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Toronto STA 302 H1F - STA 302 / 1001 H Test 1

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STA 302 / 1001 H - Fall 2005Test 1October 19, 2005LAST NAME: FIRST NAME:STUDENT NUMBER:ENROLLED IN: (circle one) STA 302 STA 1001INSTRUCTIONS:• Time: 90 minutes• Aids allowed: calculator.• A table of values from the t distribution is on the last page (page 8).• Total points: 50Some formulae:b1=P(Xi−X)(Yi−Y )P(Xi−X)2=PXiYi−nXYPX2i−nX2b0=Y − b1XVar(b1) =σ2P(Xi−X)2Var(b0) = σ21n+X2P(Xi−X)2Cov(b0, b1) = −σ2XP(Xi−X)2SSTO =P(Yi−Y )2SSE =P(Yi−ˆYi)2SSR = b21P(Xi− X)2=P(ˆYi− Y )2σ2{ˆYh} = Var(ˆYh) = σ21n+(Xh−X)2P(Xi−X)2σ2{pred} = Var(Yh−ˆYh) = σ21 +1n+(Xh−X)2P(Xi−X)2r =P(Xi−X)(Yi−Y )pP(Xi−X)2P(Yi−Y )2Working-Hotelling coefficient: W =p2 F2,n−2;1−α1 2 3 4abc 4def 4ghi11. (10 points) A simple linear regression model is fit on n observed data points.(a) What is the difference between β1and b1?(b) What does it mean if R2= 1?(c) In lecture we showedPni=1ei= 0 andPni=1eiXi= 0. Show thatPni=1eiˆYi= 0. (Youmay use the results shown in class if they are helpful.)(d) Explain why the result in (c) implies that the residuals and predicted values are uncor-related and why this is useful.22. (8 points) In order to carry out linear regression analyses, in addition to the assumption thata linear model is appropriate for the data, we have made the following assumptions:• the expectation of the random errors is zero• the variance of the errors is constant• the errors are uncorrelated• the errors are normally distributedAssume that the independent variable is not random.(a) Which of these additional assumptions are necessary to show that b1is unbiased for β1?(b) Derive the formula for the variance of b1and state which of the additional assumptionsare necessary for the derivation.33. (7 points) In lecture we have considered the Snow Gauge example. In this experiment,scientists measured the number of gamma rays (the gain) that make it through 10 samplesof each of 9 densities of polystyrene. We fit a simple regression model with the logarithm ofgain (loggain) as the dependent variable and density as the independent variable to these90 points. A scientist argues that, since 10 samples were measured at each density, takingthe mean of loggain at that density will result in a better estimate and the regression shouldthen be run using the 9 resulting points. Will the least squares estimates of the slope andintercept change? Will the estimate of the error variance change? If there is a change, saywhether it is larger or smaller. Justify your answers.44. (25 points) The data analysed in this question are from a random sample of records of esalesof homes in 1993 in the U.S. city of Albuquerque. The data collected include many variablesabout the homes sold, but we will only consider how well the size of the home (in squarefeet of usable floor space, variable name: sqft) can be used to predict the selling price (inhundreds of dollars, variable name: price) of the home.Some output from SAS is given below. Note that some numbers have been replaced by letters.Descriptive StatisticsUncorrected StandardVariable Sum Mean SS Variance DeviationIntercept 116.00000 1.00000 116.00000 0 0sqft 189751 1635.78448 337777165 238134 487.98988price 123045 1060.73276 147252397 145518 381.46781Analysis of VarianceSum of MeanSource DF Squares Square F Value Pr > FModel (A) 13229494 (B) 430.28 (C)Error 114 (D) 30746Corrected Total (E) 16734535Root MSE (F) R-Square 0.7906Dependent Mean 1060.73276 Adj R-Sq 0.7887Coeff Var 16.53058Parameter EstimatesParameter StandardVariable DF Estimate Error t Value Pr > |t|Intercept 1 -76.20835 57.17689 -1.33 0.1852sqft 1 0.69504 0.03351 (G) <.0001(a) Find the 7 missing values (A through G) in the SAS output.(b) How many houses are in the sample?(c) Is the intercept statistically significantly different from 0? Justify your answer. Explainthe meaning of the intercept for a real estate agent.5(d) A house with 2000 square feet of usable space came on the market (under the samemarket conditions as the houses used in this analysis). Predict its selling price.(e) What is the standard error of the prediction in part (d)?(f) Plots of the data including the regression line and 95% confidence intervals for the meanof Y and 95% prediction intervals for Y are given below.i. Which plot is which? How do you know?ii. For the plot on the right, show how to calculate the value on the lowest curvecorresponding to X = 1500. In your answer include the numeric value.6(g) A plot of the residuals versus predicted values is below.Describe any problems you see in the residual plot. If the plot shows that any assump-tions are being violated indicate which.(h) A student hired by the real estate board to analyse these data argues that we should con-sider correlation rather than regression since the predictor variable is random. Respondto this comment.(i) Several of the homes in the random sample used in this analysis were from a new housingdevelopment. Why should this be considered in carrying out the


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Toronto STA 302 H1F - STA 302 / 1001 H Test 1

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