# Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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. The maximum likelihood estimator ofβ0is¯Y(see1
Lecture 4 normal distribution example). Henceˆβ0is the unbiased MLE (10pts). (c) To makec1Y1+c2Y2unbiased,we needE(c1Y1+c2Y2) =c1β0+c2β0=β0. Soc2= 1-c1. Now minimize the varianceVar(c1Y1+c2Y2) =c21VarY1+c22VarY2= (c21+ (1-c1)2)σ2for someσ. By differentiating it w.r.t.c1, we getc1= 1/2. Hence,¯YisMVUE whenn= 2.4. Data for the average blood lead level (blood) of children age 6 months to 5 years and the amount of lead used ingasoline production (gasoline) for ten 6-months periods are available.blood48597980959597102102107gasoline9.311.012.814.113.613.814.614.616.018.2R output for analyzing this data is given below.>blood 48 59 79 80 95 95 97 102 102 107>gasoline 9.3 11.0 12.8 14.1 13.6 13.8 14.6 14.6 16.0 18.2>summary(lm(blood˜gasoline))Call:lm(formula = blood ˜ gasoline)Residuals:Min1QMedian3QMax-11.5112 -6.6140 -0.27887.6404 10.0596Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) -14.31216.101 -0.889 0.399984gasoline7.2981.1506.344 0.000222***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 8.508 on 8 degrees of freedomMultiple R-Squared: 0.8342,Adjusted R-squared: 0.8135F-statistic: 40.25 on 1 and 8 DF, p-value: 0.0002220(a) Is there any linear relationship between blood lead level and the amount of lead used in gasoline production?What is the statistical significance of your conclusion? (20 pts).(b) What is the correlation coefficient between two variables? (10pts).Solution.(a) We test if the intercept slop is zero, i.e.H0:β1= 0(see Lecture 22 Concept 2).TheP-valueis extremely small so we conclude that there is a linear relationship with99%significance. (b) The coefficient ofdeterminationr2= 0.8442. If you take the squre root,r=±0.844. From the negative regrssion slope, we see thatthe correlation must be negative. Soρ=-0.844is the correlation.
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