M.Sc. (Final) DEGREE EXAMINATION, MAY Final Year. Statistics. Paper I STATISTICAL QUALITY CONTROL. Answer any FIVE questions.

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1 (DMSTT ) M.Sc. (Final) DEGREE EXAMINATION, MAY 0. Final Year Statistics Paper I STATISTICAL QUALITY CONTROL Time : Three hours Maximum : 00 marks Answer any FIVE questions. All questions carry equal marks.. What are warning limits on a control chart? How can they be used? Discuss the logic underlying the use of three-sigma limits on shewhart control chart. How will the chart respond if narrower limits are chosen? How will it respond if wider limits are chosen?. The viscosity of a polymer is measured hourly. Measurements for the last 0 hours are shown as follows : Test Viscosity Test Viscosity (c) Does viscosity follow a normal distribution? Setup a control chart on viscosity and a moving range chart. Does the process exhibit statistical control? Estimate the process mean and standard deviation. 3. Based on the following data, if an np chart is to be established, what would you recommend as the center line and control limits? Assume that n = 500. Day : Number of nonconforming units : A control chart for the number nonconforming is to be established, based on samples of size 400. To start the control chart, 30 samples were selected and the number non 30 conforming in each sample determined, D = 00. What are the parameters of the np chart. i= Suppose the process average fraction non conforming shifted to 0.5. What is the probability that the shift would be detected on the first subsequent sample? i

2 5. Both concentrations are measured hourly in a decimal process. Data (in ppm) for the last 3 hours are shown here (read down from left) The process target is µ 0= 75 ppm. Estimate the process standard deviation. Construct a tabular cusum for this process using standardized values of h = 5 and k =. 6. The data shown here come from a production process with two observable quality characteristics, x and x. The data are sample means of each quality characteristic, based on samples of size n = 5. Assume that mean values of the quality characteristics and the covariance matrix were computed from 50 preliminary samples x = S = Construct a T control chart using these data. Use the phase limits. Sample Number x x Consider the single-sampling plan for which p = 0. 0, α = 0. 05, p = 0. 0 and β = Suppose that 0 ts of N = 000 are submitted. Draw the ATI curve for this plan. Draw the AOQ curve and find the AOQL. 8. We wish to find a single-sampling plan for a situation where lots are shipped from a vendor. The vendor' process operates at a fallout level of 0.50% detective. We want the AOQL from the inspection activity to be 3%.

3 (c) Find the appropriate DodgeRoming plan. Draw the OC curve and the ATI curve for this plan. How much inspection will be necessary, on the average, if the vendor s process operates close to the average fallout level? What is the LTPD protection for this plan? 9. The density of a plastic part used in a cellular telephone is required to be atleast 0.70 g/cm 3. The parts are supplied in large lots, and a variables sampling plan is to be used to sentence the lots. It is designed to have P = 0. 0, P = 0. 0, α = 0. 0, and β = The variability of the manufacturing process is unknown but will be estimated by the sample standard deviation. Find an appropriate variables sampling plan. Suppose that a sample of the appropriate size was taken and x = 0. 73, Should the lot be accepted or rejected? (c) S = Sketch the OC curve for this sampling plan. Find the probability of accepting lots that are 5% defective. 0. An electronics manufacture buys memory devices in lots of 30,000 from a vendor. The vendor has a long record of good quality performance. With an average fraction defective of approximately 0.0%. The quality engineering department has suggested using a conventional acceptance-sampling plan with n = 3, c = 0. Draw the OC curve of this sampling plan. If lots are of a quality that is near the Vendor s long-term process average, what is the average total inspection at that level of quality? (c) Consider a chain-sampling plan with n = 3, c = 0 and i = 3. Contrast the performance of this plan with the conventional sampling plan. n = 3, c= 0. (d) How would the performance of this chain-sampling plan change if we substituted i = 4 in part (c)?

4 (DMSTT ) M.Sc. (Final) DEGREE EXAMINATION, MAY 0. Final Year Statistics Paper II OPERATIONS RESEARCH Time : Three hours Maximum : 00 marks Answer any FIVE questions. All questions carry equal marks.. Write the procedure of simplex method to solve LPP. Solve the following LP problem using simplex method. Maximize z = 0x + 5x + 0x3 Subject to x 3x + 4x + 9x x, x + 6x + 6x 3 3 and x Write the procedure for two phase method to solve LPP. Solve the following LPP using the result of its dual problem : Minimize z = 4x + 30x Subject to x 4x 6x + 3x + 9x + 6x x and x 0. 0, 5, 0 3. An automobile factory manufactures a particular type of gear with in the factory. This gear is used in the final assembly. The particulars of this gear are : demand rate r = 4,000 units/year production rate K = 35,000 units/year, set-up cost C o= Rs. 500 per set-up and carrying cost C C = Rs. 5/unit/year. Find the economic batch quantity (EBC) and cycle time. A textile mill buys its raw material from a vendor. The annual demand of the raw material is 9000 units. The ordering cost is Rs. 00 per order and the carrying cost is 0% of the purchase price per unit per month, where the purchase price per unit Re.. Find the following : (i) Economic Order Quantity (EOQ) (ii) Total cost w.r.t. EOQ. (iii) Number of orders per year. (iv) Time between consecutive two orders.

5 4. Annual demand for an item is 6000 units ordering cost is Rs. 600 per order inventory carrying cost is 8% of the purchase price/unit/year. The price breakups are as shown below : Quantity Price (in Rs.) per unit 0 Q < Q < Q 3 9 Find the optimal order size. A company orders for three items jointly. The details of the situation are given in table. The fixed ordering cost is Rs. 00. The inventory carrying charge is % of the purchase prices of items in the grap. Find the EOQ in units as well as the interval multiple of each of the items. Data Item i Annual demand in units Price/unit pi (Rs.) Marginal cost of ordering item i (Rs.), , , Players A and B play a game in which each player has 3 coins (0 p, 5 p and 50 p) each of them selects a coin without the knowledge of the other person. If the sum of the values of the coins is an even number, A wins B s coin. If that sum is an odd number. B win A s coin. (i) Develop a payoff matrix with respect to player A. (ii) Find the optimal strategies for the players. Write the dominance property? 6. Write the Algorithm for n game. Consider the payoff matrix of player A as shown in table and solve it optimally using the graphical method. Player B Player A Write the (M/M/) : ( GD / / ) model. The arrival rate of customer at the single window booking counter of a two wheeler agency follows Poisson distribution and the service time follows exponential (negative) distribution and hence, the service rate also follows Poisson distribution. The arrival rate and the service rate are 5 customer per hour, respectively. Find the following : (i) Utilization of the booking clerk. (ii) Average number of waiting customer in the queue.

6 (iii) Average number of waiting customers in the system. (iv) Average waiting time per customer in the queue. (v) Average waiting time per customer in the system. 8. There are three clerks in the loan section of a bank to process the initial queries of customers. The arrival rate of customers follows Poisson distribution and it is 0 per hour. The service rate also follows Poisson distribution and it is 9 customers per hour. Find the following : (i) Average waiting number of customers in the queue as well as in the system. (ii) Average waiting time per customer in the queue as well as in the system. A weighing station has single weighing bridge. The arrival rate of the vehicles coming to the weighing station follows Poisson distribution and it is 45 vehicles per hour. The service rate also follows Poisson distribution and it is 55 vehicles per hour. In front of the weighting bridge, the waiting space is sufficient for a maximum of 0 vehicles. Find the following : (i) Average waiting number of vehicles in the queue in front of the weighting bridge as well as in the weighing station. (ii) Average waiting time per vehicle in front of the weighing bridge as well as in the weighing station. 9. Consider table summarizing the details of a project involving 4 activities. Project details Activity Immediate predecessor(s) Duration (months) A B 6 C 4 D B 3 E A 6 F A 8 G B 3 H C, D 7 I C, D J E 5 K F, G, H 4 L F, G, H 3 M I 3 N J, K 7 (c) Construct the CPM network. Determine the critical path and project completion time. Compute total floats and free floats for non-critical activities.

7 0. The details of a project consisting of activities A to K are summarized in below table : Activity Data Immediate predecessor(s) Duration (weeks) a m b A B 9 C 4 7 D A 3 E A, B 9 F C 5 9 G C 8 H E, F I E, F J D, H 5 4 K I, G 8 (c) (d) Construct the project network. Find the expected duration and the variance of each activity. Find the critical path and the expected project completion time. What is the probability of completing the project on (or) before 5 weeks?

8 (DMSTT 3) M.Sc. (Final) DEGREE EXAMINATION, MAY 0. First Year Statistics Paper III ECONOMETRICS Time : Three hours Answer any FIVE questions. All questions carry equal marks. Maximum : 00 marks. Write a note on Monte Carlo experiments. Write about the properties of least-squares estimators.. Consider the following formulations of the two-variable PRF : Model I : Y X + u i = β + β Model II : Y i = + α( Xi X ) + ui i α. i Find the estimators of β and α. Are they identical? Are their variances identical? Find the estimators of β and α. Are they identical? Are their variances identical? (c) What is the advantage, if any, of model II over model I? 3. Consider the following models. Model A : Yi = α + α X t + α 3X 3t + ut Model B : ( Yi X t ) = β + βx t + β3x 3t + ut Will OLS estimates of α and β be the same? Why? Will OLS estimates of α 3 and β 3 be the same? Why? (c) What is the relationship between α and β? (d) Can you compare the R terms of the two models? Why or why not? 4. From the following data estimate the partial regression coefficients. their standard errors and the adjusted and unadjusted R values. Y = , X = , X 3= Σ Σ Σ Σ Σ Σ ( Yi Y ) = , ( X i X ) = ( X 3 i X 3 ) = , ( Yi Y )( X i X ) = ( Yi Y )( X 3 i X 3 ) = ( X X )( X X ) i, n = 5. 3i 3 =

9 5. Write about The t-test approach and The F-test approach of restricted least squares. 6. Write about the chow test. 7. Consider the following correlation matrix. X X3... Xk R = X r3... rk X3 r3... r3k... Xk rk rk3... How would you find out from the correlation matrix whether There is a perfect collinearity, There is less than perfect collinearity and (c) The X s are uncorrected. 8. Write about Koenker-Bassett test Bartlett s homogeneits of variance test. 9. Explain the methods for detecting autocorrection. 0. Write about LOGIT and PROBIT models.

10 (DMSTT 4) M.Sc. (Final) DEGREE EXAMINATION, MAY 0. Final Year Statistics Paper IV MULTIVARIATE ANALYSIS Time : Three hours Maximum : 00 marks Answer any FIVE questions. All questions carry equal marks.. Prove that if the joint (marginal) distribution of X and X is singular (that is, degenerate), then the joint distribution of X, X and X 3 is singular. Let µ = 0 and Σ = (i) Find the conditional distribution of X and X 3 given X = x. (ii) What is the partial correlation between X and X 3 given X? P. Prove that x and S have efficiency [( ) ] ( P + / N ) / N for estimating µ and. Show that N ( N ) N ( x x ) ( x x ) = ( x x ) ( x x ) α β α β α α α< β N α= (Note : when P =, the left- hand side is the average squared differences of the observations). 3. Let x be the yield of a process and x a quality measure. Let o z =, z = 0 (temperature) ± relative to average) z 3 = ±0. 75 (relative measure of flow of one agent), and z 4 = ±. 50 (relative measure of flow of another agent). Three observations were made on x and x for each possible triplet of values of z, z3 and z 4. The estimate of β is ˆβ = ; S = 3.090, S=. 69 and r = can be used to compute S or Σˆ. Formulate an analysis of variance model for this situation Find a confidence region for the effects of temperature (i.e β ). β, (c) Test the hypothesis that the two agents have no effect on the yield and quantity.

11 4. Let T = N x S x, where x and S are the mean vector and covariance matrix of a sample of N from ( µ, Σ) ( τ, 0,... ) N. Show that λ = 0, where τ = µ Σ µ, and Σ is replaced by I. Verify that r = s /( s) multiplied by ( ) / T is distributed by same when µ is replaced by and N degrees of freedom and non centrality parameters N has the non central F-distribution with N τ. 5. Let l and l p be the largest and smallest characteristic roots of S, respectively. Prove ε l λ. ε l λl and p p σ Let ( ) ( ) σ 0 Σ =, K =. σ Σ 0 H where that in H = α α and α has P components. Show that α can be chosen so I p K Σ K σ = Hσ ( ) σ ( ) H H Σ H H σ ( ) has all 0 components except the first. 6. For P = 3, m= and = λ 3, prove ij = ( λi / ψ ii ) θ. i= Identification by 0 s let ( ) 0 C C = λ( ) ( ), C =. C C where C is non singular. Show that C ( ) 0 = λ( ) ( ) implies C C C = 0 C if and only if ( ) is of rank m. 7. If r ( n) is the sample correlation coefficient of a sample of ( = n + ) [ ] ( ) distribution with correlation ρ, then n r( n) ρ / P [( or) N r( n) the limiting distribution N ( 0, ). N from a normal [ ]/( P ) ] ρ has Find the significance points for testing ρ = 0 at the 0.0 level with N = 5 observations against alternatives. (i) ρ 0 (ii) ρ > 0 and (iii) ρ < 0.

12 8. Use Fisher s Z to test the hypothesis ρ = 0. 7 against alternatives ρ 0. 7 at the 0.05 level with r = 0. 5 and N = 50. Prove that when N = and = 0 9. Explain the concept of Cluster analysis. Pr r =. ρ, { = } = Pr( r = ) Write the similarity measures of cluster analysis. 0. Explain single linkage, complete linkage and average linkage. Explain K-means method.

Total No. of Questions : 10] [Total No. of Pages : 02. M.Sc. DEGREE EXAMINATION, DEC Second Year STATISTICS. Statistical Quality Control

(DMSTT21) Total No. of Questions : 10] [Total No. of Pages : 02 M.Sc. DEGREE EXAMINATION, DEC. 2016 Second Year STATISTICS Statistical Quality Control Time : 3 Hours Maximum Marks : 70 Answer any five