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Sampling Distribution 1 UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION BSc. MATHEMATICS COMPLEMENTARY COURSE CUCBCSS 2014 Admission onwards III Semester STATISTICAL INFERENCE Question Bank 1. 2. 3. 4. The number of possible samples of size n out of N population units with replacement is a. N2 b. n2 c. d. N! Simple random sample can be drawn with the help of a. random number tables b. Chit method c. roulette wheel d. all the above Which of following statement is true a. more the SE, better it is b. less the SE, better it is c. SE in always zero d. SE is always unity If the samples values are 1, 3, 5, 6, 9, the SE of the sample mean is a. SE = 5. 2 b. SE = 1/ 2 c. SE = 2.0 d. SE = 1/2 Student’s ‘t’ distribution was discovered by a. G.W. Snedecor b. R.A. Fisher c. W.Z. Gosset c. Karl Pearson 2 6. Statistics If X N (0, 1) and Y x/ 7. 8. 9. 10. 11. 12. 13. 14. Sampling Distribution 2 (n), the distribution of the variate y / n follows a. Cauchy’s distribution b. Fisher’s t distribution c. Student’s t distribution d. none of the above The degrees of freedom for student’s ‘t’ based on a random sample of size n is a. n 1 b. n c. n 2 d. (n 1)/2 The relation between the mean and variance of 2 with n df is a. mean = 2 variance b. 2 mean = variance c. mean = variance d. none of the above Chi square distribution curve is a. negatively skewed b. symmetrical c. positively skewed d. None of the above Chi square distribution is used to test a. goodness of fit b. hypothetical value of population variance c. both (a) and (b) d. neither (a) nor (b) Fisher’s Z is closely related to a. Helmert X2 b. Snedecor’s F c. Fisher’s t d. all the above F distribution was invented by a. R.A. Fisher b. G.W. Snedecor c. W.Z. Gosset d. J. Neymann The range of F - variate is a. to + b. 0 to 1 c. 0 to d. to 0 If X 2 (n1), Y 2 (n2) and if X and Y are independent X + Y follows a. Chi square distribution with n df 15. 16. 17. 18. 19. 20. 21. 3 b. Student’s t distribution with n1 + n2 df c. Chi square distribution with n1 + n2 df d. None of the above Student’s t curve is symmetric about a. t = 0 b. t = c. t = 1 d. t = n An estimator is a function of a. population observations b. sample observations c. Mean and variance of population d. None of the above Estimate and estimator are a. synonyms b. different c. related to population d. none of the above The type of estimates are a. point estimate b. interval estimate c. estimates of confidence region d. all the above The estimator x of population mean is a. an unbiased estimator b. a consistant estimator c. both (a) and (b) d. neither (a) nor (b) Factorasation theorem for sufficiency is known as a. Rao - Blackwell theorem b. Cramer Rao theorem c. Chapman Robins theorem d. Fisher - Neymman theorem If the sample mean x is an estimate of population mean , then x is a. unbiased and efficient b. unbiased and inefficient c. biased and efficient 4 22. 23. Statistics d. biased and inefficient Sample standard deviation as an estimate of population standard deviation is a. unbiased and efficient b. unbiased and inefficient c. biased and efficient d. biased and inefficient b. t2 is also consistent estimator for 25. 26. 27. 28. 29. a. c. 30. If t is a consistent estimator for , then a. t is also a consistent estimator for 2 24. Sampling Distribution c. t2 is also consistent estimator for 2 d. none of the above Least square theory was propounded by whom and in which year? a. Gauss in 1809 b. Markov in 1900 c. Fisher in 1920 d. None of these The credit of inventing the method of moments for estimating parameters goes to a. R.A. Fisher b. J. Neymann c. Laplace d. Karl Pearson The concepts of consistency, efficiency and sufficiency are due to a. J. Neymann b. R.A. Fisher c. C.R. Rao d. J. Berkson For an estimator to be consistent, the unbiasedness of the estimator is a. necessary b. sufficient c. necessary as well as sufficient d. neither necessary nor sufficient The maximum likelyhood estimates are necessarily a. unbiased b. sufficient c. most efficient d. unique. The notion of confidence interval was introduced and developed by 31. 32. R.A. Fisher Karl Pearson 5 b. J. Neymann d. Gauss The 100(1 )% confidence interval for of N (, ) when unknown, using a sample of size less than 30 is a. x t / c. x t s 2 n 1 s n 1 b. x t / d . x t 2 s n s n A random sample of 16 housewives has an average body weight of 52kg and a standard deviation of 3.6kg. 99% confidence limits for body weight in general are a. (54.66, 49.345) b . (52.66, 51.34) c. 55.28, 48.72) d . none of the above A confidence interval of confidence coefficient 1 is best which has a. smallest width b . Vastest width c. upper and lower limits equidistant from the parameter d . one sided confidence interval 33. 34. 35. 36. Formula for the confidence interval for the ratio of variances of the two normal population involves a. 2 distribution b . F distribution c. t distribution d . none of the above An ‘hypothesis’ means a. assumption b. a testable preposition c. theory d. supposition A wrong decision about H 0 leads to a. One kind of error b. Two kinds of error c. Three kinds of error d. Four kinds of error The idea of testing of hypothesis was first set forth by a. R.A. Fisher b. J. Neymann 6 37. 38. 39. 40. 41. 42. 43. 44. Statistics c. E.L. Lehman d. A. Wald Rejecting H0 when H0 is true is a. Type I error b. Standard error c. Sampling error d. Type II error Power of a test is related to a. Type I error b. Type II error c. both (a) and (b) d. neither (a) nor (b) Level of significance is also called a. size of the test b. size of the critical region c. producer’s risk d. all the above Area of critical region depends on a. size of type I error b. size of type II error c. value of the statistic d. sample size n A test which maximises the power of the test for a fixed is known as a. optimum test b. randomised test c. Bayes test d. likelihood ratio test Neymann Pearson lemma provides a. an unbiased test b. a most powerful test c. an admissible test d. minimax test Every test statistic is a. an estimate b. a random variable c. a fixed value d. None of these Large sample tests are conventionally meant for a sample size b. Z > 2.58 c. Z 1.96 d. Z 2.58 To test H0 : 500 against H0 : 500, we use a. one sided left tailed test b. one sided right tailed test c. two-tailed test 48. d. all the above Testing H0 : 200 against H0 : 500 leads to a. left tailed test b. right tailed test c. two-tailed test d. none of these 49. To test an hypothesis about proportions of success in a class, the usual test is a. t-test 50. b. F-test c. Z-test d. None of these Standard error of the difference of proportions p1 p 2 in two classes under the hypothesis H0 : p1 p 2 with usual notations is a. 1 1 p *q* n1 n 2 c. p * q * 1 1 p* n1 n 2 d. p1q1 p 2 q2 n1 n2 a. central limit theorem a. 12.0 c. n 30 51. 1 1 n1 n 2 b. A parametric test is performed as a large sample test using b. n < 30 d. n = 100 b. Techebysheff inequality c. Weak law of large numbers 46. 47. a. Z < 2.58 7 It is expected that 50% people of a city are cinema goers. A survy of 1600 people revealed that 35% people go to cinema. The value of Z statistic is a. n = 20 45. Sampling Distribution d. none of these 52. For a two failed test with = 0.05, the best critical region of a Z test is c. 12.58 d. 12.0 A normal population has a mean of 0.5 and SD = 6.0. The probability that the sample mean of 625 items of a sample will be negative is a. 0.0188 53. b. 6.0 b. 0.365 c. 0.4812 d. 0.135 The claimed average life of electric bulbs is 2000 hours with a SD = 250 hours. To make 95% sure that the bulbs should not fall below the claimed average life by more than 5%, the sample size should be 8 54 55. Statistics a. 24 b. 16 a. 2.73 b. 0.97 c. 41 d. none of these c. 3.30 d. 0.41 Student’s ‘t’ test was invented by 57. 58. 59. 60. 61. G.W. Snedcor b. R.A. Fisher a. 1 to +1 b. to c. W.Z. Gosset d. W.G. Cochran c. 0 to d. 0 to 1 Student’s t-etst is applicable only when the variate values are independent the variable is distributed normally the sample is not large all the above 62. paired b. correlated c. equal in number d. all the above 2 -test b. t-test c. F-test d. Z-test Range of the variance ratio F is a. 1 to +1 b. to c. 0 to d. 0 to 1 Numerator is less than the denominator b. Numerator is greater than the denominator c. Numerator is equal to the denominator d. None of these Given the following 8 sample values 4, 3, 3, 0, 3, 3, 4, 4 the value of student’s t test H0: = 0 is F-test b. Z-test c. 2 -test d. t-test 64. Statistic 2 to test H 0 : 2 02 based on a sample of size n has degrees of freedom equal to a. n1 b. n c. n+1 d. none of these 65. Degrees of freedom for 2 in case of contingency table of order 43 are a. 12 b. 9 c. 8 d. 6 66. When d.f. for 2 are 100 or more, 2 is approximated to While performing the F-est, observe that in the F ratio whether a. a. Degrees of freedom for statistic 2 in case of 22 contigency table is a. 3 b. 4 c. 2 d. 1 Equality of two population variances can be tested by a. The hypothesis that popultion variance has a specified value can be tested by 63. Paired ‘t’ test is applicable when the observations in the two samples are a. 9 Range of t statistic is a. a. b. c. d. 56. Sampling Distribution a. c. 67. t-distribution Z-distribution b. F-distribution d. none of these If all frequencies of classes are the same, the value of 2 is a. 1 b. c. zero d. none of these 10 68. Statistics Sampling Distribution The range of statistic a. 1 to +1 2 is b. to 0 to d. 0 to 1 Contingency table having a zero count is called a. a complete contingency table b. an incomplete contingency table c. abnormal contingency table d. none of these If the calculated value of 2 is greater than its degrees of freedom, then c. 69. 70. a. null hypothesis be accepted directly b. null hypothesis be rejected straight-away c. 2 table be consulted to arrive at a decision about the null hypothesis d. all the above ANSWERS (1) (6) (11) (16) (21) (26) (31) (36) (41) (46) (51) (56) (61) (66) c c c b a b a b b c d d b c Prepared by: (2) d (7) a (12) c (17) a (22) d (27) b (32) a (37) a (42) b (47) a (52) a (57) c (62) c (67) c (3) (8) (13) (18) (23) (28) (33) (38) (43) (48) (53) (58) (63) (68) b b d d c b a b b c b c d c (4) (9) (14) (19) (24) (29) (34) (39) (44) (49) (54) (59) (64) (69) a c c c a b b d c c c b a b (5) (10) (15) (20) (25) (30) (35) (40) (45) (50) (55) (60) (65) (70) c c a d d a b a a a d d d c Dr. K.X.Joseph, Director, Academic Staff College,University of Calicut. 11