Linear Regression

Question 1

Linear Model Equation

Answer 1

ŷ = bₒ + b₁ x

Question 2

Model

Answer 2

An equation or formula that simplifies and represents reality.

Question 3

The points (x, ? ) all lie exactly on the fitted line.

Answer 3

The value of ŷ found for a given x-value in the data such that the points (x, ŷ ) all lie exactly on the fitted line.

Question 4

Residuals

Answer 4

Difference between the data values and the corresponding values predicted by the regression model,
Observed value - Predicted value:

y - ŷ

Question 5

Regression Line

Answer 5

The equation of this line satisfies the least squares criterion.

Also called the line of best fit.

Question 6

Line of best fit

Answer 6

The equation of this line satisfies the least squares criterion.

Also called the regression line.

Question 7

Slope Formula for b₁

Answer 7

b₁ = r Sy / Sₓ

Question 8

Intercept Formula

Answer 8

bₒ = ȳ - b₁ x̄

Question 9

Regression to the mean

Answer 9

This happens because the correlation (r) is always less than 1.0 in magnitude:

each predicted ŷ tends to be fewer standard deviations from its mean than its corresponding x was from its mean:

ẑ_y = r z_x .

Question 10

Standard Deviation of the residuals Formula

Answer 10

s_e = √ (Σe²
___
n-2)

Question 11

Correlation Squared, R², gives…

Answer 11

This gives the fraction of the variability of y accounted for by the least squares linear regression on x.

Question 12

Does the Plot Thicken? Condition

Answer 12

Check the scatterplot of residuals vs. the x-values for this condition

(should be horizontal and shapeless for, equally scattered for all predicted values).