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' MTE -11 $ Assignment Booklet Bachelor’s Degree Programme Probability and Statistics School of Sciences Indira Gandhi National Open University New Delhi 2003 & % Dear Student, We hope you are familiar with the system of evaluation to be followed for the Bachelor’s Degree Programme. At this stage you may probably like to re-read the section of assignments 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 two tutor-marked assignments for this course. Both these assignments are 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 a 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) These assignments are to be submitted to the Study Centre as follows: • Assignment 1 should be submitted to the Study Centre Coordinator within 6 weeks of receiving this booklet. • Assignment 2 should be submitted to the Study Centre Coordinator within 12 weeks of receiving this booklet. Answer sheets received after the due date shall not be accepted. We strongly suggest that you retain a copy of your answer sheets. Wish you good luck. 2 ASSIGNMENT-1 (To be done after studying Blocks 1 and 2.) Course Code: MTE -11 Assignment Code: MTE-11/TMA-1/2003 Maximum Marks: 100 1) The following table gives the monthly wages of 200 employees working in different companies of a city. Monthly wages (Rs. in ‘000) Number of Employees 10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49 12 18 25 40 54 28 15 08 Calculate the average monthly wage of the employee of this group. Also find the limits for the monthly wages (lower and upper) under which 50% of the employees fall. (10) 2) a) The average sales (in units) and the variance (in units) of a product in three different stores in a specific locality is given below. Store Average sales Variance Number of days A 12 05 12 B 10 07 10 C 08 03 09 Calculate the average sales and variance of all the three groups combined together. b) 6 observations on (x, y) yielded the following data: P P xi = 30, P x2i = 200, yi = 180, P Σxi yi = 1000 y2i = 5642 i) Determine the correlation coefficient between x and y. ii) Given x = 10, what will be the predicted value of y. iii) Given y = 15, predict x. 3) a) (5) (5) The following is the record about the number of defectives observed in the manufacturing process during 10 days period of a manufacturing firm. 12 19 13 18 14 10 29 15 11 17 Compute the first four moments about the arithmetic mean and the coefficient of kurtosis for the data. (7) 3 4) b) In a certain community, 10% of all people above 50 years of age have diabetes. A health service in this community correctly diagnoses 95% of all persons with diabetes as having the disease, and incorrectly diagnoses 5% of all persons without diabetes as having the disease. Find the probability that a person randomly selected from among all people of age above 50 and diagonised by the health service as having diabetes actually has the disease. (3) a) Calculate the Product moment correlation coefficient between the two variables sales (x) and the advertising expenditure (y) for the following data. Sales(x) 0 − 1000 1000 − 2000 3000 − 4000 Adv. Expenditure (y) 100-150 150-200 200-250 1 7 13 10 15 16 15 20 19 (8) b) Let X be a r.v. with p.m.f. given by the following table −2 −1 0 1 2 3 20 4 20 6 20 4 20 3 20 Compute E(X2 ) 5) (2) The following table shows the data on the monthly income and the expenditure on entertainment (figures in 100 rupees) of 100 families in a specific locality. Income Expenditure 40 2 45 3 54 4 61 5 65 5 63 6.5 70 7 Fit a regression line which can be used to estimate the expenditure for the known income. Also estimate the likely expenditure if a family has an income of Rs. 7500. (10) 6) a) There are only two candidates A and B who are competing for a post of Company Executive (CE) in a pharmaceutical company. The respective probabilities for A and B to be selected as CE are 0.6 and 0.4. The company wants to introduce a new drug after recruiting the CE. From the interview of both the candidates it has been observed that the chances of introducing the drug by A is 45% and by B is 55%. Calculate the probability that the drug will be introduced. (5) b) The probability of having a male child is 0.4 in a family of 4 children. Find the probability that i) there are 2 sons ii) there are 4 daughters If 100 families, each with 4 children are interviewed, find how many families would you expect to have c) i) 2 sons ii) 4 daughters (3) For a Poisson distributed random variable X, P[X = 4] = P[X = 5]. What is the mean and variance of the distribution. (2) 4 7) a) A box contains m white and n black balls the total number of balls being m + n. Suppose r balls are drawn one by one (without being returned to the box), what is the dispersion of the number of white balls drawn? (10) b) An investor wishes to invest in equity shares of three companies A, B and C. He calculated that his expected earnings, variance and co-variances are as follows: Company A B C Expected earnings(in%) 10 15 08 Variances 40 85 12 Co-variances Cov(A, B) = 30 Cov(B, C) = −25 Cov(A, C) = 2 He invests 0.5 percent of his capital in the shares of company C, 0.3 percent of his capital in A and 0.2 percent of his capital in B. Find his total expected earnings and variance of the earnings. (3) 8) c) A chemical plant holds for all its employees to decide whether or not to accept a new pay deal. 80% of the employees voted and it is known that 60% of the employees are union members. The union ascertains that 90% of the employees are either union members or voted (or both). What is the probability that an employee selected at random is a union member who voted? (2) a) From records of 10 Russian army corps kept over 20 years, the following data was obtained showing the number of deaths caused by the kicks of a horse. Determine the average number of deaths per army corps per annum, and calculate the theoretical Poisson frequencies and compare with the observed frequencies. Number of deaths per 0 1 2 3 4 Army corps per annum (8) Frequency of occurrence 109 65 22 3 1 b) The mean and the standard deviation of family income (in Rs.) in a year of 2 regions A and B are as follows: Region A B Mean 320 370 Standard deviation 54 60 Which of these 2 regions shows a greater disparity in family income? Justify your answer. (2) 9) a) Reena enters into three independent ventures. The probability of success in these three ventures are 0.5, 0.25 and 0.1 respectively. What is the probability that she will succeed in exactly three ventures. (5) b) 5 books are selected at random from a shelf having 4 books of History, 7 books of Economics and 10 books of English, what is the probability that i) no English book is selected? ii) one book of History, two books of Economics and two books of English will be selected. (5) 10) Which of the following statements are true? Give reasons for your answers. i) ii) 2 5 The abscissa of the point of intersection of the two Ogives of less-than type and more-than type gives the mean of the data. If P(A) = 0.5, P(A ∪ B) = 0.7 and A and B are independent events, then P(B) = 5 iii) The least-square line always passes through the point (x̄, ȳ). iv) For two events A and B, it is known that P(A) = 0 and P(B) > 0, then P(A|B) > 0 v) If x1 , x2 , . . . , xn are n observations of a variable x, then n n X i=1 6 xi ≥ n X i=1 xi !2 (10) ASSIGNMENT-2 (To be done after studying Blocks 3, and 4.) Course Code: MTE -11 Assignment Code: MTE-11/TMA-2/2003 Maximum Marks: 100 1) a) Suppose a random variable X has the following density function f(x) = ( ae−ax for x ≥ 0 and a > 0 0 otherwise Find the moment generating function. Also obtain the mean and variance of the distribution. b) (4) Given that the joint density of two random variables X and Y is f(x, y) = 2 , (1 + x + y)3 x > 0, y > 0 Find c) i) fX (x) ii) fY|X (y|x) (3) Show that for any two random variables X and Y with Var.(X) < ∞ Var.(X) = E[Var.(X|Y)] + Var[E(X|Y)] 2) a) (3) A variable X has the density f(x) = 24x−4 , x≥2 Compute the upper bound for P[|X − µ| > δ] using chebychev’s inequality where δ is a constant. b) 3) a) It is known that 40% of the population die within a year. 30 persons are vaccinated of which only 5 die in a year. The inventor of the vaccine claims that the vaccine has reduced the mortality by 20% i) Discuss whether the vaccine is effective. ii) n Find the smallest value of n i.e. sample size such that if die after vaccination, his claim 6 is justified. (5) The dynamo bulbs manufactured for cycles have an average life of 400 hours with a standard deviation of 250 hours. Out of 500 bulbs thus manufactured i) how many bulbs would have burnt more than 600 hours? ii) less than 300 hours? (Assume normal distribution) b) (5) (5) Let X1 , X2 , X3 , . . . Xn denote random sample of size n from uniform distribution with probability density function 1 1 ≤ x ≤ θ + and − ∞ < θ < +∞ f(x, θ) = 2 2 0, elsewhere 1, θ − 7 Obtain Maximum Likelihood Estimator for θ. 4) a) (5) Let p be the probability that a coin will fall head in single toss in order to test H0 : P = 1 2 against H1 : p = 3 4 The coin is tossed 5 times and H0 is rejected if more than 3 heads are obtained. Find the probability of Type I error and the power of the test. (5) b) Consider the function f(x) given by f(x) = x<0 0≤x<1 0, x, 1 , 1≤x<2 2 x≥2 0, i) Sketch f(x) and show that it satisfies the condition to be a probability density function of a continuous random variable X. ii) Find the cumulative distribution function F(x) iii) Calculate the mean and the variance of X. 5) a) (5) Survey has been conducted to study the effectiveness of a specific advertising media (Television) for a product. Data have been collected before and after the advertisement on number units sold in a given locality from a sample of 10 customers as given below. Before After : : 10 12 12 11 5 7 4 6 15 14 9 7 7 12 10 15 13 14 10 11 Test the null hypothesis that the advertisement is effective i.e. H 0 : µ1 − µ2 = 0 against the (6) alternative, H1 : µ1 − µ2 > 0 at level α = 0.05. b) 6) Fifty measurements of the acceleration due to gravity g, had a mean value of 9.8 ms −2 and s.d. of 0.75 ms−2 (assuming that the gravity g follows normal distribution) i) What is the 95% confidence interval of g? ii) How many measurements would be necessary to reduce the 95% confidence interval to 9.7 < µ < 9.9? [Given that Z0.025 = 1.96, Z0.05 = 1.647] (4) a) A sample of 20 students taken from a larger group of students showed the variability in understanding a specific concept in mathematics as 12.73. Predict 90% confidence interval for the variance of group consisting of all the students. (4) b) Suppose (X, Y) has the joint probability density function f(x, y) = Find E(X|Y). x2 exp(−x(y + 1)), 0 x ≥ 0, y ≥ 0 otherwise (6) 8 7) a) The joint probabiliy density function of X and Y is given by f(x, y) = ( k(1 − x − y), k a constant, when x + y ≤ 1, 1 ≥ y ≥ 0 and 1 ≥ x ≥ 0 0 elsewhere. Find i) the marginal probability density function of X ii) the value of k. iii) the probability density function of Y when X = b) 1 2 A bus is due at a bus-stop at 11.10 a.m. Its actual time of arrival, Y is normally distributed with a mean of 11.14 a.m. and standard deviation of 4 minutes. A regular passenger arrives at the bus-stop at a time X, which is normally distributed with a mean of 11.09 a.m. and standard deviation of 2 minutes. i) What is the probability that the passenger arrives after 11.10 a.m.? ii) What is the probability that the bus arrives before 11.10 a.m.? iii) What is the probability that the passenger misses the bus? 8) (6) a) If the population correlation √ is zero (i.e. ρ = 0) the show that for the sample correla√ coefficient tion coefficient r, t = r( n − 2)/( 1 − r2 ) is distributed as student t distribution with (n − 2) degrees of freedom. (8) b) Let (X, Y) have the following p.d.f f(x, y) = ( 6xy2 , 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 0, elsewhere Find P(2Y + X ≤ 1) 9) (4) a) (2) Let X1 , X2 , X3 and X4 be independent random variables such that E(Xi ) = µ and Var.(Xi ) = σ 2 for i = 1, 2, 3, 4. Three different statistics are defined as Y= X1 + X 2 + X 3 + X 4 4 Z= X1 + X 2 + X 3 + X 4 5 T= X1 + 2X2 + X3 − X4 4 i) Examine whether Y, Z and T are unbiased estimators of µ. ii) What is the efficiency of Y relative to Z? iii) Find the correlation coefficient between Y and Z using the efficiency parameter. 9 (5) b) From a population of 540, a sample of 60 individuals is taken. From this sample, the mean is found to 6.2 and standard deviation is 1.368. i) Find the estimated standard error of mean. ii) Construct 96% confidence interval for the population mean. (5) 10) Which of the following statements are true and which are false? Give reasons for your answers. i) If X is a beta random variable with the parameters α and β, then 1 − X is a beta random variable with parameters β and α. ii) If T = θu where θ is an unknown parameter and u is a random variable having a chi-square T distribution with 2 degrees of freedom, then is an unbiased estimator of θ. 2 iii) If X has the distribution N(10, 9) and if Y = 5X + 3, then Y has distribution N(53, 81). iv) In a problem of testing of simple hypothesis against a simple alternative, if the probability of type 1 error is known to be 0.06, then the power of the test will be 0.94. v) If γxy = 0.62, u = 5 + 6x and v = 7 − 3y, then γuv = 0.62 10 (10)