Maximum Likelihood Estimation and Confidence Intervals, Assignments of Statistics

Download Maximum Likelihood Estimation and Confidence Intervals and more Assignments Statistics in PDF only on Docsity! 7-29. ! )( x exf xλλ− = ∏ ∏ = − = − ∑= == n i i ixnn i i x x e x eL n ii 1 1 !! )( 1λλλ λλ n x nx x n xn d Ld xxenL n i i n i i n i i n i i i n i n i i ∑ ∑ ∑ ∑ ∑∑ = = = = == = = =+−= ≡+−= −+−= 1 1 1 1 11 ˆ 0 01)(ln !lnlnln)(ln λ λ λ λλ λ λλλ 7-35 a) ∫ ∫ +∞ ∞− ∞ − == 0 2 3 22 2 )()( dxexdxxfxXE x θ θ ∫ ∞ −= 0 2 dyye yθ )2(2 Γ×= θ !12 ×= θ θ2= ∑ = == n i iXn XE 1 22 12)( θ so ∑ = = n i iXn 1 2 2 1Θ̂ b) ∑ ∏ ∑ ∑ −= ∂ ∂ −−== = − θθθ θ θ θ θ θ θ θ nxL nxxLexL i n i i i x i i 2 2 1 22 2 1)(ln ln 2 )ln()(ln)( 2 Setting the last equation equal to zero, the maximum likelihood estimate is ∑ = = n i iXn 1 2 2 1Θ̂ and this is the same result obtained in part (a) c) )2ln(2)5.0ln(2 15.0)( 0 2/2 θθ θ =−= −==∫ − a edxxf a a We can estimate the median (a) by substituting our estimate for θ into the equation for a. d) 809.4 62 1ˆ 6 1 2 = × =Θ ∑ =i iX ∫ ==<=< × − 5.4 0 81.42 878.0 81.4 )5.4() 10000 45000( 2 dxexXPXP x 8-1 a) The confidence level for nxnx /14.2/14.2 σμσ +≤≤− is determined by the value of z0 which is 2.14. From Table III, Φ(2.14) = P(Z<2.14) = 0.9838 and the confidence level is 2(0.9838-0.5) = 96.76%. 8-7 a) The 99% CI on the mean calcium concentration would be longer. b) No, that is not the correct interpretation of a confidence interval. The probability that μ is between 0.49 and 0.82 is either 0 or 1. c) Yes, this is the correct interpretation of a confidence interval. The upper and lower limits of the confidence limits are random variables. 8-33 a) The data appear to be normally distributed based on examination of the normal probability plot below. b) 95% two-sided confidence interval on mean comprehensive strength 235022502150 99 95 90 80 70 60 50 40 30 20 10 5 1 Data P er ce nt Normal Probability Plot for Strength ML Estimates - 95% CI Mean StDev 2259.92 34.0550 ML Estimates