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 :
Q.1. A simple linear regression model is to be fit: Y i = 0 + 1 X i + I. The data are as follows: Complete the following parts in matrix form (Note: SSTO=70): X Y 0 0 0 4 6 4 10 4 8 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
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) = 0 + 1 X 1 + X + 3 X 3 SSE 1 = 1800 Model: E(Y)= 0 + 1 X 1 + X + 3 X 3 + 11 X 1 + X + 33 X 3 + 1 X 1 X + 13 X 1 X 3 + 3 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
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 0 1 1 3 1 0 1 1 ANOVA Model 1 ANOVA Model 3 df SS MS F df SS MS F Regression 1 3344.973 3344.973 166.5618 Regression 3 6078.609 06.03 90.5746 Residual 153 307.619 0.0847 Residual 151 338.98.44915 Total 154 6417.591 Total 154 6417.591 Coefficientsandard Erro t Stat Coefficientsandard Erro t Stat Intercept 1.807034 0.63188.86457 Intercept 0.105335 0.96108 0.355733 prodcostc 1.39948 0.107931 1.90588 prodcostc 0.11536 0.06881 1.674438 distcostc.50036 0.094358 6.4983 ANOVA Model prodxdist -0.0095 0.010111-0.9156 df SS MS F Regression 6078.419 3039.09 136.018 ANOVA Model 4 Residual 15 339.173.31401 df SS MS F Total 154 6417.591 Regression 1 5099.65 5099.65 59.0188 Residual 153 1317.94 8.613999 Coefficientsandard Erro t Stat Total 154 6417.591 Intercept 0.164311 0.15601 0.76109 prodcostc 0.101986 0.05155 1.979341 Coefficientsandard Erro t Stat distcostc.485 0.0709 34.99987 Intercept -0.04937 0.43133-0.11667 totcostc 1.0738 0.044111 4.33144 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
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 1 0.56 0.64 77 1.40 1.46 78 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 = = - 4.67 + 4.00X X Y Ybar Y-hat Pure Error Lack of Fit 3 5 4 8 4 1 6 18 6 Source df SS MS F F(0.05) Lack-of-Fit Pure Error
Critical Values for t,, and F Distributions F Distributions Indexed by Numerator Degrees of Freedom CDF - Lower tail probabilities df t.95 t.975.95 F.95,1 F.95, F.95,3 F.95,4 F.95,5 F.95,6 F.95,7 F.95,8 1 6.314 1.706 3.841 161.448 199.500 15.707 4.583 30.16 33.986 36.768 38.883.90 4.303 5.991 18.513 19.000 19.164 19.47 19.96 19.330 19.353 19.371 3.353 3.18 7.815 10.18 9.55 9.77 9.117 9.013 8.941 8.887 8.845 4.13.776 9.488 7.709 6.944 6.591 6.388 6.56 6.163 6.094 6.041 5.015.571 11.070 6.608 5.786 5.409 5.19 5.050 4.950 4.876 4.818 6 1.943.447 1.59 5.987 5.143 4.757 4.534 4.387 4.84 4.07 4.147 7 1.895.365 14.067 5.591 4.737 4.347 4.10 3.97 3.866 3.787 3.76 8 1.860.306 15.507 5.318 4.459 4.066 3.838 3.687 3.581 3.500 3.438 9 1.833.6 16.919 5.117 4.56 3.863 3.633 3.48 3.374 3.93 3.30 10 1.81.8 18.307 4.965 4.103 3.708 3.478 3.36 3.17 3.135 3.07 11 1.796.01 19.675 4.844 3.98 3.587 3.357 3.04 3.095 3.01.948 1 1.78.179 1.06 4.747 3.885 3.490 3.59 3.106.996.913.849 13 1.771.160.36 4.667 3.806 3.411 3.179 3.05.915.83.767 14 1.761.145 3.685 4.600 3.739 3.344 3.11.958.848.764.699 15 1.753.131 4.996 4.543 3.68 3.87 3.056.901.790.707.641 16 1.746.10 6.96 4.494 3.634 3.39 3.007.85.741.657.591 17 1.740.110 7.587 4.451 3.59 3.197.965.810.699.614.548 18 1.734.101 8.869 4.414 3.555 3.160.98.773.661.577.510 19 1.79.093 30.144 4.381 3.5 3.17.895.740.68.544.477 0 1.75.086 31.410 4.351 3.493 3.098.866.711.599.514.447 1 1.71.080 3.671 4.35 3.467 3.07.840.685.573.488.40 1.717.074 33.94 4.301 3.443 3.049.817.661.549.464.397 3 1.714.069 35.17 4.79 3.4 3.08.796.640.58.44.375 4 1.711.064 36.415 4.60 3.403 3.009.776.61.508.43.355 5 1.708.060 37.65 4.4 3.385.991.759.603.490.405.337 6 1.706.056 38.885 4.5 3.369.975.743.587.474.388.31 7 1.703.05 40.113 4.10 3.354.960.78.57.459.373.305 8 1.701.048 41.337 4.196 3.340.947.714.558.445.359.91 9 1.699.045 4.557 4.183 3.38.934.701.545.43.346.78 30 1.697.04 43.773 4.171 3.316.9.690.534.41.334.66 40 1.684.01 55.758 4.085 3.3.839.606.449.336.49.180 50 1.676.009 67.505 4.034 3.183.790.557.400.86.199.130 60 1.671.000 79.08 4.001 3.150.758.55.368.54.167.097 70 1.667 1.994 90.531 3.978 3.18.736.503.346.31.143.074 80 1.664 1.990 101.879 3.960 3.111.719.486.39.14.16.056 90 1.66 1.987 113.145 3.947 3.098.706.473.316.01.113.043 100 1.660 1.984 14.34 3.936 3.087.696.463.305.191.103.03 110 1.659 1.98 135.480 3.97 3.079.687.454.97.18.094.04 10 1.658 1.980 146.567 3.90 3.07.680.447.90.175.087.016 130 1.657 1.978 157.610 3.914 3.066.674.441.84.169.081.010 140 1.656 1.977 168.613 3.909 3.061.669.436.79.164.076.005 150 1.655 1.976 179.581 3.904 3.056.665.43.74.160.071.001 160 1.654 1.975 190.516 3.900 3.053.661.48.71.156.067 1.997 170 1.654 1.974 01.43 3.897 3.049.658.45.67.15.064 1.993 180 1.653 1.973 1.304 3.894 3.046.655.4.64.149.061 1.990 190 1.653 1.973 3.160 3.891 3.043.65.419.6.147.058 1.987 00 1.653 1.97 33.994 3.888 3.041.650.417.59.144.056 1.985 1.645 1.960 --- 3.841.995.605.37.14.099.010 1.938