1.Short answer questions

(a). Which method is used to estimate the regression coefficients in the multiple regression?

(b). Will removing a predictor from a multiple regression model affect the effects of other predictors on the response variable?

(c). For a regression model with 30 data points and 4 predictors,what is the degree of freedom for regression? What is the degree of freedom for error?

(d) What are null and alternative hypotheses of the overall ANOVA F test for the multiple regression?

2.Use the below R output to answer questions.

> summary(reg1)

Call:

lm(formula = y ~ X1+X2+X3+X4)

Residuals:

Min1QMedian3QMax

-60.046-6.7680.97213.94746.332

Coefficients:

EstimateStd. Errort valuePr(>|t|)

(Intercept)-94.659109 211.509584-0.4480.656731

X1-0.4800800.693711-0.6920.492628

X2-0.0081950.152358-0.0540.957353

X322.6100826.3145773.5810.000866 ***

X4-0.4641520.579104-0.8020.427249

---

Signif. codes:

0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 26.34 on 43 degrees of freedom

Multiple R-squared:0.8787,Adjusted R-squared:0.8618

F-statistic: 51.91 on 4 and 43 DF,p-value: < 2.2e-16

(a) How many data points are in the data set?

(b) Write out the estimated regression model.

(c) Interpret the effect of predictor X₃ on Y.

(d) Find the estimate of error variance.

(e) Interpret the residual standard error 26.34.

(f) Is the regression model overall useful?Justify your answer clearly.

(g) Is X₁ is significantly useful to predict Y given other predictors in the model? Justify your answer clearly.

(h) Is X₃ is significantly useful to predict Y given other predictors in the model? Justify your answer clearly.

(i) Find the partial correlation between X₃ and Y (based on the relationship between r and t).

(j) Find the proportion of the sample variability in Y that is explained by the estimated model with X₁, X₂, X₃, and X₄.

(k) Construct a 95% confidence interval for

3. An urban planner is studying Y=per capita property tax base for various neighborhoods (in$1000s) as a function of X₁=average age of homes and X₂=average size of homes. Data are

available for a sample of 120 neighborhoods, in which TSS=17136. Here is information on twomodels.

Full model:Regress Y on X₁ and X₂.The resulted R²= 0.365

Reduced model: Regress Y on X₂. The resulted R²= 0.303

(a) Find SSR in the full model.

(b) Find SSR in the reduced model.

(c) Does the full modelfit significantly better than the reduced model, assuming =0.05? (Equivalently speaking, conduct a partial F test to see if X₁ is significantly useful for predicting Y given X₂ already in the model.)