STA 410 Practise set b For all significance tests, use = 0.05 significance level. S.1. A linear regression model is fit, relating fish catch (Y, in tons) to the number of vessels (X 1 ) and fishing pressure (X ) for a lake over a sample of n=16 years. The model also contains an intercept. 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 ), then SSR(X 1 )+SSR(X X 1 ) = SSTO SSE(X 1,X ) TRUE or FALSE S.3. In simple linear regression, then (X X) -1 is x. TRUE or FALSE S.4. In a multiple linear regression model with predictors (X 1 and X ), R ( X ) + R = R ( X, X ) TRUE or FALSE 1 Y 1 1 S.5. A multiple regression model is fit, relating Y to X 1, X, and X 3. The regression sums of squares include: SSR(X 1 ) = 400 SSR(X ) = 600 SSR(X 3 ) = 800 SSR(X 1,X ) = 700 SSR(X 1,X 3 ) = 1000 SSR(X,X 3 )=900 SSR(X 1,X,X 3 )= 100 SSR(X 3 X 1,X ) = SSR(X X 1,X 3 ) = SSR(X 1,X X 3 ) = S.6. A researcher reports that for a linear regression model, the regression sum of squares is three times larger than the error sum of squares. Compute R for this model R = S.7. In multiple regression, when predictor variables are highly correlated, the model is said to display multicollinearity. Effects of multicollinearity include (select all that are appropriate): i) Decreased t-statistics for some of the tests of H 0 : k = 0 (k=1,,p-1) ii) Wider confidence intervals for some of the k (k=1,,p-1) iii) Inflated standard errors for the least squares estimates of some of the b k (k=1,,p-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=8): X Y 0 14 0 10 3 9 3 7 6 6 6 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 height (Y, in cm) to hand length (X 1, in cm) and foot length (X, in cm) for a sample of n=75 adult females. The following results are obtained from a regression analysis of: Regression Statistics R Square Y = 0 + 1 X 1 + X + ~ NID(0, ) ANOVA df SS MS F* F(0.95) Regression 1105.5 Residual #N/A Total 1793.85 #N/A #N/A #N/A Coeff StdErr t* t(.975) Intercept 74.41 7.97 #N/A #N/A #N/A X1.38 0.49 #N/A X 1.73 0.37 #N/A p..a. Complete the tables. p..b. The first woman in the sample had a hand length of 19.56cm, a foot length of 5.70cm, and a height of 160.60cm. Obtain her fitted value and residual. Fitted value = Residual = p..c. Obtain simultaneous 95% Confidence Intervals for 0, 1, and (Hint: z(.9917).395)
Q.3. Regression models are fit, relating bursting strength of knit fabric (Y) to yarn count (X 1 ) and stitch length (X ). The following 5 models were fit on centered yarn counts and stitch lengths to reduce collinearity. { } { } { } { } { } Model 1: E Y = b + b x Model : E Y = b + b x Model 3: E Y = b + b x + b x 0 1 1 0 0 1 1 Model 4: E Y = b + b x + b x + b x + b x + b x x Model 5: E Y = b + b + b x + b x + b x where: 0 1 1 3 1 4 5 1 0 0 1 1 3 1 x = X - X x = X - X 1 1 1 ANOVA Model1 Model Model3 df SS MS df SS MS df SS MS Regression 1 0.509 0.509 1 17.4997 17.4997 34.4914 17.457 Residual 54 18.7953 0.3481 54 1.5466 0.3990 53 4.5548 0.0859 Total 55 39.046 55 39.046 55 39.046 ANOVA Model4 Model5 df SS MS df SS MS Regression 5 35.7314 7.1463 3 35.594 11.8647 Residual 50 3.3148 0.0663 5 3.450 0.0664 Total 55 39.046 55 39.046 Complete the following parts (all parts are based on the centered values). p.3.a. Based on model 3, test whether either centered yarn count (x 1 ) and/or centered stitch length (x ) are associated with bursting strength. H 0 : H A : Test Statistic: Rejection Region: p.3.b. Compute ( ) ( ) SSR x = SSR x x = R = 1 1 Y 1 ( ) ( ) SSR x = SSR x x = R = 1 Y1 p.3.c. Based on models 4 and 5, test whether after controlling for yarn count, stitch length, and squared yarn count, that neither squared stitch length or the cross-product between yarn count and stitch length are associated with bursting strength. That is H 0 : 4 = Test Statistic: Rejection Region:
Q.4. You obtain the following spreadsheet from a regression model. The fitted equation is Y ^ = -.67 + 3.75X Conduct the F-test for Lack-of-Fit. n = c = X Y Ybar(Grp) Y-hat Pure Error Lack of Fit 3 5 4 1 4 16 6 18 6 0 Source df SS MS F F(0.05) Lack-of-Fit Pure Error Q.5. A firm that produces technical manuscripts is interested in the relationship between cost of correcting typographical errors (Y, in dollars) and the total number of galleys (pages, X). They wish to determine whether a regression-through-the-origin model is appropriate. You are given the following results for the model Y = X + : Y X Y-hat e 18 7 16.19 1.81 13 1 16.3-3.3 191 10 180.7 10.73 178 10 180.7 -.7 50 14 5.38 -.38 446 5 450.67-4.67 540 30 540.81-0.81 457 5 450.67 6.33 34 18 34.48-0.48 177 10 180.7-3.7 å å å å å å( ) å( ) å( )( ) å n = X = Y = X = Y = XY = 10 161 904 3163 108088 57019 X - X = Y - Y = X - X Y - Y = e = 571 904 1065 14 { } b = MSE = s b = 95% CI for b : 1 1 1
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