STA 4210 Practise set 2a

Transcription

1 STA 410 Practise set a For all significance tests, use = 0.05 significance level. S.1. A multiple linear regression model is fit, relating household weekly food expenditures (Y, in $100s) to weekly income (X 1, in $100s) and the number of people living in the household (X ). Assuming the model has an intercept, and is based on a sample of n=40 households. Give the appropriate degrees of freedom. df Total = df Regression = df Error = S.. In a multiple linear regression model with predictors (X 1 and X ), if X 1 and X are uncorrelated, SSR(X 1 ) = SSR(X 1 X ). TRUE or FALSE S.3. In simple linear regression, the hat-matrix is x. TRUE or FALSE S.4. In a multiple linear regression model with predictors (X 1 and X ), SSR(X 1 ) + SSR(X X 1 ) = SSR(X ) + SSR(X 1 X ). TRUE or FALSE S.5. In a multiple linear regression model with 3 predictors (X 1, X, and X 3 (where each has 3 levels)), a researcher wishes to include all -variable interactions, and all squared terms. Give the appropriate degrees of freedom, assuming there is 1 observation at all combinations of the levels of each predictor, and 3 extra points when each factor is at its mid-level. df Total = df Regression = df Error = S.6. A researcher reports that for a linear regression model, the regression sum of squares is twice as large as the error sum of squares. Compute R for this model R = S.7. A simple linear regression model is fit, based on a sample of n=10 observations. You obtain the following estimates: b 0 = 1.0 s{b 0 } = 3.0 b 1 = 5.0 s{b 1 } =.0. Use Bonferroni s method to obtain simultaneous 90% Confidence Intervals for 0 and 0 : 1 :

2 Q.1. A simple linear regression model is to be fit: Y i = X i + I. The data are as follows: Complete the following parts in matrix form (Note: SSTO=70): X Y p.1.a. X= Y = p.1.b. X X = X Y = p.1.c. (X X) -1 = b = ^ p.1.d. Y = e = p.1.e. MSE = s {b} = p.1.f. Complete the following tables: ANOVA Regression Residual Total df SS MS F Coefficientsandard Err t Stat Intercept X

3 Q.. A regression model is fit, relating a response (Y) to 3 predictors (X 1, X, X 3 ) based on n=30 individuals. Two models are fit: Model1: E(Y) = X 1 + X + 3 X 3 SSE 1 = 1800 Model: E(Y)= X 1 + X + 3 X X 1 + X + 33 X X 1 X + 13 X 1 X X X 3 SSE =1400 Test whether none of the quadratic terms or interaction terms contributes above and beyond the effects of X 1, X, and X 3. p..a. H 0 : p..b. Test Statistic: p..c. Reject H 0 if the test statistic falls above / below p..d. Compute R Y, X 1, X, X 3, X 1 X, X 1 X 3, X X 3 X 1, X, X 3

4 Q.3. Regression models are fit, relating total revenues (Y, in millions of dollars) for n=155 films released in the 1930s to production costs (X 1, in millions of dollars) and distribution costs (X, in millions of dollars). { } = b0 + b1 1 { } = b0 + b1 1 + b { } = b + b + b + b { } = b + b ( + ) Model 1: E Y X Model : E Y X X Model 3: E Y X X X X Model 4: E Y X X ANOVA Model 1 ANOVA Model 3 df SS MS F df SS MS F Regression Regression Residual Residual Total Total Coefficientsandard Erro t Stat Coefficientsandard Erro t Stat Intercept Intercept prodcostc prodcostc distcostc ANOVA Model prodxdist df SS MS F Regression ANOVA Model 4 Residual df SS MS F Total Regression Residual Coefficientsandard Erro t Stat Total Intercept prodcostc Coefficientsandard Erro t Stat distcostc Intercept totcostc Complete the following parts. p.3.a. Based on model 3, test whether there is an interaction between production (X 1 ) and distribution (X ) costs. H 0 : H A : Test Statistic: Rejection Region: p.3.b. Compute ( ) ( ) SSR X1 = SSR X X1 = R Y 1 = p.3.c. Based on models and 4, test whether the effects of increasing production and distribution costs by 1 unit are equal. H 0 : H A : Test Statistic: Rejection Region: p.3.d. Based on model, give the predicted total revenues (in dollars) for a movie that had production costs of $500,000 and distribution costs of $50,000

5 p.3.e. Conduct the Brown-Forsyth test, to determine whether the variances for the low predicted values (group=1) and ~ ~ the high predicted values (group ) are equal. dij = eij - ei i = 1, j = 1,..., ni ei = median ( ei1,..., ein ) i Group Mean(D) Var(D) n p.3.e.i. Test Statistic: p.3.e.ii. Rejection Region Q.4. You obtain the following spreadsheet from a regression model. The fitted equation is Y ^ Conduct the F-test for Lack-of-Fit. n = c = = X X Y Ybar Y-hat Pure Error Lack of Fit Source df SS MS F F(0.05) Lack-of-Fit Pure Error

6 Critical Values for t,, and F Distributions F Distributions Indexed by Numerator Degrees of Freedom CDF - Lower tail probabilities df t.95 t F.95,1 F.95, F.95,3 F.95,4 F.95,5 F.95,6 F.95,7 F.95,