STA 4210 Practise set 2b

Transcription

1 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).

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=8): 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 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 = X 1 + X + ~ NID(0, ) ANOVA df SS MS F* F(0.95) Regression Residual #N/A Total #N/A #N/A #N/A Coeff StdErr t* t(.975) Intercept #N/A #N/A #N/A X #N/A X #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 cm. Obtain her fitted value and residual. Fitted value = Residual = p..c. Obtain simultaneous 95% Confidence Intervals for 0, 1, and (Hint: z(.9917).395)

4 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 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: x = X - X x = X - X ANOVA Model1 Model Model3 df SS MS df SS MS df SS MS Regression Residual Total ANOVA Model4 Model5 df SS MS df SS MS Regression Residual Total 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:

5 Q.4. You obtain the following spreadsheet from a regression model. The fitted equation is Y ^ = X Conduct the F-test for Lack-of-Fit. n = c = X Y Ybar(Grp) Y-hat Pure Error Lack of Fit 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 å å å å å å( ) å( ) å( )( ) å n = X = Y = X = Y = XY = X - X = Y - Y = X - X Y - Y = e = { } b = MSE = s b = 95% CI for b : 1 1 1

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,