predicted by the regression line
Residual = observed y – predicted y
The mean of the least-squares residuals is always zero
If you don’t get exactly zero, it’s due to round-off error

11. Residual plot
A scatterplot of the regression residuals against the explanatory variable
Helps assess the fit of a regression line
Uniform scatter indicates the regression line fits well

12. Curved pattern--relationship is not linear
Individual points with large residuals are outliers in the vertical (y) direction because they lie far from the line
Individual points that are extreme in the x direction may not have large residuals

13. Influential Observations
If removing the observation would markedly change the result of the calculation, it is influential
Points that are outliers in the x direction are often influential

14. Practice Problems
pg. 176 #