Linear Regression Problems

Transcription

1 Linear Regression Problems Q.1. A multiple regression model is fit relating a response Y to 4 predictors: X1, X, X3, X4. We fit models (each based on a sample of n=0 cases): i) E[ Y] X X 0 1X1 X 3X 3 4 X 4 ii) E[ Y] For model i), the error sum of squares is 50. for model it is 300. Test H0: at the =0.05 significance level. Q.. A multiple regression model is fit relating a response Y to 6 predictors: X 1, X, X 3, X 4, X 5, X 6. We fit models (each based on a sample of n=34 cases): i) E[ Y] X X X 0 1X1 X 3X 3 4 X 4 5 X 5 6 X 6 ii) E[ Y] For model i), the error sum of squares is For model it is Test H 0: at the =0.05 significance level. Q.3. A regression model is fit, relating breaking strength of a fiber (Y) to the amount of an additive applied to it (X 1), the amount of time it is heated (X ), and the temperature (X 3) at which it is heated. The fitted equation and coefficient of multiple determination are given below (n=4): ^ Y X X 0.010X 3 R 0.75 Give the predicted breaking strength of a fiber with X 1=10 units of additive, heated for X =15 minutes, at X 3=300 degrees Test H 0: at the =0.05 significance level. Q.4. A linear regression model is fit, relating breaking strength of steel bars to thickness, length, and material type with 3 nominal levels (no interaction terms or polynomial terms are included in the model). The model is fit based on 50 experimental units and includes an intercept term. DF(Regression) DF(Error) DF(Total) Q.5. For any regression model, Adjusted-R will be larger than R. Q.6. If we continue adding new predictors to a regression model, the error sum of squares will never increase, but the error mean square may increase. Q.7. It is not possible to get a negative F-statistic when (properly) conducting a Complete versus Reduced F- test, when we are testing to determine whether one set of predictors is not associated with Y, after controlling for another set of predictors.

2 Q.8. A researcher states that her regression model explains 75% of the variation in her dependent variable. This means that SSE/TSS = 0.5, where SSE is the Error sum of squares and TSS is the Total sum of squares Q.9. A study was conducted to determine whether company size (X1=#Employees) and presence/absence of an active safety program (X=1 if yes, 0 if no) were related with the lost work hours by employees in a 1-year period (Y). The sample consisted of 40 firms, 0 used the safety program, 0 did not. The model fit is: Y = X1 + X + p.9.a. Complete the following tables ( and Coefficients) for the multiple regression: Source df SS MS F(obs) F(0.05) Regression Error #N/A #N/A Total #N/A #N/A #N/A Estimates Predictor Coefficien SE(Coef) t(obs) t(.05) Intercept #N/A #N/A X X p.9.b. Controlling for company size, firms with the safety program are estimated on average to have more / less lost work hours in 1 year than firms without the safety program. p.9.c. What proportion of variation in lost work hours is explained by the variables X1 and X p.9.d. Give the predicted number of lost work hours in 1 year for a firm with 10,000 workers and the safety program. p.9.e. Based on Backward elimination with Significance Level to Stay (SLS) of 0.05, would you drop either X1 or X from the model?

3 Q.10. A study considered failure times of tools (Y, in minutes) for n=4 tools. Variables used to predict Y were: cutting Speed (X1 feet/minute), feed rate (X), and Depth (X3). The following models were fit (TSS=3618): Model 1: E(Y) = X1 + X + X3 SSE1 = 874 Model : E(Y) = X1 + X + X3 + X1*X + X1*X3 + X*X3 + X1*X*X3 SSE = 483 p.10.a. Treating Model as the Complete Model with all potential predictors, compute Cp for Model 1 p.10.b. Test whether all of the interaction terms can be removed from the model, controlling for all main effects. That is, test H0: at significance level: p.10.b.i. Test Statistic: p.10.b.ii. Reject H0 if the test statistic falls in the range: p.10.b.iii. Based on this test, are we justified in dropping all interaction terms from the model? Yes / No Q.11. A regression model is fit with p=6 predictors (and an intercept), based on n=0 experimental units. How large does R for the model need to be to reject H0: at significance level? Q.1. A regression model is fit relating Peak Power Load (Y, in megawatts) to daily high temperature (X, in Degrees F) for a sample of n=10 days. The analyst believes that Peak Power Load will increase with temperature, and that the RATE of change will increase as temperature increases. The model to be fit is (along with estimates and standard errors): Model 1: E(Y) = X + X Model : E(Y) = W + W (where W is centered value of X, that is: W = X-91.5) Coeff StdErr t Stat P-value Coeff StdErr t Stat P-value Intercept Intercept X W X^ W^ p.1.a. Test H0: vs HA: based on Model 1 with = 0.05: p.1.a.i. Test Statistic: p.15.a.ii. Reject H0 if test stat falls in range: p.1.b. Obtain a 95% Confidence Interval for 1 for Model 1. Does it contain 0? p.1.c. Obtain a 95% Confidence Interval for 1 for Model. Does it contain 0?

4 p.1.d. The correlation between X and X for Model 1 is and between W and W for Model is Obtain the Variance Inflation Factors for Models 1 and (Note that since there are only predictors for each model, there will only be one VIF for each model). Model 1: VIFX = VIFX = Model : VIFW = VIFW = p.1.e. Obtain predicted Peak Power Load when X=90 for Model 1, and when W = = -1.5 for Model : Model 1: Model : Q.13. A simple linear regression was fit relating number of species of arctic flora observed (Y) and July mean temperature (X, in Celsius). The results of the regression model, based on n=19 temperature stations is given below. df SS MS F ignificance F Regression Residual Total Coefficientsandard Erro t Stat P-value Lower 95%Upper 95% Intercept JulyTemp SS_XX X-bar p.13.a. What proportion of the variation in number of species is explained by mean July temperature? p.13.b. Compute a 95% Confidence Interval for the population mean number of species, with mean July temperature of 6 degrees. p.13.c. Compute a 95% Prediction Interval for the number of species, at a single station with mean July temperature of 6 degrees. Q.14. An experiment was conducted, relating the penetration depth of missiles (Y) to its impact factor (X). The results from the regression, and the residual versus fitted plot are given below (n=5). df SS Regression Residual Total Coefficientsandard Erro -0. Intercept impact Residuals vs Fitted Values Residuals p.14.a. Test H 0: 1 = 0 (Penetration depth is not associated with impact factor) based on the t-test. Test Statistic: Rejection Region:

5 p.14.b. Test H 0: 1 = 0 (Penetration depth is not associated with impact factor) based on the F-test. Test Statistic: Rejection Region: p.14.c. The residual plot appears to display non-constant error variance. A regression of the squared residuals on the impact factors (X) is fit, and the is given below. Conduct the Breusch-Pagan test to test whether the errors are related to X. Do you reject the null hypothesis of constant variance? Yes or No df SS Regression Residual Total Test Statistic: Rejection Region: Q.15. An experiment was conducted to measure air permeability of fabric (Y) as a function of the following factors: warp density (X 1), weft density (X ), and Mass per unit area (X 3). There were n=30 observations, and 4 models are fit: E Y X X X X X X X X X X X X SSE E Y X X X X X X X X X SSE E Y X X X SSE E Y X X X X X SSE p.15.a. Use the first two models to test H 0:. Test Statistic: Rejection Region: p.15.b. Use the 3 rd and 4 th models to test whether the weft-mass interaction is significant, controlling for all main effects. Test Statistic: Rejection Region: Q.16. A regression model was fit, relating the share of big 3 television network prime-time market share (Y, %) to household penetration of cable/satellite dish providers (X = MVPD) for the years (n=5). The regression results and residual versus time plot are given below. df SS MS F Regression Residual Total Coefficientsandard Erro t Stat P-value Intercept mvpd Residuals Residuals p.16.a. Compute the correlation between big 3 market share and MVPD. p.16.b. The residual plot appears to display serial autocorrelation over time. Conduct the Durbin-Watson test, with null hypothesis that residuals are not autocorrelated. 5 t e e d n p d n p , 5, , 5, t t1 L U Test Statistic: Reject H 0? Yes or No

6 p.16.c. Data were transformed to conduct estimated generalized least squares (EGLS), to account for the auto-correlation. The parameter estimates and standard errors are given below. Obtain 95% confidence intervals for 1, based on Ordinary Least Squares (OLS) and EGLS. Note that the error degrees of freedom are 3 for OLS and for EGLS (estimated the autocorrelation coefficient). beta-egls SE(b-egls) OLS 95% CI: EGLS 95% CI: Q.17. Regression analyses were fit, relating various chemical levels to age for stranded bottlenose dolphins in South Carolina and Florida. This plot gives the quadratic fit, relating mercury/selenium molar ratio (Y) to age (X) for the Florida dolphins. Complete the following parts. Note: The data were NOT centered. The model fit was: E(Y) = 0 + 1X + X n = Predicted value when age = 15 Test H 0: 1 = = 0 Test Statistic: Rejection Region: Q.18. A study was conducted to determine which factors were associated with percent release (Y) of hydroxypropyl methylcellulose (HPMC) tablets. The factors were: X1 = Carr s compressibility index, X = angle of repose, X3 = solubility, X4 =molecular weight, X5 = compression force X6 = apparent viscosity of 4% (w/v) HPMC. The sample size was n=18, and the authors reported the fit of the following models.

7 p.18.a. Complete the table in terms of AIC and SBC (BIC). Predictors p' SSE AIC SBC X1,X,X3,X4,X5,X X1,X,X3,X4,X X1,X,X3,X X1,X3,X4,X X1,X3,X X,X3,X X3,X4,X p.18.b. Which model is best based on AIC: BIC: p.18.c. R for the complete model was Compute the total (corrected) sum of squares (TSS): Q.19. A study was conducted to relate construction plant maintenance cost (pounds) to 3 categorical predictors (industry: coal/slate, machine type: front shovel/backacter, and attitude to used oil analysis: regular use/not regular use) and one numeric predictor (machine weight, in tons). Due to the distributions of costs and machine weight (skewed), both are modeled with (natural) logs. Thus, the variables are (based on a sample of n=33 construction plants): Y = ln(costs) X 1 = 1 if coal industry, 0 if slate X = 1 if machine type = front shovel, 0 if backacter X 3 = 1 if attitude to used oil analysis = regular use, 0 if not X 4 = ln(machine Weight) They fit Models: Model 1: E{Y} = X 1 + X + X 3 + X 4 Model : E{Y} = X 1 + X + X 3 + X 4 + X 1X 4 + X X 4 + X 3X 4 Model1 Model df SS MS F ignificance F df SS MS F ignificance Regression Regression Residual Residual Total Total Coefficientsandard Erro t Stat P-value Coefficientsandard Erro t Stat P-value Intercept Intercept X X X X X X X X X1*X X*X X3*X

8 p.19.a. Based on Model 1, give the fitted value for a firm that is a coal industry, has machine type=backacter, is a regular user of used oil, and ln(machine Weight) = 4.0. Note that the units of the fitted value is ln(costs). p.19.b. Test whether any of the categorical predictors interact with machine weight. H 0:. Test Statistic: Rejection Region: Q.0. Consider the following models, relating Stature (Y) to foot dimensions (RFL = Right Foot Length, RFB = Right Foot Breadth) and Age among the Rajbanshi of North Bengal. We observe the following statistics, based on several regressions among n = 175 adult males. Complete the following table. Note: The total sum of squares is TSS = , and for C p, use s = MSE(RFL,RFB,Age) = All models contain an intercept ( 0). Predictors p' SSE R-square Cp AIC BIC(SBC) RFL RFB Age RFL,RFB RFL,RFB,Age p.0.a. What is the best model based on C p? p.0.b. What is the best model based on AIC? p.0.c. What is the best model based on BIC (SBC)? Q.1. A linear regression was (inappropriately) run, relating success in throwing a frisbee through a hula-hoop (Y=1, if good, 0 if not) to children s eye-color (X = 1 if dark, 0 if light). Note that this should be conducted as a chi-square test, or logistic regression (it was a very old paper). The authors report that r =.030 (well, that s what it should be) from a sample of n = 136 children. Based on the regression : E(Y) = X, test H 0: 0 vs H A: 0 based on the F- test. Q.. A simple linear regression model and Analysis of Variance (Completely Randomized Design) model were fit, relating stretch percentage of viscose rayon to specimen length (there were 4 lengths: 5, 10, 15, 0 inches). The results from each model are given below: Regression Model df SS MS F Regression Residual Total Coefficientsandard Erro t Stat P-value Intercept X Variable Completely Randomized Design (1-Way )

9 df SS MS F Treatments (Lengths) Error Total Below are the fitted values (regression) and sample means (1-way ), and sample sizes: Length Yhat(Reg) Ybar(Grp) n(grp) Conduct the Goodness-of-Fit F-test, where H 0 states that the relationship between stretch percentage and specimen length is linear. Hint: All numbers (sums of squares and degrees of freedom can be obtained (implicitly) from the tables). Test Statistic: Rejection Region: Q.3. A study was conducted, relating Total Medical Waste (Y, in kg/day) to hospital type (Government: X 1=1, Education and Non- Education: X =1, University: X 3=1, Private is the reference type), Hospital Capacity (X 4 = # of beds), and Occupancy Rate (X 5 = % of Beds in Use). Consider the following two models: Model 1: E Y X X X X X Model : E Y X X SUMMARY OUTPUT Model 1 SUMMARY OUTPUT Model df SS MS F ignificance F df SS MS F ignificance Regression Regression Residual Residual Total Total Coefficientsandard Erro t Stat P-value Coefficientsandard Erro t Stat P-value Intercept Intercept govtype beds eduntype occ_rate univtype beds occ_rate p.3.a. Test whether there is a Hospital type effect (controlling for beds and occupancy rate): H 0: Test Statistic: Rejection Region: Reject H 0? Yes or No p.3.b. The first hospital in the sample is University type, has 15 beds, and an occupancy rate of 47%. Based on Model 1, compute its fitted (predicted) value. Its observed value was 30. Compute its residual. Fitted Value: Residual: p.3.c. What proportion (or percentage) of the Variation in Total Medical waste is explained by number of beds and occupancy rate? Answer:

10 Q.4. A study related Personal Best Shot Put distance (Y, in meters) to best preseason power clean lift (X, in kilograms). The following models were fit, based on a sample of n = 4 male collegiate shot putters: Model 1: E Y X SSE R.686 Y X X ^ 0 1 Model : E Y X X SSE R.79 Y X, X X X p.4.a. Use Model to test H 0: (Y is not related to X) ^ Test Statistic: Rejection Region: Reject H 0? Yes or No p.4.b. Use Models 1 and to test H 0: (Y is linearly related to X) Test Statistic: Rejection Region: Reject H 0? Yes or No p.4.c. Obtain the predicted value from each model for a man with best power clean of 175 kg. Model 1: Model : Q.5. A regression model is fit relating weight (Y, in lbs) to height (X, in inches) among n=139 WNBA players (treating this as a random sample of all female athletes from sports such as basketball and volleyball). The results from the regression, and the residual versus fitted plot are given below. df SS Regression Residual Total e Coefficientsandard Erro Intercept Height p.5.a. Test H 0: 1 = 0 (Weight is not associated with Height) based on the t-test. Test Statistic: Rejection Region: Reject H 0? Yes or No p.5.b. Test H 0: 1 = 0 (Weight is not associated with Height) based on the F-test. Test Statistic: Rejection Region: Reject H 0? Yes or No p.5.c. The residual plot appears to display non-constant error variance. A regression of the squared residuals on height (X) is fit, and the is given below. Conduct the Breusch-Pagan test to test whether the errors are related to X. Do you reject the null hypothesis of constant variance? Yes or No df SS Regression Residual Total Test Statistic: Rejection Region: Reject H 0? Yes or No

11 Q.6. A data set consisted of n = 3 observations on the variables Y, X1, X, X3, and X4. Error Sum of Squares = SSE for each of all possible models. For each model, the variables that are in the model are also shown. Use this information to answer the questions following the table. The Total Sum of Squares = SSTO = #Variables SSE Cp AIC SBC=BIC Vars in Model X #N/A #N/A #N/A X #N/A #N/A #N/A X #N/A #N/A #N/A X4 55 X,X4 84 #N/A #N/A #N/A X,X3 90 #N/A #N/A #N/A X1,X3 95 #N/A #N/A #N/A X1,X 40 #N/A #N/A #N/A X1,X4 445 #N/A #N/A #N/A X3,X X1,X,X #N/A #N/A #N/A X1,X,X #N/A #N/A #N/A X,X3,X #N/A #N/A #N/A X1,X3,X X1,X,X3,X4 p.6.a. Complete the table by computing C p, AIC, and SBC=BIC for the best models with 1,,3, and 4 independent variables. p.6.b. Give the best model (in terms of which independent variables to be included), based on each criteria. C p: AIC: SBC=BIC: Q.7. A linear regression model was fit, relating the electric potential (Y) to the current density (X) for a series of n=18 consecutively observed pairs of X and Y. Model: Y X ~ N 0, t 0 1 t t t t 1 t t p.7.a. The estimated regression coefficients and standard errors for ordinary least squares are given below. Obtain a 95% Confidence Interval for the slope coefficient for current density ( 1). Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) <e-16 *** x <e-16 *** Lower Bound: Upper Bound: p.7.b. The Durbin-Watson test is used to test H 0: = 0. The test statistic and P-value are given below. Do you conclude that the errors are serially correlated (not independent)? Yes or No lag Autocorrelation D-W Statistic p-value p.7.c. The estimated regression coefficients and standard errors for estimated generalized least squares are given below. Obtain a 95% Confidence Interval for the slope coefficient for current density ( 1) Coefficients: Value Std.Error t-value p-value (Intercept) x Lower Bound: Upper Bound:

12 Q.8. A study considered failure times of tools (Y, in minutes) for n=4 tools. Variables used to predict Y were: cutting Speed (X1 feet/minute), feed rate (X), and Depth (X3). The following models were fit (SSTO=3618): Model 1: E(Y) = X1 + X + X3 SSE1 = 874 Model : E(Y) = X1 + X + X3 + X1*X + X1*X3 + X*X3 + X1*X*X3 SSE = 483 Compute Cp, AIC, and SBC=BIC for each model. Based on each criteria, which model is selected? Criteria Model1 Model Best Model Cp AIC SBC=BIC Q.9. A study compared n = 43 island nations with respect to various demographic and transportation measures. The authors fit a linear regression relating Vehicles/road length (Y, cars/km) to GDP (X, $1000s/capita). The regression output and residual versus predicted plot are given below. df SS Regression Residual Total e Coefficients Standard Error Intercept GDP_K p.9.a. Test H 0: 1 = 0 (Car density is not related to GDP) based on the t-test. Test Statistic: Rejection Region: p.9.b. Test H 0: 1 = 0 (Car density is not related to GDP) based on the F-test. Test Statistic: Rejection Region: p.9.c. The residual plot appears to potentially display non-constant error variance. A regression of the squared residuals on GDP (X) is fit, and the is given below. Conduct the Breusch-Pagan test to test whether the errors are related to X. Do you reject the null hypothesis of constant variance? Yes or No df SS Regression Residual Total Test Statistic: Rejection Region:

13 Q.30. A regression model was fit, relating Weight (Y, in kg) to Height (X 1, in m) and Position for English Premier League Football Players. Position has 4 levels (Forward, Midfielder, Defender, and Goalkeeper). Thus, 3 dummy variables were generated: X = 1 if Forward, 0 otherwise; X 3 = 1 if Midfielder, 0 otherwise; X 4 = 1 if Defender, 0 otherwise. Goalkeepers were the reference position. The following regression models were fit, based on data for n = 441 league players. Model 1: E Y X X X X X X X X X X Model : E Y X X X X Model 3: E Y X TSS 4847 SSR 100 SSR 1184 SSR p.30.a. We wish to test whether the slopes relating weight to height is the same among the positions, allowing the intercepts to differ among positions. Conduct this test for an interaction between height and position. H 0: H A: Test Statistic: Rejection Region: p.30.b. Assuming the interaction is not significant, test whether there is a position effect, after controlling for height. H 0: H A: Test Statistic: Rejection Region: Q.31. A regression model was fit, relating points scored in 014 WNBA games by Skylar Diggins (Y) to whether the game was a Home game (X 1 = 1 if Home, 0 if Away) and the number of minutes she played (X ) over a season of n = 34 games. The regression results are given below for the model: Y = 0 + 1X 1 + X + df SS MS F F(0.05) R^ Regression Residual #N/A #N/A #N/A Total #N/A #N/A #N/A #N/A p.31.a. Complete the table. Do you conclude that Skylar s average point total is associated with the game being at home, and/or the number of minutes she played? Yes No p.31.b. Conduct the Durbin-Watson test, with null hypothesis that residuals are autocorrelated. 5 t e e d n p d n p , 34, , 34, 1.58 t t1 L U Test Statistic: Reject H 0? Yes or No

14 p.31.c. Data were transformed to conduct estimated generalized least squares (EGLS), to account for potential autocorrelation. The parameter estimates and standard errors are given below. Obtain 95% confidence intervals for, based on Ordinary Least Squares (OLS) and EGLS. Note that the error degrees of freedom are 34-3=31 for OLS and 30 for EGLS (estimated the autocorrelation coefficient). OLS OLS EGLS EGLS Parameter Estimate StdError Estimate StdError Intercept Home Minutes OLS 95% CI: EGLS 95% CI: Q.3. A model was fit, relating optimal solar panel tilt angle (Y, in degrees) to a city s latitude (X, in degrees) for a sample of n = 35 cities in the Northern Hemisphere. Consider the following 3 models (the X values have NOT been centered): Model 1: E Y X SSE Y X ^ 0 1 Model : E Y X X SSE Y X X ^ 1 3 Model 3: E Y X X SSE Y 1.445X X Note that Model 3 does not have an intercept (this is the model the authors fit). p.3.a. Give the predicted optimal solar tilt for a location at a latitude of 30 degrees for each model. ^ Model 1: Model : Model 3: P.3.b. Use Models (Full Model) and Model 3 (Reduced Model) to test H 0: 0 = 0 (Given Lat and Lat are in model). Test Statistic: Rejection Region: P.3.c. Use Models (Full Model) and Model 1 (Reduced Model) to test H 0: = 0 (Given an intercept is in model). Test Statistic: Rejection Region: Q.33. A multiple regression model was fit, relating consumer s satisfaction of culinary (food) event (Y) to the following predictor variables/scales: food tasting (X 1), food/beverage prices (X ), ease of coming/going to event (X 3), and convenient parking (X 4). The regression model was fit, based on n=77 respondents, with R = The regression coefficients are given below, for the model: E(Y) = X 1 X X 3 X 4 Parameter Estimate StdError t Constant X X X X p.33.a. Test H 0: Customer satisfaction is not associated with any of the predictors ( ).

15 p.33.a.i. Test Statistic: p.33.a.ii. Reject H 0 if the test statistic falls in the range p.33.b. Test H 0: Controlling for all other predictors, customer satisfaction is not associated with food/bev prices ( 0). p.33.b.i. Test Statistic: p.33.b.ii. Reject H 0 if the test statistic falls in the range Q.34. A linear regression model is fit, relating Y to 3 independent variables. The researcher is interested in determining whether any interactions are significant among the 3 independent variables. She fits the following models (n = 30): Model 1: E Y X X X R Model : E Y X X X X X X X X X R p.34.a. Compute the test statistic for testing H 0: No interactions among X 1, X, and X 3 ( ) p.34.b. Reject H 0 if the test statistic falls in the range Q.35. A regression model was fit, relating estimated cost of de-commissioning oil platforms (Y, in millions of $) to predictors: Total number of piles/legs (X 1) and water depth (X, in 100s of feet). The model was fit, based on n = 17 oil platforms. Consider the models: Model 1: E Y X X X X X X Model : E Y X X Model 1 Model df SS df SS Regression 7397 Regression 7163 Residual 551 Residual 785 Total Total Coefficientsandard Error Coefficientsandard Erro Intercept Intercept totpiles totpiles wtrdph wtrdph tp wtrdph tp*wd p.35.a. Test whether the linear (main effects) model is appropriate. H 0: 11 = = 1 = 0 p.35.b. For each model, obtain the predicted value and residual for rig 17 (Y = 78.5, X 1 = 44, X = 13) Model 1: Predicted = Residual = Model : Predicted = Residual =

16 Q.36. A study related subsidence rate (Y) to water table depth (X 1) for 3 crops: pasture (X = 0, X 3 = 0), truck crop (X = 1, X 3 = 0), and sugarcane (X = 0, X 3 = 1). Note the total sum of squares is TSS = , and n = 4. p.36.a. Give a model that allows separate intercepts for each crop type, with a common slope for water table depth among crop types. Sketch the graph for this model. For this model, SSE = Give the error degrees of freedom. p.36.b. Give a model that allows separate intercepts for among crop types, with separate slopes for water table depth among crop types. Sketch the graph for this model. For this model, SSE = Give the error degrees of freedom. p.36.c. Test the null hypothesis that the (simpler) model in p..a. is appropriate. That is, the extra parameters in the second model are not significantly different from 0. p.36.d. For the model in p..a., compute R Q.37. A study investigated meteorological effects on condition of wheat yield in Ohio (Y), based on a series of n = 4 years. The predictors were: Average October/November temperature (X 1), September precipitation (X ), October/November precipitation (X 3), and percent September sunshine (X 4). The best 1-,-,3-, and 4-variable models (minimum SSE) are given below. Model p' SSE Cp AIC BIC X X,X X1,X,X X1,X,X3,X p.37.a. Complete the table. Use MSE of the full (X 1,X,X 3,X 4) models as the estimate of when computing C p p.37.b. Give the best model based on each criteria. C p AIC BIC p.37.c. The following output gives the regression coefficients for the X 1,X,X 3 model. Give the fitted value and residual for the first year (Y = 9, X 1 = 46, X = 1.6, X 3 = 6.3). Coefficients Intercept tempon.x rains.x.38 rainon.x3.96 Fitted Value Residual p.37.d. For the model in p.3.c., we obtain: 4 t e e d n p d n p 306 4, , t t1 L U Test H 0: Errors are not autocorrelated versus H A: Errors are autocorrelated Circle the Best Answer Reject H 0 Accept H 0 Test is inconclusive

17 Q.38. A simple linear regression model was fit relating Weight (Y, in pounds) to Height (X, in inches) for a random sample of n=5 National Hockey League players. The total sum of squares, TSS = 937, and R = 0.6. p.38.a. Complete the following table. Source df SS MS F F(0.05) Regression Residual #N/A #N/A Total #N/A #N/A #N/A p.38.b. Complete the following table and use it to conduct the F-test for Lack-of-Fit (there are c = 9 distinct heights). H : X j 1,..., c H : X 0 j 0 1 j A j 0 1 j c ^ c j j j LF j 1 j PE j1 j1 SSLF n Y Y df c SSPE n S df n c Height n Y-bar Y-hat n*(yb-yh)(n-1)s^ Sum 5 #N/A #N/A Test Statistic: Reject H 0 if Test Stat falls in Range: Q.39. A simple linear regression model is fit, relating Orlando June Total Precipitation (Y) to Mean Temperature (X) over an n = 45 year period. The following table gives the results. df SS Regression Residual Total Coefficientsandard Erro Intercept meantemp p.39.a. Use the t-test to test H 0: 1 = 0 vs H A: 1 0 at = 0.10 significance level Test Statistic: Reject H 0 if Test Stat falls in range:

18 A plot of the residuals displays a possible case of unequal variances. The regression of the squared residuals on X is given below: 10.0 e e df SS Regression Residual Total p.39.b. Conduct the Breusch-Pagan test to test H 0: Equal Variances H A: Variance is related to X. (Use = 0.05): Test Statistic: Reject H 0 if Test Statistic falls in the Range: Q.40. A multiple regression model is fit with 3 predictors and an intercept, based on a sample of n=5 observations. How large must R /(1-R ) be to reject H 0: 1 =? Q.41. It is possible to fail to reject H 01: 1 = 0 and H 0: = 0 based on t-tests in multiple linear regression model with p > predictors, but still reject H 01: 1 = = 0, controlling for the remaining p- predictors. True or False Q.4. A multiple linear regression model is fit relating a dependent variable to a set of 3 numeric predictor variables, based on a sample on n=0 experimental units. How large does R need to be so that the null hypothesis H 0: will be rejected? Q.43. An experiment is conducted with 3 numeric predictors and categorical predictors, one with 3 levels, the other with levels. There are no interaction or polynomial terms in the model, and the sample size is n = 30. Give the degrees of freedom for Regression and Error. Df Reg = Df Error = Q.44. It is possible for a dataset to reject H 0: p = 0 based on the F-test, but fail to reject H 0: i = 0 for i=1,,p based on the individual t-tests. True or False

19 Q.45. A linear regression model is fit, relating salary (Y) to experience (X 1), gender (X =1 if female, 0 if male) and an experience/gender interaction term to employees in a large law firm. The fitted equation is ^ Y X 1000X 100X X 1 1. Give the predicted salaries for the following groups of individuals. Males with 0 experience Females with 0 experience Males with 10 experience Females with 10 experience Q.46. A regression model was fit, relating blood alcohol elimination rate measurements (Y, in grams/litre/hour) to Gender (X 1=1 if female, 0 if male), breath alcohol elimination measurements (X in mg/l/h) and a gender/breath interaction term. The sample was 59 adult Austrians. Y X X X X p.46.a. Complete the following Analysis of Variance Table and test H 0. 0 : 1 3 df SS MS F F(.05) Regression Residual #N/A #N/A Total #N/A #N/A #N/A p.46.b. Is the P-value for the test Larger or Smaller than 0.05? p.46.c. What proportion of the variation in Blood alcohol elimination rate measurements is explained by the model? p.46.d. The regression coefficient estimates are given below. Test H : 0 : 0 for each coefficient. 0 i H A i Coefficients Standard Error t_obs t(.05) Reject H0? Intercept #N/A #N/A #N/A female breath f*breath

20 Q.47. Monthly mean temperatures for Boston (Y, in Fahrenheit) for the years are fit using a linear regression model to Year (X 1=Year-190) and 11 monthly dummy variables (X = 1 if January, 0 otherwise,, X 1 = 1 if November, 0 otherwise, Note that December is the reference month). The table and regression coefficient estimates are given below. df SS MS F ignificance F Regression Residual Total Coefficientstandard Erro t Stat P-value Intercept year month month month month month month month month month month month p.47.a. Give the predicted temperatures for December 190, June (Month 6) 190, December 010, and June 010. December June p.47.b. Compute a 95% Confidence Interval for the change in annual mean temperature, controlling for month. Lower Bound: Upper Bound: 1140 p.47.c. Compute the Durbin-Watson statistic. e DW = t t et p.47.d. What proportion of the variation in temperature is explained by the model?

21 Q.48. A response surface model is fit, relating potato chip moistness (Y) to 3 factors: drying time (X 1), frying temperature (X ), and frying time (X 3). There were n = 0 experimental runs (observations). The following 3 models were fit: Model 1: E Y X X X SSR 475. SSE Model : E Y X X X X X X X X X SSR SSE Model 3: E Y X X X X X X X X X X X X SSR SSE 1.4 p.48.a. Test whether any of the -way interaction effects are significantly different from 0, controlling for main effects. H 0: Test Statistic: Rejection Region: P-value: > or < 0.05 p.48.b. Test whether any of the quadratic effects are significantly different from 0, controlling for main effects and - factor interactions. H 0: Test Statistic: Rejection Region: P-value: > or < 0.05 Q.49. A regression model was fit, relating Price (Y, in $1000s) to acceleration rate (X 1) and Miles per gallon (X ) for a sample of n = 5 models of hybrid compact cars. The fitted equation and summary model statistics are given below. ^ Y X 0.0X SSR 957 SSE p.49.a. Test whether price is related to either acceleration rate and/or Miles per gallon. H 0:. Test Statistic: Rejection Region: P-value: > or < 0.05 p.49.b. A plot of the residuals versus predicted values suggests a possible non-constant error variance. A regression of the squared residuals on X 1 and X yields SS(Reg * ) = Test: H : Equal Variance Among Errors i H : Unequal Variance Among Errors h X X 0 i A i 1 i1 i Residuals versus Predicted Test Statistic: Rejection Region: P-value: > or < 0.05

22 Q.50. A regression model is fit, relating energy consumption (Y) to 3 predictors: area (X 1), age (X ), and effective number of guest rooms (X 3 = rooms*occupancy rate) for a sample of n = 15 hotel rooms. p.50.a. Complete the following table of C p, AIC, and BIC for all possible regressions involving X 1, X, and X 3. Model p* SSE C_p AIC BIC X X X X1,X X1,X X,X X1,X,X p.50.b. Which model is selected based on each criteria? C p: AIC: BIC: p.50.c. To check for issues of multicollinearity, a regression relating each predictor on the other predictors is fit. The largest R of the 3 regressions is when X 1 is regressed on X and X 3. That R 1 value is Compute the Variance Inflation Factor VIF for X 1, where VIF1 1/ 1 Ri. Does it exceed 10? Q.51. A study of legitimacy theory was conducted after the Exxon Valdez oil spill in the late 1980s among a sample of n = 1 firms in Alaska. The response variable was the change in the number of environmental disclosures (Y, ) and the predictor variables were the firms log revenues for 1989 (X 1) and an indicator variable of whether the firm was E Y X X part owner of the Aleyska pipeline (X = 1 if Yes, 0 if No) p.51.a. Complete the following Analysis of Variance Table and test H 0:. Source df SS MS F F(.05) Regression 13.7 Error #N/A #N/A Total 7.95 #N/A #N/A #N/A p.51.b. Is the P-value for the test Larger or Smaller than 0.05? p.51.c. Give the residual standard error for the model (estimate of ).

23 p.51.d. The theory states that the regression coefficients for both firm size and part ownership of the pipeline should be positive. The regression coefficient estimates are given below. Test H 0: i = 0 vs H A: i > 0 for each coefficient. Coeffs Std. Err. t_obs t(.05) Reject H0? Intercept #N/A #N/A #N/A X X Q.5. Monthly mean temperatures for Houston (Y, in Fahrenheit) for the years are fit using a linear regression model to Year (X 1=Year-1947) and 11 monthly dummy variables (X = 1 if January, 0 otherwise,, X 1 = 1 if November, 0 otherwise, Note that December is the reference month). The table and regression coefficient estimates are given below. df SS MS F Regression Residual Total Coefficientsandard Erro t Stat P-value Intercept Year E-09 Month E-06 Month Month E-48 Month E-139 Month E-6 Month E-87 Month E-307 Month Month E-66 Month E-165 Month E-41 p.5.a. Give the predicted temperatures for December 1947, June (Month 6) 1947, December 007, and June 007. December June p.5.b. Compute a 95% Confidence Interval for the change in annual mean temperature, controlling for month. Lower Bound: Upper Bound: p.5.c. Compute the Durbin-Watson statistic. e DW = 816 t t et

24 Q.53. A response surface model is fit, relating lipstick color characteristic b* (yellow/blue) (Y) to 3 factors: liposoluble dye amount (X 1), titanium dioxide (X ), and pigment (X 3). There were n = 17 experimental runs (observations). The following 3 models were fit: Model 1: E Y X X X SSR SSE Model : E Y X X X X X X X X X SSR SSE Model 3: E Y X X X X X X X X X X X X SS R3 SSE p.53.a. Test whether any of the -way interaction effects or quadratic terms are significantly different from 0, controlling for main effects. H 0: Test Statistic: Rejection Region: P-value: > or < 0.05 p.53.b. What proportion of variation in Y is explained by each model? Model 1 Model Model 3 Q.54. A regression model was fit, relating Total Points (Y = 3*Wins + Draws) to Total Salary (X, in millions of pounds) for the n = 0 English Premier Football League 1998/1999 season. The fitted equation and summary model statistics are given below. ^ Y X SSR 050 SSE 1674 p.54.a. Test whether Total Points is related to Total Salary. H 0:. Test Statistic: Rejection Region: P-value: > or < 0.05 p.54.b. A plot of the squared residuals versus predicted values suggests a possible non-constant error variance. A regression of the squared residuals on X yields SS(Reg * ) = Test: H : Equal Variance Among Errors i H : Unequal Variance Among Errors h X 0 i A i 1 i e^ versus Y-hat Test Statistic: Rejection Region: P-value: > or < 0.05

25 Q.55. A regression model for language diversity was fit, relating log of the number of languages spoken (Y) to 3 predictors: log of area (X 1), log of population (X ), and mean growing season (X 3, in months) for n = 54 countries. p.55.a. Complete the following table of C p, AIC, and BIC for all possible regressions involving X 1, X, and X 3. Model p* SSE C_p AIC BIC X X X X1,X X1,X X,X X1,X,X p.55.b. Which model is selected based on each criteria? C p: AIC: BIC: p.55.c. To check for issues of multicollinearity, a regression relating each predictor on the other predictors is fit. The largest R of the 3 regressions is when X 1 is regressed on X and X 3. That R 1 value is Compute the Variance Inflation Factor VIF for X 1, where VIF1 1/ 1 Ri. Does it exceed 10? Q.56. Backward Elimination and Forward Selection based on using the model AIC will always choose the same model. True / False Q.57. In multiple regression it is possible to have a significant F-test for the full model, while none of the individual t-tests for the partial coefficients are significant. True / False Q.58. A researcher states that her regression model explains 75% of the variation in her dependent variable. This means that SSE/TSS = 0.5, where SSE is the Error sum of squares and TSS is the Total sum of squares True / False