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MTE-11 Assignment Booklet Bachelor’s Degree Programme (B. Sc.) Probability and Statistics (01 July, 2008 –30 June, 2009) School of Sciences Indira Gandhi National Open University New Delhi Dear Student, Please read the section on assignment in the Programme Guide for elective Courses that we sent you after your enrolment. A weightage of 30 per cent, as you are aware, has been earmarked for continuous evaluation, which would consist of one tutor-marked assignment for this course. This assignment is in this booklet. Instructions for Formating Your Assignments Before attempting the assignment please read the following instructions carefully. 1) On top of the first page of your answer sheet, please write the details exactly in the following format: ROLL NO.:…………………………………………… NAME :…………………………………………… ADDRESS :…………………………………………… …………………………………………… …………………………………………… COURSE CODE: ……………………………. COURSE TITLE : ……………………………. ASSIGNMENT NO. ………………………….… STUDY CENTRE: ………………………..….. DATE.………………………….………. PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND TO AVOID DELAY. 2) Use only foolscap size writing paper (but not of very thin variety) for writing your answers. 3) Leave 4 cm margin on the left, top and bottom of your answer sheet. 4) Your answers should be precise. 5) While solving problems, clearly indicate which part of which question is being solved. 6) This assignment is valid only upto 30 June, 2009. If you have failed in this assignment or fail to submit it by 30 June, 2009, then you need to get the assignment for the year 2009 – 2010 and submit it as per the instructions given in the programme guide. We strongly suggest that you retain a copy of your answer sheets. Wish you good luck. 2 TUTOR MARKED ASSIGNMENT (To be done after studying all the four Blocks) Course Code: MTE-11 Assignment Code: MTE-11/TMA/2008-09 Maximum Marks: 100 Q.1 (a) The following table shows the number of children, out of a sample of 500, with the heights in the given range: Height in cm. 105-108 108-111 111-114 114-117 117-120 120-123 123-126 126-129 129-132 132-135 Number of Children 50 35 82 105 66 48 24 16 48 26 Draw a cumulative frequency ogive curve to represent the given data. From your curve estimate (i) the median height, (ii) the upper and lower quartile heights, (iii) the number of children in this sample with height 120 cm or more. (b) The weight in grams of 50 apples picked out at random from a consignment is as follows: 106, 107, 76, 82, 109, 107, 115, 93, 187, 95, 123, 125, 111, 92, 86, 70, 126, 68, 130, 129, 139, 119, 115, 128, 100, 186, 84, 99, 113, 204, 111, 141, 136, 123, 90, 115, 98, 110, 78, 90, 107, 81, 131, 75, 84, 104, 110, 80, 118, 82, (i) Form the grouped frequency table by dividing the variate range into intervals of equal width, each corresponding to 20 grams in such a way that the mid-value of first class corresponds to 70 grams. (ii) Find the mean, median, standard deviation and mean deviation of the weights of the apples. (iii) Also, find the proportion of the apples with weight more than 150 grams. Q.2 (a) (5) (10) Calculate the coefficient of correlation from the following data: X: 1 Y: 9 2 8 3 10 4 12 5 11 6 13 7 14 8 16 9 15 Also write down the equations of the lines of regression and obtain an estimate of Y which corresponds the X = 6.2. (7) 3 (b) The face cards are removed from a full pack of cards. Out of the remaining cards, 4 cards are drawn at random. What is the probability? (i) that they belong to different suits? and (ii) that the 4 cards drawn belong to different denominations? (c) (d) Q.3 (a) (b) (3) A Company has two plants to manufacture an item. Plant I manufactures 70% of the items and Plant II manufactures 30%. At Plant I, 80% of the items are rated standard quality and at Plant II, 90% of the items are rated standard quality. An item is picked up at random and is found to be of standard quality. What is the probability that it has come from Plant I? (3) A sample space has five outcomes S e1 , e 2 , e 3 , e 4 , e 5 with the probabilities of elementary events 2 1 1 1 1 P(e1 ) , P(e 2 ) , P(e 3 ) , P(e 4 ) , P(e 5 ) . 6 12 6 3 12 Let A = {e3, e4} and B= {e1, e4}, then show that A and B are statistically independent. (2) Using the distribution found in Q.1 (b), compute (i) moment coefficient of skewness. (ii) moment coefficient of kurtosis. A discrete random variable has the probability distribution X : 0 1 2 3 4 P (X = x) : a 3a 5a 7a 9a (6) 5 11a 6 13a 7 16a Determine (i) a (ii) P (X 5 ) (iii) The value of X such that P (X x ) 0.7 . (c) Q.4 (a) (6) Let X 1 and X 2 are independent Poisson variates with means 1 and 2 respectively, find P (X1 X 2 4). (3) The marks of a class are normally distributed with means = 200 and s. d. =10. (i) Find the number of students in a class of 100 getting marks between 200 and 220, between 190 and 215, above 230, below 180. (ii) Assuming that there are only 84 seats in the next class, calculate the minimum number of marks a candidate should secure in order to be eligible for promotion. (6) (b) The joint probability density function of the two-dimensional random variable (X, Y) is given by : 8 x y , 1 x y 2 f ( x , y) 9 , elsewhere 0 (i) Find the marginal density functions of X and Y, (ii) Find the conditional density functions of Y given X = x, and conditional density function of X given Y = y. (4) 4 (c) If X is the number scored in a throw of an unbiased die, show that the chebychev’s inequality gives P [| X | > 2.5] < 0.47, where is the mean of X, while the actual probability is zero. (5) Q.5 (a) A six faced die is thrown and the expectation that in 10 throws it will give five even numbers is twice the expectation that it will give four even numbers. How many times in 10, 000 sets of 10 throw each, would you expect it to give no even number? (5) (b) A genetical studies says that children having one parent of blood group M and the other parent of blood group N will always be one of the three blood groups M, MN, N, ; and that the average number of children in these groups will be in the ratio 1 : 2 : 1. The reports on an experiment states as follows : “Of 162 children having one M parent, and one N parent, 28 % were found to be of group M, 42% of MN and the rest of the group N”. Test the hypothesis that the data in the report confirms to the expected genetic ratio 1: 2: 1 at the 5% level of significance? (5) (c) The heights of 10 males of a given locality are found to be 70, 67, 62, 68, 61, 68, 70, 64, 64, 66 inches. Is it reasonable to believe that the average height is greater than 64 inches? Test at 5% significance level, assuming that for 9 degree of freedom P (t >1.83) = 0.05. (5) Q.6 (a) X 1 , X 2 , and X 3 is a random sample of size 3 from a population with mean value and variance 2 . T1 , T2 , T3 are the estimator’s used to estimate mean value , where T1 X1 X 2 X 3 , T2 2X1 3X 3 4X 2 , and T3 (X1 X 2 X 3 ) / 3 (i) (ii) (iii) (iv) (b) Are T1 and T2 unbiased estimators? Find the value of such that T3 is unbiased estimator for . With this value of is T3 a consistent estimator? Which is the best estimator? Given the probability density function f(x : ) [{1 (x ) 2 }] 1; x , show that the Cramer-Rao lower bound of variance of an unbiased estimator of is where n is the size of the random sample from this distribution. (c) (5) 2 , n (5) Prove that the maximum likelihood estimate of the parameter of a population having density function: 2 ( x ), 0 x 2 for a sample of unit size is 2x, where x is the sample value. Also, show that the estimate is biased. (5) Q.7 State whether the following statements are true or false? Give reasons for your answers. (i) The regression coefficient of X on Y is 3.2 and that of Y on X is 0.8. (ii) A certain dice was thrown 600 times and a 3 or 4 was obtained 205 times. On the assumption of random throwing, this data indicate an unbiased die. (iii) For a Poisson distribution with parameter , 1 is consistent estimator of 1 , where X is X the mean of a random sample for the given population. 5 (iv) From the population consisting of 5 items, the total number of possible samples of size 2 when the sampling is without replacement, when ordering is important and when it is ignored are 60 and 10, respectively. (v) If the coefficients of variation of two series are 75% and 90%, with S. D. 15 and 18 respectively, then both the series have same mean. (10) 6