STA 302 H1F / 1001 HF – Fall 2010TestOctober 21, 2010LAST NAME: FIRST NAME:STUDENT NUMBER:INSTRUCTIONS:• Time: 90 minutes• Aids allowed: calculator.• All of the formulae below can be taken as known unless a question indicates otherwise.• Total points: 50Some formulae:b1=P(xi−x)(yi−y)P(xi−x)2=Pxiyi−nxyPx2i−nx2b0= y − b1xVar(ˆβ1|X) =σ2P(xi−x)2Var(ˆβ0|X) = σ21n+x2P(xi−x)2Cov(ˆβ0,ˆβ1|X) = −σ2xP(xi−x)2SST =P(yi− y)2RSS =P(yi− ˆyi)2SSReg = b21P(xi− x)2=P(ˆyi− y)2Var(ˆy|X = x∗) = σ21n+(x∗−x)2P(xi−x)2Var(Y − ˆy|X = x∗) = σ21 +1n+(x∗−x)2P(xi−x)2r =P(xi−x)(yi−y)√P(xi−x)2P(yi−y)2SXX =P(xi− x)2=Px2i− nx2hij=1n+(xi−x)(xj−x)SXXhii>4nDFBETASik=bk−bk(i)s.e.(bk)> 1 or2√nDFFITSi=ˆyi−ˆyi(i)s.e.(ˆyi)> 1 or 2q2nDi=P(ˆyj(i)−ˆyj)22S2>4n−21abc 1d 2 3abc 3def 3gh, 411. Consider the example we have been examining in lecture in which we have been using simplelinear regression to model how the atmospheric concentrations of CFC-11 (in parts per trillion)were changing as a function of time (in years) before the implementation of the MontrealProtocol. The data consist of 153 measurements of CFC-11 taken monthly from 1977 to theend of 1989 and the date on which the measurements were taken.(a) (2 marks) State the simple linear regression model being used. Which terms in the modelare random variables?(b) (3 marks) State the Gauss-Markov conditions for the model in part (a).(c) (4 marks) Assume that the Gauss-Markov conditions and the usual distributional as-sumptions hold. State fully the distributions of the random variables in the model. Howdo the distributions change with time?2(Question 1 continued.)(d) Suppose that the regression model has been fit to the data and the usual statistics havebeen calculated.i. (3 marks) Show thatPni=1ˆeixi= 0.ii. (3 marks) Show that estimator of the slope of the regression line is unbiased.32. Suppose a simple linear regression is carried out to investigate the relationship between adependent variable Y and an independent variable X. The data consist of n pairs of observedvalues of X and Y , (xi, yi), i = 1, . . . , n.(a) (2 marks) What is the first step you should carry out in the regression analysis? Whatdo you hope to accomplish in this step?(b) (1 mark) As part of the output for the simple linear regression analysis, SAS gives theresults of the statistical test with null hypothesis H0: β1= 0 and alternative hypothesisHa: β16= 0. Why is this test of particular interest?(c) You suspect that there is a strong linear relationship between Y and X.i. (2 marks) For the test with null hypothesis H0: β1= 0, do you expect the teststatistic to be large or small? Explain.ii. (2 marks) For the test with null hypothesis H0: β1= 0, do you expect the p-valueto be large or small? Explain.43. In a paper published in the British Medical Journal in 1965, Lea looked at data from countiesin regions of Great Britain, Norway, and Sweden. He was interested in how the mean annualtemperature (in degrees Fahrenheit) affected the mortality index for breast cancer. (Themortality index is a measure of the death rate for women diagnosed with breast cancer. Theindex Lea used measures death rate relative to the average death rate for England and Wales.On his scale, England and Wales was given the value of 100. Mortality indices greater than100 indicate a higher death rate than that of England and Wales.)Here are some quantiles from t-distributions which may be useful for some of the questionsthat follow.Degrees of Upper-tail probabilityfreedom 0.005 0.010 0.025 0.05 0.1014 2.977 2.624 2.145 1.761 1.34515 2.947 2.602 2.131 1.753 1.34116 2.921 2.583 2.120 1.746 1.337Some SAS output is given below and on the next page for the analysis Lea carried out.The REG ProcedureDescriptive StatisticsUncorrected StandardVariable Sum Mean SS Variance DeviationIntercept 16.00000 1.00000 16.00000 0 0temperature 713.50000 44.59375 32285 31.17663 5.58360mortality 1333.50000 83.34375 114535 226.42929 15.04757Dependent Variable: mortalityAnalysis of VarianceSum of MeanSource DF Squares Square F Value Pr > FModel 1 2599.53358 2599.53358 (A) <.0001Error 14 796.90580 56.92184Corrected Total 15 3396.43938Root MSE 7.54466 R-Square 0.7654Dependent Mean 83.34375 Adj R-Sq 0.7486Coeff Var 9.05246Parameter EstimatesParameter StandardVariable DF Estimate Error t Value Pr > |t|Intercept 1 -21.79469 15.67190 -1.39 0.1860temperature 1 2.35769 0.34888 (B) <.00015Output StatisticsDependent Predicted Std Error Std Error StudentObs Variable Value Mean Predict Residual Residual Residual1 102.5000 99.1550 3.0053 3.3450 6.920 0.4832 104.5000 95.8543 2.6429 8.6457 7.067 1.2233 100.4000 96.0900 2.6674 4.3100 7.057 0.6114 95.9000 94.2039 2.4779 1.6961 7.126 0.2385 87.0000 92.5535 2.3270 -5.5535 7.177 -0.7746 95.0000 90.9031 2.1929 4.0969 7.219 0.5687 88.6000 89.7243 2.1093 -1.1243 7.244 -0.1558 89.2000 84.5373 1.8944 4.6627 7.303 0.6389 78.9000 87.3666 1.9779 -8.4666 7.281 -1.16310 84.6000 77.4642 2.0772 7.1358 7.253 0.98411 81.7000 82.4154 1.8912 -0.7154 7.304 -0.098012 72.2000 80.7650 1.9244 -8.5650 7.295 -1.17413 65.1000 77.9358 2.0489 -12.8358 7.261 -1.76814 68.1000 72.9846 2.4305 -4.8846 7.142 -0.68415 67.3000 53.1800 4.8457 14.1200 5.783 2.44216 52.5000 58.3669 4.1494 -5.8669 6.301 -0.931Cook’s Hat Diag CovObs -2-1 0 1 2 D RStudent H Ratio DFFITS1 | | | 0.022 0.4697 0.1587 1.3329 0.20402 | |** | 0.105 1.2475 0.1227 1.0544 0.46663 | |* | 0.027 0.5965 0.1250 1.2558 0.22544 | | | 0.003 0.2298 0.1079 1.2895 0.07995 | *| | 0.031 -0.7621 0.0951 1.1744 -0.24716 | |* | 0.015 0.5533 0.0845 1.2092 0.16817 | | | 0.001 -0.1497 0.0782 1.2538 -0.04368 | |* | 0.014 0.6244 0.0630 1.1668 0.16209 | **| | 0.050 -1.1789 0.0687 1.0164 -0.320310 | |* | 0.040 0.9826 0.0758 1.0874 0.281411 | | | 0.000 -0.0944 0.0628 1.2358 -0.024412 | **| | 0.048 -1.1915 0.0651 1.0082 -0.314313 | ***| | 0.124 -1.9327 0.0738 0.7555 -0.545414 | *| | 0.027 -0.6703 0.1038 1.2090 -0.228115 | |**** | 2.093 3.1052 0.4125 0.6507 2.602016 | *| | 0.188 -0.9264 0.3025 1.4632 -0.6100Questions related to this output begin on the next page.6(Question 3 continued.)(a) (5 marks) What are the values of each of the following:- the number of observations- the number replaced by (A) in the SAS output- the number replaced by (B) in the SAS output- the estimate of the correlation betweenmortality index and mean annual temperature- the estimate of the