chap 8

For a multiple regression test of H0: β1 = β2 = ... = βp = 0, the test statistic is ______.

F = (SSR/p)/(SSE/(n-p-1))

In a multiple linear regression model, it is assumed that the errors ε1ε1, ..., εnεn are random and independent, each satisfying εi ~ εi ~ _____.

N(0, σ2)

Under the assumptions of a multiple regression model yi = β0 + β1x1i + ... + βpxpi + εiyi = β0 + β1x1i + ... + βpxpi + εi, the y coordinates satisfy yi ~yi ~ _____.

N(β0 + β1x1i + ... + βpxpi, σ2)

Match the multiple regression diagnostic plot on the left with the assumptions it checks on the right.

Residuals vs. fitted values, {(ei, ŷŷi)} matches Errors are random and independent (with no dependence on fitted value) with the same variance. Normal probability plot of residuals matches Errors are normally distributed. Residuals versus order in which observations were made matches Errors are independent (with no dependence on time). Residuals versus each independent variable matches Errors are random and independent (with no dependence on any independent variable) with the same variance.

The analysis of variance identity says that ______.

SST = SSR + SSE

Under the assumptions of multiple linear regression (and for i in {0,1,...,p}), the quantity β̂i-βisβ̂iβ̂i-βisβ̂i follows a ______.

Student's t distribution with n - (p + 1) degrees of freedom

One method for selecting a best regression model from among those found by best subsets regression uses a statistic called the _______ R2 = R2 - (kn-k-1)kn-k-1(1 - R2).

adjusted

Select all that apply Under the assumptions of a multiple regression model yi = β0 + β1x1i + ... + βpxpi + εiyi = β0 + β1x1i + ... + βpxpi + εi, the y coordinates _____.

are random have variance σ2yi = σ2

Select all that apply In a multiple regression model (as in a simple linear model), it is assumed that the errors ε1ε1, ..., εnεn _____.

are random and independent all have the same variance σ2σ2 all have mean 0 are normally distributed

While a polynomial regression model is not linear in its independent variables, it is linear in its __________ and is therefore a special case of the multiple regression model.

coefficients

Suppose the full model yi = β0 + β1x1i + ... + βkxki + βk+1xk+1 i + ... + βpxpi + εi is correct. To test H0: βk+1 = ... = βp = 0, the test statistic is ______.

f = ((SSEreduced-SSEfull)/(p-k)/(SSEfull/(n-p-1))

Select all that apply The multiple regression model has multiple ______.

independent variables coefficients

Adjusted R2 is used instead of (plain) R2 in selecting a single model from best subsets regression because it adjusts for the number of ______.

independent variables in the model

An _________ in a multiple regression model is the product of two (or more) of its independent variables.

interaction

Select all that apply In the context of multiple regression, R2 ______.

is the coefficient of determination =regression sum of squares / total sum of squares is the proportion of variance in y explained by regression

The equation ŷŷ = β̂β̂0 + β̂β̂1 x1 + ... + β̂β̂p xp is called the ______ equation or fitted regression equation.

least-squares

Select all that apply Which are correct names for the expression ŷŷ = β̂β̂0 + β̂β̂1 x1 + ... + β̂β̂p xp?

least-squares equation fitted regression equation

The model of the form yi = β0 + β1x1i + ... + βpxpi + εi, in which x1, ... ,xp are independent variables (with p > 1) and yi is a dependent variable, is called the ________ regression model.

multiple

In multiple regression, the denominator used in the estimated error variance s2 is ______.

n - (p + 1), the number of observations minus the number of estimated parameters

In the polynomial regression model, the independent variables are all ______________ of a single variable.

powers

The fitted regression equation is found by minimizing the sum of the squares of the differences between _____.

the observed y values and the y values given by the equation

Select all that apply If multiple regression diagnostic plots indicate that the assumptions made for errors are not met, it is reasonable to ______.

try transforming one or several variables abandon the model

Select all that apply Which combinations of x1x1 and x2x2 are independent variables in a quadratic regression model in x1 and x2?

x1 x1^2 x1x2 x2 x2^2

In a quadratic regression model in x1 and x2, which combination of x1 and x2 is an interaction?

x1x2

Select all that apply In a multiple regression test of H0: β1 = β2 = ... = βp = 0, if H0 is not rejected, then ______.

y has a linear relationship with none of x1, ... , xp the model may not be useful for predicting y from x1, ... , xp F = (SSR/p)SSE/(n−p−1) is not in the tail of the Fp, n-p-1 distribution

Select all that apply Suppose the full model yi = β0 + β1x1i + ... + βkxki + βk+1xk+1 i + ... + βpxpi + εi is correct. If a test of H0: βk+1 = ... = βp = 0 gives P-value 0.0017, then ______.

y probably depends linearly on at least one of the variables xk+1, ... ,xp the reduced model yi = β0 + β1x1i + ... + βkxki is probably insufficient

Which of these models is not considered to be a linear model?

yi = β0 + β1x1i +β2xi^2

Select all that apply Select the multiple regression models that are quadratic in the variables x1x1 and x2x2.

yi= β0+ β1x21i+ β2x1ix2i+ β3x22i yi=β0+β1x1i+β2x1ix2i (two longest)

In a multiple regression model with p > 1 independent variables, a level 100(1 - α)% confidence interval for βi is given by ______.

β̂β̂i ± tn-p-1,α/2sβ̂i

Match each expression on the left with its description on the right.

∑i=1n(ŷi-y)2 matches Regression sum of squares ∑i=1n(yi-ŷi)2 matches Error sum of squares ∑i=1n(yi-y)2 matches Total sum of squares R2 matches Coefficient of determination

The coefficients β̂β̂0, β̂β̂1, ...,β̂β̂p in the least squares equation ŷŷ = β̂β̂0 + β̂β̂1 x1 + ... + β̂β̂p xp are chosen to minimize ______, where ei = yi - ŷŷi.∑i=1nei2

∑i=1nei2

In multiple regression, the error variance σ2 is estimated as s2 = ______.

∑ni=1(yi-ŷi)2 / (n - p - 1)