Descriptive Statistics and Probability Theory — DU B.A. (Prog.) with Statistics as Major Discipline Previous Year Question Papers

1. (a) Answer the following : (i) Geometric mean of the observations 2, 0 and 8 with frequencies 1, 2 and 3 respectively is - (ii) Median = - Decile. (iii) Two uncorrelated variables are always independent (True/False). (iv) The number of elements in the sample space when a die is rolled thrice is (v) The sum of the probabilities of all mutually exclusive and exhaustive events in a sample space is always (b) Average marks in statistics of 10 students of a class was 68. A new student took admission with 72 marks whereas two existing students left the college. the remaining students. If the marks of these students were 40 and 39, find the average marks of (c) Prove that algebraic sum of deviations of a set of values from their arithmetic mean is zero. (d) Differentiate between pairwise independence and mutual independence of events. (e) Can Y = 5 + 1.4X and X = 3 + 0.57 be the estimated regression equations of Y on X and X on Y respectively? Explain your answer with suitable arguments. (1) For a frequency distribution xi/fi, i = 1,2,..., n prove that B, > 1. (2*5) 2. (a) Show that in finding the arithmetic mean of a set of readings on a thermometer, it does not matter whether we measure temperature in Centigrade or Fahrenheit, but that in finding the geometric mean, it does matter which scale we use. (b) The marks (out of 100) obtained by 50 students are summarized below : (i) The mean and variance of the distribution using the method of change of origin and scale. (ii) The median and mode of the distribution. (a) Explain any three measures of dispersion along with their merits and demerits. (b) Define moments. For a distribution, the mean is 10, variance is 16, y, = +1 and B, = 4. Obtain the first four moments about the origin. Comment upon the nature of the distribution. (c) Explain any two approaches to defining probability. Also discuss their merits and demerits. 4. (a) For any three events A, B and C, show that P(A U B U C) ≤ P(A) + P(B) + P(C). If three machines A, B and C in a factory can fail on a given day with corresponding probabilities of failure being P(A) = 0.2, P(B) = 0.3 and machine fails on that day. P(C) = 0.1, then find the upper bound for the probability that at least one (b) A factory has three machines, A, B, and C, producing 25%, 35%, and 40% of the total products, respectively. The percentages of defective items produced by them are 5%, 4%, and 2%. If a product is found to be defective, determine the probability that it was produced by machine A. (c) From a city population, the probability of selecting (i) a male or a smoker is 7/10, (ii) a male smoker is 2/5 and, (iii) a male, if a smoker is already and (c) a smoker, if a male is first selected. selected is 2 /3. Find the probability of selecting (a) non-smoker (b) a male 5. (a) Define Spearman's rank correlation coefficient. Prove that Spearman's rank correlation coefficient (p) for non-repeated ranks is given by :