Statistics

(Statistics) FIME ALLOWED: 3 HOURS MAXIMUM MARKS: 100 Note. Attempt FIVE questions in all, including QUESTION NUMBER 8 which is COMPULSORY. All questions carry equal marks. Statistical tables will be provided. 1. (a) Describe the classical, relative frequency and subjective concept of 08 probability. (b) Three applicants are to be selected at random out of 4 boys and 6 12 girls. What is the probability of selecting (i) all girls (ii) all boys and (iii) at least one boy. 2. (a) If the universal set U is all numbers from 1 to 25, specify the elements 10 in the following sets: (i) A is all prime numbers in u. (ii) B is all numbers evenly divisible by 4. (iii) C is all numbers containing the digit 2. (b) The probability that an air line passenger is served by attendant A or 10 B is 0.75 or 0.25 respectively. From past experience it is known that attendant A get the correct beverage order 98 in l00 times while attendant B is correct 80 in 100 times. A passenger who has just received an incorrect beverage order and does not remember which attendant took the order, has blamed attendant A for the mistake because this person took most of the orders. Which attendant has the larger probability of making the mistake? 3. (a) Define a discrete random variable and its probability distribution. What are the two basic properties of all probability distributions.(8) (b) A large store places its last 15 clock radios in a clearance sale. Unknown to anyone, 5 of the radios are defective. If a customer tests 3 different clock radios selected at random, what is the probability distribution of X = number of defective radios in the sample.(12) 4. (a) Describe the binomial distribution and derive its mean and variance. 08 (b) Suppose that the number of insurance claims closely approximates a 12 Poisson distribution with mean equal to 0.05. Find the probability of (i) no claim (ii) one or fewer claims. 5. (a) The mean scores of 1000 students appearing for an examination is 34.4 and the standard deviation is 16.6. How many candidates may be expected to obtain marks between 30 and 60 assuming the normality of the distribution?(10) (b) A certain drug is claimed to be effective in curing cold. In an experiment on 164 people with colds, half of them were given the drug and half of them were given sugar pills. The patients’ reaction to the treatment are recorded in the following table: Category Helped Harmed No effect Drug 52 10 20 Sugar 44 12 26 Test the hypothesis at 5% level of significance that the drug is no better than sugar pills for curing colds.

6. (a) Let X1, X2, .... , Xn be a random sample from a normal population with parameters µ and 2. Find the MLE for (i) µ when 2is known (ii) 2 when µ is known. (b) To verify whether a course in statistics improve performance, a similar test was given to 12 participants both before and after the course. The original grades of the participants recorded before and after the test were as following: Participant 1 2 3 4 5 6 7 8 9 10 11 12 Before 44 40 61 52 32 44 70 41 67 72 53 72 After 53 38 69 57 46 39 73 48 73 74 60 78 Was the course useful as measured by performance on the test, assuming the 12 participants as a sample from a population. (10) 7. (a) Find least square estimates of parameters in the model Y¡ = a + ßX¡ + e¡, where e¡s, are distributed independently with zero mean and constant variance.(10) (b) The owner of a retailing organization is interested in the relation between price at which a commodity is offered for sale and the quantity sold. The following sample data have been collected: (10) Price 25 45 30 50 35 40 65 75 70 60 Quantity Sold 118 105 112 100 111 108 95 88 91 96 (i) Determine the least square equation for the estimated regression line. (ii) Verify that e¡=0 COMPULSORY QUESTION 8. Select the correct answer by writing (a) or (b) or (c) or (d) in the answer book. 20 Don’t reproduce questions. (1) Which of the following cannot be a value of the probability of an event: (a) 0.005 (b) Zero (c) 1.5 (d) 1 (2) Based on past marketing data, there is 0.10 chance of selling more than 1000 new cars. The statement is an example of: (a) classical (b) frequency (c) subjective probability (d) none of these. (3) The probability of a sample space is always: (a) 1 (b) less than 1 (c) greater than 1 (d) zero. (4) If A and B are independent events than P(AIB) is equal to: (a) P(A) (b) P(B) (C) P(A).P(B) (D) P(A)/P(B)

(5) The probability density function (Pdt) is a function of: (a) parameters (b) probability (c) random variable (d) none of these. (6) For a Poisson distribution with S. D. equal to 2, mean of the distribution is: (a) 2 (b) 4 (c) 1 (d) zero. (7) For a continuous r,v,x ( -oC_< x< oC ), the value of p(x=0) is: (a) zero (b) 1 (c) 0.5 (d) mean of x. (8) A r.v.x with pdf (1/2)xQ,4)x, X = 1,2,3,... is: (a) discrete (b) continuous (c) neither a nor b. (9) The value of 6 P3 is equal to: (a) 18 (b) 20 (c) 120 (d) 108. (10) A binomial distribution with n=100 and p = 0.5 is: (a) symmetrical (b) asymmetrical (c) skewed to right (d) skewed to left. (11) The value of the mean of normal distribution lies between: (a) O and 1 (b) O andcC (c) -oC and 0 (d) -oC and oC. (12) The degrees of freedom in a continuous distribution is considered as: (a) statistics (b) parameter (c) random variable (d) none of these. (13) For a X2(v) ,mean of the distribution is equal to: (a) v/2 (b) 2v (c /2v (d) v (14) The mean of a t-distribution with 10 d.f will be: (a) 5 (b) 10 (c) 9 (d) zero (15) The mean of F(5, 10) is equal to: (a) 7.5 (b)10/8, (c) 5/ 8(d) 8/5 (16) The analysis of variance is used to test the equality of: (a) means (b) variances (c) standard deviations (d) none of these.

(17) The probability of accepting a false Ho is equal to: (a) a (b) 1-a (c) ß (d) 1-ß (18) The equality of two population variances is tested by: (a) Z (b) t (c) x² (d) F (19) If the correlation coefficient between X and Y is r and r> 0, then the slope ß between X and Y is: (a) Zero (b) greater than zero (c) 1 (d) negative (20) To test Hp: þ= po, use: (a) Z-test (b) t-test (c) x² -test (d) F-test