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Unformatted text preview: Indian Institute of Technology Kharagpur Dept. of Mathematics, SPRING-END—SEM Exam. 2008—2009 ‘ Probability and Statistics, Subject No. MA20104 IInd Yr. B.Tech. and M.Sc. Time: 3 Hrs, Max Marks: 50 Instructions: Answer Question No. 1 and any SIX from the rest. Statistical Tables may be used 1. Examine each part of this question and carry out your solution as usual. Write ONLY the answer for each part of this question on the first page of your answerscripts. (21) Suppose er N(i1,,02). State the distribution of with its proper param— eter value. (b) Let 52 be the sample variance of a random sample of size 26 from NM, 16), Find P(9.35 < 52 < 24.09). (0) Let Xi, i = 1,2, . . . , 10 be independent random variables. Each X; is an uni— formly distributed random variable over [0,1]. Using central limit theorem evaluate PE; Xi > 6). (d) Let X follows exponential distribution with A = 1 Find the density function of Y = X 2. (e) The time to failure in months, X, of the light bulbs produced at two manufacturing plants obey exponential distributions with means 5 and 2 months, respectively. Plant B produce three times as many bulbs as plant A. The bulbs are indistinguishable to eye intermingled and sold. What is the probability that a bulb purchased at random will burn at least 5 months. (f) Choose the correct answer The critical region is the region where the null hypothesis is ........ ..(rejected/ accepted) if the value of the test statistic falls in this region. (g) Suppose XN NM, 02) and both it, and 02 unknown. Classify each of the hypotheses simple or composite Ho : /.L = O, 02 2 1(2'2') HO : u > 3, 0‘2 = H0 : u > 3 (h) Let a random sample of size 16 from N(u, 02) yield 23 = 4.7 and .92 = 5.76. Deter— mine 90% confidence interval for u. (8 X l) 2.(a) Let X1 N N(3, 9), X2 N N(0, 1) and X3 N N(l, l) are independent random variables. Further let Z1 = 2X1 + 3X2 ~ 4X3 and 22 2 3X1. — 2X2 + X3. Find the correlation coefficient between Z1 and Zg. (b) Let the random variable X~P()\) (Poisson random variable with parameter A). (P.T.O) Find the probability that X takes even numbers {0, 2, 4, ...., _ , (c) It is known that the probability of a user’s being able to log on a computer from a remote terminal at any given time is 0.7. Suppose that if user has not been able to log on in 4 attempts, then what is the probability that he will need 3 more attempts to be able to log on. (3 + 2 + 2) 3.(a) Evaluate the probability P{]X ~ ul > 20} for a Poisson distribution with pa- rameter A = 4. and compare it with the corresponding value obtained by Chebyshev’s inequality. (b) Given that my) = was“); a: 2 0, y > 0. Find E[Y|X = (c) A manufacturer offers the following deals to buyers. A buyer inspects 4 items from a large lot. If he find no defective items he pays Rs. 100000 for the lot. If he find one or 'more defectives, he pays Rs. 800.00 for the lot. If the actual proportion of defectives is ‘ 5%, what is the expected amount the manufacture can make per lot. (2 + 2 + 3) 4.(a) Consider the following probability density function == once—52mg, :6 > 0 where or > 0 and [3 > 0 are constants. Find 05 and fl given that E (X ) = 4. I (b) Suppose the time “X” (in hours) between orders for a given part at a large ware- house is gamma random variable with parameters r and A. Find by the method of moment the estimate of 7" and /\ based on the sample data 15.5, 4.5, 6.8, 46.0, 34.5, 4.7, 20.9, 8.2, 14.9, 17.7. . (3 + 4) 5.(a) A certain type of thread is manufactured with mean tensile strength of 78.3 kg and a standard deviation of 5.6 kg. How is the variance of the sample mean changed when the sample size is increased from 64 to 196? (b) If 312 and S3 represent the variances of independent random samples of size n; = 8 and n; = 12, taken from two normal populations with equal variances, find P < 4.89). '2 (c) Show that the sample variance 3% = —— X)2 is not an unbiased estimator of population Variance.- . - - " r r _: ' ‘ (2—1—23: 3-) ; 6(a) Thepweather bureau measured the followigy-Omnellwels units) i L ~ _- cations ma particular village before and after a cool front moved area. Is there a significance drop in ozone level? Decide at the level of significance 0.05. State clearly your hypotheses, test statistic and critical value on the basis of which you support your decision. (b) Independent random samples from two normal populations with common variance gave the following results: Can you conclude that the mean of the first population is greater than the second at a = 0.05? " (3 + 4). 7.(a) Suppose X1, X2, .....Xn is a random sample from the probability density function f(7;,)\) , where f(:r, A) = fiemfl—f , x > 0, A > 0. Find the maximum likelihood estimator of A. (b) If the expected value of this density function is 2/\. Do you think your estimator is an unbiased one? (c) If 8.6,, 6.4, 8.4, 9.8, 7.2, 7.8, 7.5 are the observed values of a random sample of size 9 from N (n, 02). Construct a 95% confidence interval for 02. (2+2+3). 8.(a) Suppose we want to compare fasting serum cholesterol levels among Asian immi— grants with general female population of United states. It is assumed that cholesterol levels in women aged 21—40 in USA are approximately normally distributed with mean 170 mg/dL and standard deviation 40mg/dL. It is beleived that the cholesterol levels among Asians (of same age group) are also normally distributed with mean 185mg/dL and the same standard deviation as of USA women. One hundred blood samples of female Asians immigrants gave a sample (2' = 181.25 mg/dL . Test the hypothesis that the mean cholesterol level of Asian immigrants is higher than that of USA female population at a = 0.05. State the critical region of the test procedure. Compute the power of the test when M1 = 185mg/dL. (b) Suppose it is desired to conduct a test at 1% level of significance with power of 90% for the two hypotheses that you have considered in part (a). How large should you take the sample size for the test procedure? (4 +3). M—The Paper Endsw» ...
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