Chapter 13

Question 1

Y hat = Ŷ

Answer 1

Ŷ

Question 2

Define a Simple Linear Regression

Answer 2

AKA: The Prediction Line

• Ŷ_i* = predicted value of Y for observation i
• X_i* = value of X for observation i
• b₀* = sample Y intercept
• b₁* = sample slope

Question 3

Review the Computational Formula for the Slope, _b1

Answer 3

See page 433 and 434 in book

Question 4

When using a regression model, only use what?

Answer 4

The relevant range of the independent variable in making predictions

i..e. you can interpolate with the relevant range of X, but you cannot extrapolate

Question 5

Symbols for Y Intercept and Slope in Regression Testing

Answer 5

Y intercept = b₀

Slope = b₁

Question 6

Define the Least-squares method

Answer 6

A method that minimizes the sum of the squared differences between the actual values (Y_i) and the predicted values (Y_i)

Question 7

Define the three measures of Variation

Answer 7

Needed when using the least-square method to determine the regression coefficients

• SST - Total Sum of Squares
• SSR - Regression Sum of Squares (This is the explained variation)
• SSE - Error Sum of Squares(This is the unexplained variation)

SST = SSR + SSE

Question 8

Define the Total Sum of Squares (SST)

Answer 8

Question 9

Define the Regression Sum of Squares (SSR)

Answer 9

Question 10

Define the Error Sum of Squares (SSE)

Answer 10

Question 11

Define the Coefficient of Determination

Answer 11

Measures the proportion of variation of Y that is explained by the variation in the independent variable X

The range of r² is from 0-1

The greater the value, the more the variation in Y in the regression model can be explained by the variation in X

Question 12

Review Computational Forumlas p.440

Do I need these equations?

Answer 12

Question 13

Define the Standard Error of the Estimate

Answer 13

The standard deviation around the prediction line (i.e. Measures the variability of the observed Y values from the predicted Y values)

Question 14

What are the 4 Assumptions of Regression

Answer 14

L.I.N.E.

1. Linearity (Rel’t b/t the variables is linear)
2. Independence of errors (requires that the errors, ε_i, are independent of one another)
3. Normality of error (requires that the errors, ε_i, be independent of each other)
4. Equal variance (requires that the variance of the errors, ε_i, by constant for all values of X)

Question 15

Define Residual Analysis

Answer 15

Visually evaluates the assumption of a set of data to determine whether the regression model selected is appropriate

The residual, or estimated error value, ei, is the difference b/t the observed (Y_i) and predicted (Ŷ_i) values of the dependent variable for a given value of X_i

Question 16

Evaluate the Assumption of Linearity

Answer 16

• Plot the residuals on the vertical axis against the corresponding X, values of the independent variable on the horizontal axis
• If it’s appropriate, there will be no apparent pattern in the plot
• If there is, you will need another model (e.g. quadratic or curvilinear)

(See attached for examples of not appropriate plots

Question 17

Evaluate the Assumption of Independence

Answer 17

• Evaluate the assumption of independence of the errors by plotting the residuals in the order or sequence in which the data were collected.
• If the show a cyclical pattern, then the assumption does not fit

Question 18

Evaluate the Assumption of Normality

Answer 18

Evaluate the assumption of normality in the errors by constructing a histogram or normal probability plot.

Standard normality

Question 19

Evaluate the Assumption of Equal Variance

Answer 19

• You can evaluate the assumption of equal variance from a plot of the residuals with X_i.
• There should appear to be no difference in the variability

For an example where there is a difference in variability, see image