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# Probability and statistics 2006 Tronc Commun Université de Technologie de Belfort Montbéliard

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Examen du Supérieur Université de Technologie de Belfort Montbéliard. Sujet de Probability and statistics 2006. Retrouvez le corrigé Probability and statistics 2006 sur Bankexam.fr.
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SQ 28exam Endterm th MondayJune 262006 For this test, you may use an electronic calculator and the Tables of Statistics. Length2 hours. I.Test of a process A factory makes, in mass production, circular parts whose mean diameter, which has a normal distribu tion, has to be M = 5 (cm), with a standard deviations= 0.24 . A 36sized sample is randomly taken out of the production. The diameter of each part is measured and the control service takes the following deci sion. If the mean diameter of this sample is lower than 4.92 or is higher than 5.08, then the process is to be checked and the machine adjusted to 5. Otherwise the machine is left without intervention. 1–1 : What is the risk to stop unnecessarily the process when M is actually equal to 5 ? What is the name of this risk ? 1-2 : What is the probability to decide that the process works properly if M= 5 ? 1- 3 :       if What isactually M = 5.05 ? the name of this risk ? 1-4 : What is the power of the test if M = 4.95 ?
II.Wrapping fast : Two candidates A and B to a position in a wrapping service must wrap 20 parcels. The time, in min utes, taken by A and B to wrap 20 parcels of the same kind are collected in the following table : A 5,17,3 5,4 5,2 6,86 5,73.7 5.3 2.8  8.43.4 4.4 2.9 3.3 6.1 5.2 5.6 4.1 4.3 B 5.33.1 5.6 9.5 7.6 8.58 5.48.2 4.5  3.45.2 7 5.2 6 4.95.9 6.79 7.8 2  1Determine estimators of the mean and the variance of the random variables X = wrapping time for A and Y = wrapping time for B.Calculate the point estimations for these four parameters. 2-2 Assumingthat X and Y are Gaussian (Normal) and that the wrapping times are independent, is there evidence to say, with anarisk = 0,05,that the variances of the two candidates are different ? If not, determine an estimation of the common variance. 2-3The management decides to hire A because they say A is significantly faster than B. Put in place the test which may accept or reject this decision and conclude. …/…
III.needle : Buffon’s We consider a parquet floor whose boards are 2a >0 wide. A 2blong needle (with b< a) is thrown on the floor and we are interested in the probability p(A) = probability that the needle lays on two boards of the parquet floor (across a line between two boards). Let I be the middle of the needle, Y the distance between I and the first lower line andqthe angle of the needle and the axis of the boards. p S V For symmetrical reasons, we may assume that the universeis :W =(y,q) / 0£y£2a, 0£ q £, the T 2 probability being uniform onW.3  1Prove that the needle meets a line if and only ify+b sinq ³2a or y£b sinq.3  2Represent in a orthonormal coordinates system (0,q, y) the universeWand the points where the needle meets a line (set A). 2b 3  3Prove thatp(A)=. pa 3  4Let X be the random variable : number of trials until the needle crosses a line. We suppose that 2a = 12 (cm) and 2b = 6 (cm). a) Whatis the distribution of X ? Give its expectation and its variance. b) Asurvey on 100 trials yields the following outcome. Values of X1 2 3 4 5 6 7³8 Number of trials28 21 189 8 4 210 Test, with a 0.95 level of significance, that these data fit with the distribution found in the previous question. c) Whatdo you think of the method to find an approximation of a famous number ? I want money, simply to be rich. John LENNON (ENGLAND 1940-1980 )