M.Sc. (Final) DEGREE EXAMINATION, MAY Final Year STATISTICS. Time : 03 Hours Maximum Marks : 100

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1 (DMSTT21) M.Sc. (Final) DEGREE EXAMINATION, MAY Final Year STATISTICS Paper - I : Statistical Quality Control Time : 03 Hours Maximum Marks : 100 Answer any Five questions All questions carry equal marks 1) a) Explain how a control chart helps to control the quality of a manufactured product. Explain standard deviation and C.V. charts and give examples one each. b) Account for the following procedures in control chart analysis, stating clearly the assumptions involved. i) A sample of 5 is ordinarily used in control charts ii) The range is used instead of standard deviation and iii) The control limits can be used to improve the specification limits. 2) a) Explain mid range control chart. Discuss the working of the chart in controlling measurable characteristic during a production process. What are the causes of presence of trends and cycles in the chart? b) Define average run length. Discuss its merits and demerits. Give any four examples of situations where individual measurements arise for process monitoring. 3) a) Explain natural tolerance limits and specification limits and their use in deciding whether a process needs adjustment or not. b) Discuss the assumptions and approximations involved in the calculation of control limits for C- Chart. Discuss the adaptations of the C-Chart (i) to variations in the area of opportunity for a defect (ii) to quality rating. 4) a) A p-chart indicates that the average is If 50 items are inspected each day, what is the probability of detecting a shift of 0.04 (i) on the first day after the shift and (ii) by the end of the third day after the shift. State the important steps involved while establishing the control limits for future production giving the practical way of overcoming the difficulties that arise due to varying inspection number. b) Discuss the theoretical basis of p and np charts. Under what situations in industry would you prefer the one to the other? 5) a) Explain the procedure of constructing CUSUM Chart with two-sided decision rule. b) Explain Chi-square and Hotelling T2 control charts with suitable examples.

2 6) a) Distinguish between EWMA and Moving average chart. Discuss their applications two each. b) Discuss ANOM for X and P Charts with examples. 7) a) Explain the terms LTPD, consumer s risk and producer s risk, process average fraction defectives, average amount of inspection, A.O.Q and AOQL. b) Explain the single sampling plan. Given lot size (N), L.T.P.D (Pt), Consumer s risk ( P c) and process average ( p ) derive the most economical single sampling plan for acceptance purposes. 8) a) Explain the notion of sequential sampling by attributes. Briefly discuss A.O.Q.L and L.T.P.D plans of Dodge and Roaming. b) Describe the method of double sampling plan and derive its O.C, A.O.Q, A.S.N and A.T.I. 9) a) Explain the advantages and disadvantages of variable sampling b) Explain the notion of sequential sampling by variables. A lot of 500 items is submitted for inspection. Suppose that we wish to find a plan from MIL STD 414, using inspection level II. If the AQL is 4% find the procedure / sampling plan from the standard. 10) a) Discuss the role of multilevel sampling plan with an example. b) Explain (i) Skiplot sampling plan (ii) TQM.

3 (DMSTT22) M.Sc. (Final) DEGREE EXAMINATION, MAY Final Year STATISTICS Paper - II : Operations Research Time : 03 Hours Maximum Marks : 100 Answer any Five questions Each question are of equal marks 1) a) Explain duality. State and prove its properties. Explain its advantages b) Solve the following linear programming problem Minimize : Z = 3x 1+ 2x2+ 3x3 Subject to : x + x + x x + 4x2+ 2x3 1 x 1, x2, x3 0 2) a) Define the general linear programming problem. Explain the two-phase method b) Use dual simplex method to solve : Minimize : Z = 3x1 2x2 Subject to : x +x x 1+x2 x x 2 10 x 2 3 and x 1, x ) a) What are the advantages and disadvantages of increased inventory? Explain the objectives that must be fulfilled by an inventory control system. Explain the Harris lot size formula. b) Explain the problems of replacement of items that fail completely. 4) a) Explain the EOQ models without shortages. b) Explain the problems of replacement of items that deteriorate with time.

4 5) a) Explain : i) Two person-zero-sum game ii) Saddle point iii) Maxmin and Minimax principle and iv) Dominance rule b) For any 2 2 game without a saddle point find the optimal mixed strategies and the value of the game. 6) a) Explain : i) Algebraic method and ii) Graphical method of solving a game. b) Solve the following game by using simplex method ) a) Explain M/M/C system with infinite capacity. Obtain its steady state solution. Find the average queue length and the average waiting time of an arrival. b) Explain M/G/1 system. Obtain Pollaczek-Khinchin formula. Show that M/M/1 is a particular case of M/G/1 system. 8) a) Explain M/Ek/1 system. Obtain its steady-state solution. Find the average waiting time of an arrival. b) Explain M/M/1 system. Obtain its steady state solution. Find the waiting time distribution for the system. 9) a) Distinguish between PERT and CPM. Describe the steps in PERT/CPM techniques. Critically comment on the assumptions on which PERT/CPM analysis is done. b) A project has the following details: Task Predecessor Duration in hours A - 14 B A 22 C B 10 D B 16 E B 12 F C 10 G C 6 H F,G 8 I D, E, F 24 J I 16

5 i. Draw the network diagram ii. Find the critical path iii. Calculate free and total floats for each activity. 10) a) Discuss the role of statistical technique in PERT. What are the basic assumptions underlying the expected time estimate? b) A small project consists of seven activities, the details of which are given below: Activity : A B C D E F G Predecessor : - A A B,C B D,E D Most likely duration (in days) : Optimistic duration (in days) : Pessimistic duration (in days) : i. Draw the networks. Find the critical path, the expected project completion time and the next most critical path. ii. What project duration will have 95 percent confidence of completion?

6 (DMSTT23) M.Sc. (Final) DEGREE EXAMINATION, MAY Final Year Statistics Paper - III : ECONOMETRICS Time : 03 Hours Maximum Marks : 100 Answer any Five questions All Questions carry equal marks 1) a) Explain simple linear model, log-linear and reciprocal models. Give their applications one each. Explain the decomposition of sum of squares in the simple linear model b) Develop a test statistic for testing the significance of the regression coefficient. Construct 10ν (1 α )% confidence interval for the regression coefficient 2) a) Obtain the least squares estimators of the parameters in a simple linear model. State and prove the properties of the estimators. b) Define prediction error. Obtain its variance. Construct a 95% confidence interval for the predicted value. 3) a) Explain in detail the general linear model. State and prove the Gauss-Markov theorem. b) Derive the variance of the disturbance term in the general linear model. 4) a) Obtain the distribution of the least squares estimator and develop a test statistic to test for the significance of any element of the estimator. b) Develop a test statistic for testing the significance of the joint significance of the complete set of explanatory variables. 5) a) In the general linear model discuss the estimation subjected to linear restrictions and develop a test statistic to test H o : Rβ = r. b) What are dummy variables? Explain their role in describing temporal effects. 6) a) Explain MWD test. b) Explain the tests of structural change with K-Variables. Explain Chow test.

7 7) a) Explain multicollinearity and its types. What are its consequences? How do you detect it? b) Describe the sources of non-spherical disturbances. Derive the Aitken estimator. 8) a) Discuss the problem of estimation under heteroscedasticity. Explain Gold feld-quandt test. b) How do you detect multicollinearity. Discuss the remedial approaches to multicollinearity. 9) a) What is autocorrelation? What are its consequences? Explain Durbin-Watson test. b) Explain LPM and LOGIT models. How do you estimate them. 10) a) Explain probit model. Explain its applications. How do you estimate the model. b) What are the reasons for the autocorrelated disturbances? Discuss estimation with autocorrelated disturbances.

8 (DMSTT24) M.Sc. (Final) DEGREE EXAMINATION, MAY Second Year STATISTICS Paper - IV : Multivariate Analysis Time : 03 Hours Maximum Marks : 100 Answer any Five questions All questions carry equal marks 1) a) Derive expressions for the mean and variances of the (p 1) random vector X. Define stochastic independence of random vectors. b) Define multivariate normal distribution. Prove that all subsets of a multivariate normal random vector are normally distributed. 2) a) Let X be a multivariate normal random vector. Prove that the conditional distributions of the components of X are normal. b) Obtain the maximum likelihood estimators of the mean vector and the covariance matrix of a p- variate normal distribution. 3) a) Define Hotelling s T 2 and derive its distribution b) Obtain the likelihood ratio statistic for testing the significance of the mean vector of multivariate normal. Discuss the test procedure. 4) a) Explain MANOVA for one-way classification. b) Prove that the test based on Hotelling s T 2 is equivalent to the likelihood ratio test of H o : µ= µ o against H 1 : µ µ o where µ is the mean vector of a p-variate normal. 5) a) Define the principal components. Derive the expressions for the first and second principal components. Explain the concept of dimension reduction. b) Explain orthogonal factor model. Explain the principal component method of parameter estimation. 6) a) Explain factor loadings and communalities. What are factor scores? How do you obtain them? b) Explain principal components. Prove that they are uncorrelated. Derive the i th principal component of the standardized variables.

9 7) a) Establish the relationship between Hotelling s T 2 and Wilk s lambda. Derive the null distribution of Hotelling s T 2 statistic. b) Show that Hotelling s T 2 statistic can be used to test the equality of means of corresponding variables in two multi-variate normal populations having the same dispersion matrix. 8) a) Explain the problem of classification with several multivariate normal populations. b) Discuss the problem of classification between two multivariate normal distributions. 9) a) Explain the importance of cluster analysis. Explain similarity coefficients for clustering items. b) Distinguish between hierarchical and non-hierarchical clustering methods. Explain K-means method. 10) a) Explain hierarchical clustering methods. b) What are non-hierarchical clustering methods? Discuss K-means method.