29T-STAT.pmd

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1 29T STAT Total number of pages STATISTICS Full Marks : 00 Pass Marks : 30 Time : Three hours The figures in the margin indicate full marks for the questions. All Questions are Compulsory. Total Questions : 25 Nos. Q. No. carries mark each 2 = 2 Q. No. 2 to Q. No. 7 carry 3 marks each 3 6 = 48 Q. No. 8 to Q. No. 25 carry 5 marks each 5 8 = 40 Total = 00 Contd.

2 . Answer as directed : 2=2 øúàapple«±ú ± œ Î M øapple ± (a) Write down the value of 3 2x 2 5x 3 3 2x 2 5x 3 ±Ú øúì«fl ±º. (b) Verify whether the following function is a probability density function (p.d.f ) or not. Ó Ó øapple ± Ù ÚÀÈ ± y±øªó ± ÚQ Ù Ú ŒÚ Ú Ûœé ± fl ±º f x 3x 2, 0 x (c) Establish a relation between the operators and E. fl ±fl ±n E ±ÊÓ È ± g ö± ÛÚ fl ±º (d) If A and B are two independent events, then B A? øapple A ±n B apple È ± Ó La È Ú±, ŒÓ ÀôL P B A? P (e) If Var X 4, find the value of Var 3X 2. øapple Var X 4, ŒÓ ÀôL Var 3X 2 ±Ú øúì«fl ±º (f) When a sample is considered large? Œfl øó ± È ± õ øó apple «fl Î ± ı ø Ì fl ± (g) Define Null hypothesis. ømê õ fl ä :± øapple ±º (h) If X and Y are two random variables, under what condition E XY E X. E Y? øapple X ±n Y apple È ± ±apple ø2âfl fl, øfl Ó«Ó XY E X. E Y E (i) Write a point of distinction between sampling error and non-sampling error. õ øó Ú Sn øè ±n õ øó Ú Sn øè È ± Û±Ô«fl Î À~ fl ±º 29T STAT [ 2 ]

3 (j) Define simple random sample. ±apple ø2âfl õ øó apple «:± ø ±º (k) Under what condition binomial distribution will be symmetrical? øfl Ó«Ó ø Z Ûapple ıkè Ú õ øó í ı (l) What is meant by a parameter? ë õ ± í ı ø À øfl ı ʱ 2. Write down the Newtons forward and backward interpolation formulae. ½+½=3 øúî È Ú ËªÓ œ«±n Û(±»ªÓ«œ ôlà ı«ú S apple È ± ø ±º 3. Show that 3 Œapple ª± Œ f log f x log f x x 4. What is interpolation? Write down the general quadrature formula and identify different terms in it. +2=3 ôlà ı«ú ı ø À øfl ı ʱ ±Ò±Ì ı «œfl Ì SÀÈ ± ø ± ±n ±Ó ı ª +Ó Ûapple Ûø øapple ±º 5. From the table given below, estimate f 2. 3 Ó Ó øapple ± Ó ±ø fl ± Û± f 2 ±Ú øúì«fl ±º x : f x : T STAT [ 3 ] Contd.

4 6. State the addition rule of probability for any two events. What form does this result take when the events are (i) independent and (ii) mutually exclusive? 3 ø Àfl ±ÀÚ± apple È ± È Ú± ı±à ı y±øªó ± Œ ± SÀÈ ± ø ±º øapple È Ú± apple È ± (i) Ó La ±n (ii) Û Û±ôL, ŒÓ øó ± SÀÈ ± Œfl ÀÚ í ı kx 7. If P x ; x, 2, 3 is the p.m.f of X, find k and E X. 3 3 øapple X fl y±øªó ± ˆ Ù Ú P x ; x, 2, 3, ŒÓ ÀôL k ±n ±Ú Î ø ª±º 29T STAT [ 4 ] kx 3 E X 8. Given P A, P B and P A B øapple ± ±À A Find [Î ø ª±] P, B 2 4 (i) P A B (ii) P A B P º 5 P ±n A B 9. If X has Poisson distribution and P X 0, what is E X? 2 (Given log ) 3 e øapple ±apple ø2âfl fl X Œ Û y±øªó ± ıkè Ú ±ÀÚ ±n X øfl í ı [øapple ± ±À If B n, p log ] e P 0, E X 2 ±Ú X ~, then find Var X. 3 øapple X ~ B n, p, Var X ±Ú Î ø ª±º. Define random experiment with two examples. 3 apple È ± Î apple± Ì ±apple ø2âfl Ûœé ± :± ø ±º

5 2. If A, B and C are three mutually exclusive and exhaustive events and 3 PC 2P A PB, then find the value of P C. 3 øapple A, B ±n C øó øúè ± Û Û ıø ˆ«Ó ±n Û Ì«È Ú± ±n 3 PC 2P A PB, ŒÓ ÀôL P C ±Ú øúì«fl ±º 3. What do you mean by stratified random sampling? For the following data on stratified random sampling, estimate the population mean. 2+=3 ô œfl Ó ±apple ø2âfl õ øó Ú ı ø À øfl ı ʱ Ó Ó øapple ± ô œfl Ó ±apple ø2âfl õ øó Ú Ó Ô Û± ø Ü ±Ò ±fl Ú fl ±º Stratum size [ô ±fl ±] Sample means [ õ øó apple «±Ò ] Define and discuss the level of significance. 3 ±Ô«fl Ó ± ô :± øapple ± ±n ı ± ± fl ±º 5. Write down the large sample test statistic for testing the difference between two population means. Also mention its sampling distribution. 2+=3 apple È ± ø Ü ±Ò Û±Ô«fl Ûœé ± fl ø ıõ, ı» õ øó apple «±Ô«fl Ó ± Ûœé ± õ øó apple «ÊÀÈ ± ø ±º ÀÓ Ó ± õ øó apple «Ê ıkè ÚÀÈ ± ø Ú±Mê fl ±º 6. A coin is tossed 900 times and head turns up 480 times. Test the hypothesis that the coin is unbiased. (Level of significance = 5%) 3 È ± ^± 900 ı± ÛÕ øúàé Û fl ± í ±n ± øˆ Ó Ó 480 ı± G Œ Û±ª± í º ë ^±ÀÈ ± Úøˆ ÚÓ í õ fl äàè ± Ûœé ± fl ±º [ ±Ô«fl Ó ± ô = 5%] 7. Write down two applications of the chi-square test. ½+½=3 fl ± - ı «Ûœé ± apple È ± õ À ±ÊÚœ Ó ± Î À~ fl ±º 29T STAT [ 5 ] Contd.

6 6 dx 8. What do you mean by Numerical integration. Evaluate 0 x by Simpson s 3 rule and hence find log e 7. +4=5 6 dx øú ÀÈ ± ı ª ± fl ø 0 - ±Ú x ±Rfl Ú fl Ú ı ø À øfl ı ʱ ø Û Ú 3 Î ø ª± ±n Ó ± Û± log e 7 ±Ú øúì«fl ±º 9. State the uses of Pilot survey. Show that in S.R.S.W.R., the sample mean is an unbiased estimator of the population mean. 3+2=5 øappleflƒ õ fl ä œé ± ı ª ± ÛÀfl«Î À~ fl ±º Œapple ª± Œ Û Ú ö± ÛÚ ±apple ø2âfl õ øó Ú Œé SÓ õ øó apple «±Ò ø Ü ±Ò Úøˆ ÚÓ ±fl fl º 20. Derive the mean and variance of the Poisson distribution. 2+3=5 Û ıkè Ú ±Ò ±n õ Ì øúì«fl ±º 2. (a) If X ~ N,, find the distribution of øapple, ŒÓ ÀôL X ~ N, Z X. 3 Z X ıkè ÚÀÈ ± øúì«fl ±º (b) If X ~ N 80, 5, find P 75 X øapple X ~ N 80, 5, ŒÓ ÀôL P 75 X 85 ±Ú øúì«fl ±º 22. An urn contains 4 white and 4 black balls. A second urn contains 5 white and 4 black balls. One ball is transferred from the first urn to the second urn and then a ball is drawn from the later. What is the probability that it is white? 5 È ± ŒÈ Àfl ø Ó 4È ± ı ± ±n 4È ± fl í ± ı ±Àº 2 ÀÈ ± ŒÈ Àfl ø Ó 5 È ± ı ± ±n 4È ± fl í ± ı ±Àº õ Ô ŒÈ Àfl ø Û± È ± ı 2 ŒÈ ±Õ Ú± í ±n Û±ÂÓ 2 ŒÈ Àfl ø Û± È ± ı È Ú± í º ı ÀÈ ± ı ± Œ ±ª± y±øªó ± øúì«fl ±º 29T STAT [ 6 ]

7 23. Draw all possible samples of size 2 by simple random sampling (without replacement) from a population of 4 units (2, 4, 6, 8) and show that the sample mean is an unbiased estimator of the population mean and find its standard error. 5 4È ± Ûapple È ± ø Ü (2, 4, 6, 8) Û± ±apple ø2âfl õ øó Ú [ Û Ú ö± ÛÚ] ÛX øó Z±± 2 ±fl ± fl À ± y± ı õ øó apple «Î ø ª±º Œapple ª± Œ õ øó apple «±Ò ø Ü ±Ò Úøˆ ÚÓ ±fl fl ±n ± ±Úfl Sn øè øúì«fl ±º 24. From a population, 0 students are chosen at random and their heights in inches found to be 62, 60, 6, 65, 59, 7, 70, 68, 60 and 6. Test whether the population mean height is 66''? [ Given that : t 9d.f 2 26 ] ±apple ø2âfl ˆ ±Àª Œfl ±ÀÚ± È ± ø Ü Û± øú ı«±ø Ó fl ± 0 ÊÚ Â±S Î 2 Ó ± ± Û [ ø= ø ± ÛÓ ] øapple ± ±À 62, 60, 6, 65, 59, 7, 70, 68, 60 ±n 6º ø ÜÀÈ ± Î ˇ Î 2 Ó ± 66'' ÀÚ ±Ô«fl Ó ± Ûœé ± fl ±º [ øapple ± ±À 9d.f 2 26 t ] (a) Write a short note on statistic and standard error. 3 õ øó apple «Ê ±n ±Úfl Sn øè ø ı À øè ŒÈ ±fl ± ø ±º (b) What are the different types of errors in sampling? 2 õ øó ÚÓ Ô±øfl ı Û± ø ıøˆ iß õ fl ± Sn øè øfl øfl 29T STAT [ 7 ]

8 29T STAT [8] 2 5+