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1 School of Busiess ad Ecoomics Exam: Code: Examiator: Co-reader: Busiess Statistics E_BK_BS / E_IBA_BS dr. R. Heijugs dr. G.J. Frax Date: 5 May, 08 Time: :00 Duratio: Calculator allowed: Graphical calculator allowed: hours Yes No Number of questios: 3 Type of questios: Aswer i: Ope / multiple choice Dutch or Eglish BK / Eglish IBA Remarks: You will receive a special aswer sheet for questio You will receive ormal lied paper for questios ad 3 3 Please write your ame ad studet umber 7 digits o paper ad 4 You may keep the questios 5 You ca use the statistical tables, which will be haded out Credit score: Grades: Ispectio: Number of pages: Start Questio Questio Questio Jue, 08 but we try earlier. Will be aouced o Cavas. icludig frot page ad formula sheet you may detach the formula sheet Good luck!

2 Questio 4 poits Questio cosists of 4 short sub-questios. Each sub-questio couts for 3 poits. You must give a aswer oly, o a separate special aswer sheet. Note the followig i aswerig the sub-questios: The idicatio exact meas that you have to fill i oe or more exact umbers, such as, 3 ad 3e. The idicatio decimal meas you have to fill i a umber at the specified accuracy, such as 3.0. I additio, you may have to specify additioal text, such as euro. The idicatio sigificat digits meas you have to fill i a umber at the specified accuracy, such as I additio, you may have to specify additioal text, such as euro. The idicatio text, meas you have to supply a phrase, such as There is o statioary poit. The idicatio formula meas that you have to fill i a mathematical expressio, such as a +. The idicatio choose oe meas that you have to choose oe optio, such as B. The idicatio choose oe or more meas that you have to choose oe or more optios, such as B ad D ad F. a I the aalysis of chicke farms, a box plot is show as below: Describe what the distace betwee 3 ad 8 so 8 3 = 5 meas. For istace: The variace of the populatio. text b c We roll a die 8 times ad fid a eve umber 7 times. What is the p-value for the ull hypothesis that the die is fair? all decimals We roll aother die 50 times. How is the average outcome approximately distributed? Specify the ame of the distributio, but do ot specify parameter values. Example aswers: χ or beroulli. d We use a o-parametric test to test equality of the cetral values of three groups = 8, = 0, 3 =. Fid the critical value of the usual test statistic at α = all decimals e f Same questio as d, but ow for a parametric test. all decimals We wat to test equality of three proportios. Which test do you use? Give the ame of the test with some details. Example aswers: Kruskall-Wallis with 3 groups or 3 sample z-test with equal variace or biomial test with 3 degrees of freedom.

3 g h i j k For a certai aimal species, it is kow that the weights X, i kg are ormally distributed with μ = 00 ad σ = 0. What is the probability that a radom aimal of this species has a weight of 0? 3 decimals See g. What is the probability that a radom aimal has a weight higher tha 0? 3 decimals See g. Every week, we test 6 radom aimals of this species ad calculate their average weight. Calculate the stadard deviatio of this average weight. 3 decimals See i. What is the probability that exactly aimals out of a sample of 6 have a weight higher tha 00? 3 decimals What happes if you decrease α from 0.05 to 0.0 i testig H 0 : μ = μ 0? all other thigs stay uchaged? Choose oe or more A The probability of a type I error icreases C The probability of a type I error decreases B The probability of a type II error icreases D The probability of a type II error decreases E Noe of the above l m We collect data o the umber of cars with a specific color black, blue, red i three cities Amsterdam, Brussels, Paris. We test idepedece of color ad city. What is the critical value of the usual test statistic at α = 5%? I case there are two critical values, give the smallest. all decimals See l. Which requiremet eeds to be satisfied i order to perform this test accurately? Example aswer: all variaces must be symmetric. The aswer may also be oe. I a two-sample test of equality of medias, we fid for sample size 8 a rak sum of 73, ad for sample size 5 a rak sum of 8. I what rage is the p-value of this test? Give the smallest possible iterval that ca derived from the table. Example aswer: 0.05 p-value 0.. Do ot use a ormal approximatio. Bous questio: if you miss oe of the above questios, you may still obtai maximum score by correctly aswerig the questio below. o Give are the followig pieces of iformatio: A H 0 : μ = 00 B there is evidece that μ = 00 C there is o evidece that μ 00 D do ot reject H 0 E accept H 0 F p-value>α Select all elemets that you ca use i a cosistet summary of a hypothesis test, ad place them i the logical order. Example aswer: D F A. 3

4 Questio 4 poits Questio must be aswered o the lied exam sheets. Please start at the top of a page. You must specify all steps you take ad use good otatio priciples. Pop sogs are a area of statistical research. We aalyze the duratio i secods of pop sogs, divided over two periods. Summary data are reproduced below. a b c Have the sog duratios become more uiform i the last few decades? I other words: has the variace decreased? Use the data to judge this with a appropriate hypothesis test α = 5%. Use the 5-step procedure. 0 poits Give a 95% cofidece iterval for the differece of the mea duratio i the two periods. State ad/or justify all assumptios ad/or requiremets. If you like, you may assume equal variaces still otig this assumptio i your aswer. 8 poits I a follow-up study, we defie three arrower periods: C , D ad E ad collect more data C = 30, D = 34, E = 36. Suppose we wat to test if the mea duratio is the same i every period, which test would we employ? Give the ame of the test ad write dow the first three steps of the 5-step procedure. 6 poits 4

5 Questio 3 4 poits Questio 3 must be aswered o the lied exam sheets. Please start at the top of a page. You must specify all steps you take ad use good otatio priciples. A oeologist is a perso who kows a lot about wies. a b We ask 00 oeologists to blidly idetify two wies: a Bordeaux wie B ad Châteaueuf-du- Pape wie C. Of the 50 persos that had wie B, 4 correctly idetified it; the other 8 mistook it for C. Of the 50 persos that tasted wie C, 38 made the right choice ad made a mistake. Is wie B easier to recogize tha wie C? Formulate ad test a appropriate ull hypothesis, at α = 5%. Use the five-step procedure. 0 poits We asked a sample of oeologists to rate wies of differet prices o a 0-00 scale, 0 meas awful, 00 meas delicious. Because we expect that the scores may differ by geder, we also keep track of this variable 0=male, =female. Results are below. A critical wie expert thiks that the regressio coefficiet for price is.0 or less. Test if this ca be show i a five-step procedure α = 5%. Formulate the model associated with the above computer output ad defie all o-stadard symbols. 0 poits c See b. Fid a 95% cofidece iterval for the regressio coefficiet for geder. 4 poits 5

6 Basic formulas for Statistics Jauary 08 positio P th-percetile Mea + P 00 µ = xi N x = xi or x = c j= f jm j x G = x x x Variace σ xi µ = N s xi x x = = i x x = i x i / Chebyshev: k 00% CV = σ µ s 00% CV = 00% x xi x y i y xi x yi y ρ = σ X,Y σ X σ Y r = or fj m j x Odds: P A P A P A P A Bayes: P B A = P B A P A = P A B P B P A Bi = P A B P B P A Bi P B i µ = µ X = E X = x i P X = x i Eh X = h x i P X = x i e.g. E X 3 = x i 3 P X = x i E ax = ae X σ = σ X = var X = x µ P X = x = E X µ = = E X E X = E X E X var a X = a var X σ ax = a σ X

7 σ X,Y = cov X, Y = i,j x i µ X y j µ Y P X = x i, Y = y j s X,Y = = N N x i µ X y j µ Y i= r X,Y = s X,Y s X s Y x i x y j y i= cov ax, by = ab cov X, Y µ X+Y = µ X + µ Y σ X+Y = var X + Y = σ X + σ Y + σ X,Y Variables X ad Y are idepedet if ad oly if for all x ad for all y: P X = x, Y = y = P X = x P Y = y If X ad Y idepedet: σ X+Y = var X + Y = σ X + σ Y cov X, Y = 0 Special Distributios:! C r = = r r! r! Biomial: P X = x = P r =! r! π x π x ; E X = π; var X = π π x ormal approximatio if π 5 ad π 5 Poisso approximatio if 0 ad π 0.05 Hypergeometric π = S/N: S N S P X = x = x x ; E X = π; var X = π π N N N ormal approximatio if expected frequecies i all cells are at least 5 biomial approximatio if /N < 0.05 Poisso: P X = x = λx e λ ; E X = λ; var X = λ x! ormal approximatio if λ too large for table 3

8 Geometric: P X = x = π π x ; for x =,,... E X = ; var X = π /π π Uiform discrete: P X = x = b a + for x = a, a +,..., b E X = a + b ; var X = b a + Uiform cotiuous: f x = b a Expoetial: for a x b E X = a + b ; var X = b a f x = λe λx for x 0, λ > 0 P X x = e λx for x 0 EX = σ X = λ Normal: f x = σ π exp x µ ; σ E X = µ; var X = σ Estimators. Estimator for µ : X E X = µ; var X = σ or σ N N X µ σ/ N 0, ormal popul.; or symmetric popul. ad 5; or 30 X µ S/ t ormal popul.; or symmetric popul. ad 5; or 30 Estimator for σ : S S χ σ ormal populatios Estimator for π: p formerly also p or p E p = π; var p = π π or 4 π π N N

9 Cofidece itervals: x ± z α/ σ or x ± z α/ σ N N s s N x ± t ;α/ or x ± t ;α/ N p ± z α/ p p x ± z α/ x s χ ;α/ Sample size: σ s χ ; α/ E = z α/ σ or E = z α/ π π p p N or p ± z α/ N Testig hypotheses z = x µ 0 σ or z = x µ 0 σ N N = Z α + Z β σ µ t = x µ 0 s or t = x µ 0 t = r df = r z = r S s N N z = x π 0 ± π0 π 0 χ = s σ 0 z = x x µ µ σ / + σ / or z = x π 0 ± π0 π 0 N N ormal populatios t = x x µ µ s / + s / ; df = s + s s + s 5

10 t = x x µ µ ; df = + ; s s p/ + s p = s + s p/ + z = z = p p π π p p / + p p / p p 0 ; p = x + x p p / + p p / + F = s s df, df = ; ormal populatios W = R + i ; EW + = ; var W = i= T = R i ; E T = + + ; var T = + + i= χ = f jk e jk Simple Regressio: b = SS xy SS xx = b 0 = y b x e jk ; df = r c e jk 5 xi x y i y xi x = xi y i xy x i x e i = SSE = y i ŷ σ = s = MSE = e i k DW = e t e t t= t= e t t = b β s B where s B = σ xi x ; df = k 6

11 t = r r ; df = k ŷ i ± t σ h i resp. ŷ i ± t σ + h i where h i = + x i x xi x Multiple Regressio Sum of Mea Squares df Squares Regressio ŷi y k Residual yi ŷ i k σ error Total yi y R = ŷi y e yi y = i yi y partial F leverage B Std.Error Beta t Costat b 0 s B 0 b 0 X b s B... b X b s B R adj = [ ] R = k F = SSE Restricted SSE F ull /m V IF j = R j MSE F ull F m, k sb 0 sb b sb k e i yi y MSE s bj = SS xj R j h i k + 7

12 ANOVA c colums,,,..., c observatios per colum Source of variatio Sum of Squares df Betwee SSA = j yj y c Error Withi SSE = y ij y j c Total SST = y ij y Tukey T calc = y j y k T c, c; MSE j + k Crit.Rage = T c, c MSE + j k ANOVA r rows, c colums, m observatios per cell Source of variatio Sum of Squares df ROWS A SSA = c m r yi y r COLUMNS B SSB = r m c y j y c INTERACTIONS SSAB = m r c yij y i y j + y r c Error SSE = r c m yijk y ij. rc m Total SST = r c m yijk y Kruskal-Wallis H = + c T j j 3 + df = c ; j 5 8

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