Statistik II - Exercise session &

Transcription

1 Statistik II - Exercise sessio & Ifo Classroom: SPA1 03 Time: Wedesdays, 16:15-17:45 i Eglish Assigmets o webpage (lvb>staff>pb) Cotact: Petra Burdejova petra.burdejova@hu-berli.de Office: SPA1 R400 (upo agreemet) Schedule: Date Week Exercises E1 4-11, 4-18, E1 4-11, 4-18, E 5-6, 5-10, 5-11, 5-14, E 5-6, 5-10, 5-11, 5-14, E3 5-16, 5-1, 5-3, E3 5-16, 5-1, 5-3, E4 6-3, 6-9, 6-13, E4 6-3, 6-9, 6-13, E5 7-3, 7-5, 7-6, E5 7-3, 7-5, 7-6, E6 8-1, 8-4, 8-7, E6 8-1, 8-4, 8-7, E7 TBA E8 Review for exam E8 Review for exam 1

2 Review week 9 & week 10 Slides: Theory of samplig (05 Stichprobetheorie) Estimatio procedures (06 Schätztheorie) 1 Sample distributio 1.1 Sample distributio of sample mea Distributio of X Populatio R.v. Distr. Coditio X i N(µ; ) kow Z = X µ / N(0, 1) t( 1) for 30 ukow T = X µ S/ N(0, 1) for > 30 Ukow distributio kow Z = X µ / ukow T = X µ S/ 1. Sample distributio of Sample proportio Sample fuctio: ˆΠ = X (Example: Smokers i Berli, X - umber of smokers i our sample.) Distributio of simple radom sample N(0, 1) for > 30 N(0, 1) for > 30 X B(; π) E(X) = π V ar(x) = π (1 π) Approximatio by ormal distributio: ( ) π(1 π) ˆΠ N π; ˆΠ = Estimatio procedures true parameter of populatio θ Estimator (fuctio) θ = g(x1,..., X ) MSE=Mea Square Error MSE = E[( θ θ) ] = E[{ θ E( θ)} ] + {E( θ) θ} }{{}}{{} =V ar( θ) =bias Example: θ = µ, θ = X = X i

3 .1 Maximum - Likelihood (ML) Method Likelihood-Fuctio L(θ) = L(θ x 1,..., x ) = f(x i θ) LogLikelihood-Fuctio log(l(θ)) = log(f(x i θ)) maximize maximize. Least Squares (LS) Method Quadratic Form Q(θ) = (x i E(X i )) = (x i g i (θ)) miimize.3 Cofidece iterval at level 1 Cofidece iterval for µ kow Cof. iterval P Estimator iterval Legth X i ormally distributed or distr. of populatio ukow, but 30 ( ) X z 1 µ X + z 1 = 1 [ X z 1 [ x z 1 ; X + z 1 ; x + z 1 ] ] z 1 l = e = z 1 with l = legth ad e = error from N(0; 1) sample size z 1 e Cofidece iterval for proportio π for Normal approximatio Approximative Cofidece iterval Estimator iterval X B(; π) ad ˆΠ = X/ is approximately ormally distributed ( X P z 1 ˆΠ π X ) + z 1 ˆΠ = 1 X X z 1 (1 ) X ; X X + z 1 (1 ) X x x z 1 (1 ) x ; x + z 1 z 1 from N(0, 1) ) x (1 x Sample size z 1 / 4 e 3

4 Exercises Exercise Lamps A supply of N = 000 lamps will be ivestigated by meas of a simple radom sample of size = 0. With a help of the radom variable X: umber of defective bulbs i the sample of size =0 aumber d of defective lamps i the supply is estimated. a) Give a ubiased estimator θ = f(x) for d ad show that E(θ) = d. b) I a give sample, the umber of defective bulbs is equal to 3. How may defective bulbs do you estimate i the delivery? Exercise Gamblig machie A gamblig machie has the followig probability distributio for the wi X per game (i EUR): x P(X = x) p p 1- p The producer of these machies hired a statisticia to perform a estimate for p to kow whether the value of p has chaged sice the gamblig machies starts up. a) The statisticia draws a sample of size = 6, i.e. plays with the machie 6 times ad writes dow the wi. The sample (X 1, X, X 3, X 4, X 5, X 6 ) had realizatio as follows: (-1,1,-1,0,1,1). Verbalize this sample result. b) Calculate the followig probabilities: P(X = 0),P(X = 1),P(X = 1). c) How would you determie the probabilities of wi X per game accordig to sample metioed above, if you have o iformatio about the probability distributio of X? d) What is the probability P {(X 1, X, X 3, X 4, X 5, X 6 ) = ( 1, 1, 1, 0, 1, 1)} based o the above probability distributio? e) Determie the maximum likelihood estimator for p i this problem. f) Estimate p by this sample result through the maximum likelihood method. g) Estimate p by this sample result through the least squared method. Exercise Dioxi emmisios It is believed that the dioxi emissios of a cosmetic factory pre miute are ormally distributed with the mea 5 ad st. deviatio 1 N (5kg; 1kg). a) What is the probability that the average of a sample of size = 9 is betwee 4 ad 6 kg. b) What is the area, where the average value will be with probability of 95%? c) How large should be the sample, so that the average dioxi emissios are exactly estimated with probability 95% ad est.error for e = 0.5 kg/mi? 4

5 d) Compute the cofidece itervals for the average dioxi emissios at the cofidece level 1. e) They measured 9 times of the dioxi emissios radomly (kg/mi): 7; 4; 5; 10; 9; 6; 8 ; 6.5; 7.5. Calculate the estimatio iterval at a cofidece level 1 = Exercise Kilometrage A) For a test 49 radomly draw car of the same type were fuelled with the same amout of fuel. With this amout of fuel the cars wet o average 50 km. Assume that st. deviatio is kow 7km. a) Give a explicit cofidece iterval [V L, V U ] for average kilometrage µ for this type of car at the cofidece level 1. b) Determie the iterval for µ whe 1 = 95%. c) What sample size is eeded, if the estimated iterval for µ at the same level shall have a width km? B) Some visitors of this test evet were radomly chose by jouralist ad asked about their membership i the ADAC (Germa automobile club). Amog 00 people 40 were ADAC members. Determie the iterval fro π whe 1 =99 %. C) Coffee machie was istalled at the tribue for this evet. It fills 0.l cup with coffee. Assume that the quatity is ormally distributed. Radom sample of size = 5 has followig values: 0.18, 0.5, 0.1, 0.0, 0.5. a) Give a explicit cofidece iterval [V L, V U ] for average quatity µ for this machie at the cofidece level 1. b) Determie the iterval for µ whe 1 = 95%. 5