QP-IES-ISS-26-220626-STATISTICS-PAPER-IV

ination, 2026 LES./LS.S. Examination fateq DETACHABLE STATISTICS Paper IV Time Allowed : Three Hours Maximum Marks : 200 QUESTION PAPER SPECIFIC INSTRUCTIONS Please read each of the following instructions carefully before attempting questions. There are FOURTEEN questions divided under SEVEN Sections. Candidate has to choose any TWO Sections and attempt the questions therein. All the Sections carry equal marks. The number of marks carried by a question/part is indicated against it. Wherever any assumptions are made for answering a question, they must be clearly indicated. Diagrams/Figures, wherever required, shall be drawn in the space provided for answering the question itself. F-table and Graph paper can be found at the end of both sections in the booklet. Unless otherwise mentioned, symbols and notations have their usual standard meanings. Attempts of questions shall be counted in sequential order. Unless struck off, attempt of a question shall be counted even if attempted partly. Any page or portion of the page left blank in the Question-cum-Answer (QCA) Booklet must be clearly struck off. Answers must be written in ENGLISH only. 1 SNCS-D-STS 1. (a) 1. (b) 1. (c) SECTION ‘A’ (Operations Research and Reliability) 1. Answer all of the following : 10x5=50 A paint company produces both interior and exterior paints using two raw materials M, and M). The following table provides the basic data of the problem : Tons of raw material Maximum daily per ton of availability (tons) Exterior paint Interior paint Raw Material M, 6 4 24 Raw Material M, 1 2 6 Profit per ton 50 40 (in thousand Rs.) The daily demand for interior paint cannot exceed that of exterior paint by more than 1 ton. Also the maximum daily demand for interior paint is 2 tons. Determine the optimal product mix. 10 Neon lights on the campus of a university are replaced at the rate of 100 units per day. It costs Rs. 10,000 to initiate a purchase order. A neon light kept in storage is estimated to cost about Rs. 2/day. The lead time between placing an order and receiving the order is 12 days. Determine optimal inventory policy for ordering the neon lights. Also, calculate cycle length and reorder point of the order. 10 There are five jobs to be performed by employees in a Departmental Store. The time (in hours) that each employee takes to perform each job is given in the following effective matrix : Employees I ff W IVvev Jobs A 10 5 13 15 16 B 3 18 13 6 Cc 100 7 2 2 2 D 7 WwW 9 7 12 E 7 9 10 4 = «12 \o How the jobs should be assigned, one per employee, so as to minimize the total manhours ? 10 SNCS-D-STS 2 1. (d) 1. (©) (a) Two jobs are to be performed on five machines A, B, C, D and E. Processing times (in hours) are given below : Processing Time Machines Job 1 | Sequence A B C D _E Processing Time | 3 4 2 6 2 Job 2 | Sequence B C A D _ £E 5 4 3 2 6 Use graphical method to obtain minimum elapsed time and idle times for the two jobs. Define the terms : IFR, IFRA, NBU, NBUE, DMRL Answer any two from the following : 10 10 25x2=50 Consider the following table summarizing the details of a project involving 14 activities : Activity Immediate Duration predecessor(s) (in months) A ~ 2 B - 6 Cc - 4 D B 3 E A 6 F A 8 G B 3 H Cc, D 7 I Cc, D 2 J E 5 K F,GH 4 L FGH 3 M I 13 N J,K 7 Draw the network diagram and identify the critical path of the project. 25 3 SNCS-D-STS 2. (b) i) (ii) | 2() (ii) 2d) SNCS-D-STS Write the basic characteristics of queueing model (M/M/I) : (FCFS/c0/s0). 5 A company has received a contract to supply gravel to three new construction projects located in three towns A, B, C from 3 gravel pits located in three towns X, Y and Z. The delivery cost from each pit to each project site of truckloads is given below: Project Location A B G Supply Xx 4 8 8 76 Pits Y 16 24 16 82 Z 8 16 24 77 Demand 72 102 41 Find an optimum schedule of transportation so as to minimize the total cost of transportation. 20 Discuss the replacement policy for items whose running cost increases with time and value of money remains constant during a period. 10 Suppose that a sample of 12 items were put on test and the test was terminated at the 8" failure time, and the lifetimes follow an exponential distribution with mean @>0. The failure times are, in hours 30, 57, 145, 167, 320, 440, 505, 675 (J) obtain maximum likelihood estimate of mean lifetime (II) construct 95% confidence interval for 8 [Given 7716, 0.975) = 28-845; x7 (16, 0.025) = 6-908] 15 Obtain an optimal solution using simplex method to the following linear programming problem : Maximize z = 3x, + 2x, Subject to x, +2x,<4 3x, + 2x, < 14 xX, — XS 3 xX}, X2 20 Also find an alternative optimal solution if it exists. 15 4 - (b) a) -@ - (©) (ii) When do we apply the Rule of Dominance in solving a game problem ? Solve the game whose payoff matrix is given below : Player B Player A i! I Il I 30 40 —80 Il 0 15 —20 Il 90 20 50 SECTION ‘B’ (Demography and Vital Statistics) Answer all of the following : Define Central Mortality Rate (m,) and Force of Mortality (,). Establish the relation Lu yy xth stating the necessary assumptions involved. (i) Define crude death rate with its limitations. 10 10x5=50 5 5 4 (ii) Explain standardized death rates including the methods of their computation. 6 Ascertain whether Gompertz’s or Makeham’s formula for graduation is suitable for the following data : Age x 35 40 45 50 55 60 I(x) | 89785 | 86237 | 82377 | 78018 | 72895 | 66666 10 Derive an algebraic expression relating the probability of a person surviving between age x and x + 1, p, to the force of mortality, p,. 10 Fill in the blanks in a portion of life table given below : Age (x) Ix de | Px dx | Ly T. | Px 4 90000 | 500 ? ? ? |5060200| ? 5 ? 300 ? ? ? ? ? 10 5 SNCS-D-STS 4. Answer any two from the following : 25x2=50 (a) What is vital statistics ? State the uses of vital statistics. Explain registration method and census method of obtaining vital statistics. 5+5+15=25 (b) The sex wise distribution of population and number of births with survival rates of a town in 2022 are given below: Age Population Births Survival rate | group Male Female Male Female (ntx) 15-19 6432 6210 65 64 0-92 20-24 5318 5430 120 115 0-90 25-29 4470 4580 110 105 0-87 30-34 3960 3970 80 78 0-86 35-39 3580 3600 60 65 0-84 40-44 3140 3050 15 18 0-83 45-49 2890 2705 4 3 0-81 Compute GFR, ASFR, TFR, GRR, NRR. 25 (c) In usual notations as in a life table, given /g9 = 16000 and Age (x) | 80 | 81 | 82] 83 | 84 | 85 | 86 d, 5000 | 3000 | 2500 | 2000 | 1000 | 1000 | 500 (i) Compute the values of /, and q, for x= 81, 82, ..., 86. 10 (ii) The ages of three persons A, B and C are 81, 82 and 83 respectively. Find the probabilities. (1) that A, B and C will be alive in two years’ time 5 (2) that one at least of the three will be alive in two years’ time 5 (3) that exactly one of them will be alive in two years’ time 5 (d) | What is internal and international migration ? 10 Discuss various classifications of internal and international migration. 15 SNCS-D-STS 6 SECTION ‘C’ (Survival Analysis and Clinical Trials) 5. Answer all of the following : 10x5=50 (a) (b) ) () (a) bP ‘eb p-l pp hazard function is an increasing function for p > 1 and decreasing function for p < 1. 10 For gamma lifetime model f(x, b, p) = ,b>0, p>0, x>0, show that One often hears that the death rate of a person that smokes is, at each age, twice that of a non-smoker. Does it mean that a non-smoker has twice the probability of surviving a given number of years as does a smoker of the same age ? Justify your answer. 10 Define hazard function, cumulative hazard function and mean residual life function. If T is a continuous non-negative random variable with cumulative hazard function H(T) then show that H(T) follows standard exponential distribution. 10 What is multicenter trial ? Why are multicenter trials conducted ? 10 Consider two groups of survival data with hazards 2,(d), A,(é) and survivor functions $,(0), S(t) respectively. (i) One of the assumptions for Cox model is proportional hazard, what is really meant by “proportional hazard” ? 5 (ii) Assuming the two hazard functions are not the same, examine the connection between the crossings of the two hazard functions and the crossing of the two survivor functions. 5 Answer any two from the following : 2x25=50 The survival times of two groups of breast cancer patients who had surgical treatment are given below : Survival group (in months) : 3, 77, 9,9, 11*, 16 Chemotherapy group (in months) : 8, 9, 10°, 12*, 18, 23* Apply logrank test for comparing survival distributions of the two groups at 5% level of significance using exact method. [Given x3 (0-05) = 3-841] 25 7 SNCS-D-STS (bv) Explain time censoring and number censoring (type-II censoring). State the likelihood functions in each censoring scheme. Assume the life time follows exponential distribution with failure rate Ls A> 0. Derive the maximum likelihood estimate of survival function at time t = 700 based on the following censored data : t, = 80, ty =95, t= 105, t= 180, t5=270, t= 330, t7 = 670, tg = 800, 1, > 1000, i=9, 10, ..., 20. Using the (i) actual failure time/survival time 13 (ii) number of failures observed only without considering the actual failure time. 12 (c) A study is made on the impact of regular exercise and gender on the risk of developing heart diseases amongst 55-75 year olds. A sample of people was followed from 0 if female, 1 if male, the exact age of 55 years until they either develop heart diseases or turn 75 years, whichever comes first. The Cox PH model was used for this study with the two covariates being defined as 0, if female. 0, ifrare z, (Gender) = | ifmale > 72 (exercise rate) = 1, ifregular The model results were as follows : Model fitted Maximum log likelihood 1. Null model : 1190 2. Gender only : 1177 3. Gender and Exercise : -1170 4. Gender, Exercise and Interaction : -—1166 ; (Interaction = Gender * Exercise) Covariate Parameter fitted Gender > By = 0-25 Exercise : By =-0-35 Gender * Exercise : By = —0-45 (i) Give two reasons why the Cox PH model is suitable in this data analysis. (ii) Perform a statistical test to show that the interaction term is significant in the model (Take a = 0-05). (iii) Give the hazard functions for a male who does not exercise regularly; a female who exercises regularly; a female who rarely exercises and a male who exercises regularly. SNCS-D-STS 8 (iv) Identify the baseline hazard for this model. (v) Interpret your results in (iv) with reference to the baseline hazard and the hazard function for males who rarely exercise. [Given : %%, 0.05) = 3841] 2+8+84+2+5=25 (d) | What is randomization in a clinical trial ? How should the randomization code be determined ? Discuss some common randomization methods. 5+4+16=25 SECTION ‘D’ . Answer all of the following : 10x5=50 (a) | Why do we make use of Statistical Quality Control ? Write the advantages when a process is working in a state of statistical control. 10 (b) Explain the statistical reasoning for using 3-o limits in statistical quality control. 10 (c) A Metropolitan Transit system uses the number of written passenger complaints per day as a measure of its service quality. For 10 days, the number of complaints received are given below : Day (sample) no. 1}2;);3})4;)5]6/]7)]8 49 | 10 No. of complaints/day| 4 | 8 | 2 |0}]3]91|10}0]6]| 4 Obtain the three control limits for the above data. 10 (d) Explain the differences between CUSUM and Shewhart Control Charts in terms of their uses and methodology. 10 (e) Describe Double-Sampling Inspection plan and discuss its advantages. 10 Answer any two from the following : 25x2=50 (a) A machine is set to deliver packets of a given weight. Ten samples of size 5 each were recorded and the recorded data is produced below : Sample No. | 1 2 3 4 | 5 6 | 7 8 9 | 10 Mean (x) 15 | 17 | 15 | 18 | 17 | 14] 18 | 15 | 17 | 16 Range (R) | 7|7]/4]9 {8 ]7 1/12] 4/11] 5 Can the process be regarded under control ? (Given conversion factors for 2 =5, A, = 0-58, D3 =0, Dg = 2-115) 25 9 SNCS-D-STS (b) Explain the statistical basis and construction of p and np charts. How is the choice between p and np charts made ? 25 (c) (i) It has been decided to sample 100 items at random from each large batch and to reject the batch if more than 2 defectives are found. The acceptable quality level is 1% and unacceptable quality level is 5%. Find the producer’s and consumers risks. 10 (ii) Define a single sampling plan. Consider a single sampling plan : Lot size (N) = 2000, Sample size (n) = 50 and acceptance number of defectives (c) = 2. Find the probability of accepting the lot. 15 (d) Hourly concentration (X;) data collected from a chemical process is given below : Hour X; Hour X; 1 5-50 11 6-75 2 4:50 12 3-25 3 5-25 13 5-25 4 6-0 14 5-0 5 5:25 15 4-5 6 3-50 16 6-50 7 5-75 17 7-20 8 6:25 18 6-80 9 4-50 19 6-75 10 5-0 20 6-50 If the target mean (Up) = 5-0, n= 1 and o= 1, use CUSUM Control Chart to detect the shift to uw, = 6-0 for K=0-5. 25 SECTION ‘E’ (Multivariate Analysis) 9. Answer all of the following : 10x5=50 (a) Given X~ N; (u, 2), where 2 8 2 3 M=|4|andY=/2 4 1], 3 3 1 3 (i) find the regression function of X; on X, and X3, and (ii) compute the conditional variance of X, given X, and X3. 10 SNCS-D-STS 10 (2) Show that X= (20, 20, ..., X,)' has p-variate normal distribution if and only if every linear combination (J; X, + 1,X,+...+1,X,) of X follows a univariate normal distribution. 10 (c) Let # ‎ where } ~N, (. 2) and A ~ W, (n, 2) which is independent of Y. Show that 7 = 72 for K <p. 10 () there be two populations 70] and 2. It is known that about 30% of all objects belong to 1702 and © (211) : cost incurred when a ‎ observation is incorrectly classified as 702 observation = 15; © (1|2) : cost incurred when a 22 observation is incorrectly classified as ‎ observation = 10. Suppose the two density functions f(x) and 70) (corresponding to 7, and 772) are evaluated at a new observation 0 and 71000) = 0-32, 700) = 0-56. Can the new observation be classified as coming from 70] or 7 7 10 (e) Let X= (21, 2, 703) has the correlation matrix R given by 1 42 1 1 =-- R=|‏ 42 42 1 1 =- 0 42 Obtain the first two principal components and the percentage of population variance explained by the first two principal components. 10 10. Answer any two from the following : 25x2=50 (a) (i) If Xis a random p-vector distributed as , >), then obtain the distribution of and specify its parameters. 15 (ii) If A ~ W,(u, 2), then prove that CAC'~ Wn, CXC"), where C is a ( xp) matrix of rank 50. 10 11 SNCS-D-STS (b) Define sample generalised variance based on a random sample X,(a = 1, 2, ..., N) of size N drawn from N,(u, Z) and obtain its distribution. Also find an expression for its h'" moment (h = 1, 2, ...). If Z,, Z, and Z, are independently and identically distributed as N2(Q, Z) with 25 = =() <) then obtain E 3 Biez a=l (c) Let Xj, X, ..., X, represent measurements or characteristics on one member of a twin pair and Xp Xp+2 ue Xp represent the same measurements on the other member. Assuming that X= (Xj, XQ, Xyp)'~ No (LL x), with © unknown, develop a suitable test for testing the hypothesis of equality of measurements of the twins. 25. (d) Define canonical correlations and canonical variates and obtain the characteristic equation they satisfy. Hence, or otherwise, show that multiple correlation and simple correlation are special cases of canonical correlation. 25 SECTION ‘F’ (Design and Analysis of Experiments) 11. Answer all of the following : 10x5=50 (a) In an RBD there are only two blocks. Let K be the number of treatments and Xp j=1,2 the average yield of j” block. Show that the between block sum of squares can be expressed as £5, -%) and write the ANOVA table. 10 (b) Explain the two basic ways of testing hypothesis involving contrasts of K parameters. Explain orthogonal contrasts and their use. 10 (c) | Suppose you have v varieties compared in V? plots. How will you carry out the experiment under each of the following situations ? (i) there is no fertility difference among the V? plots. (ii) the fertility changes along two perpendicular direction. Write the appropriate ANOVA table for each case. 5+5 SNCS-D-STS 12 12. (d) (e) (a) (b) (c) State the advantages of a factorial experiment over a simple experiment. Explain Yates’ method of computing factorial effect totals. 10 Define the linear model for a split-plot design with two factors replicated r times. Write the ANOVA table. Give an example of the design. 10 Answer any two from the following : 25x2=50 In the table given below are the yields of 6 varieties in a 4 replicate experiment for which one observation under treatment 2 in block 2 is missing. Estimate the missing observation and analyse the data : Treatments Blocks 1 2 3 4 5 6 1 18-5 15-7 16-2 14-1 13-0 13-6 2 11-7 - 12-9 14-4 16-9 12-5 3 15-4 16-6 15-5 20:3 18-4 921-5 4 16-5 18-6 12:7 15-7 16-5 18-0 (Given : F314 (0-05) = 3-34, Fs 14 (0-05) = 2-96) 25 Diet affects weight gain. We wish to compare nine diets : these diets are the factor level combinations of protein source (beef, pork and grain) and number of calories (low, medium and high). There are test animals nine in number that were randomly assigned to the nine diets one animal per diet. The responses (weight gain) are : Calories Source Low Medium High Beef 76-0 86:8 101-8 Pork 83:3 89-5 98-2 Grain 83-8 83-5 86-2 Using an appropriate linear model analyse the data and give the ANOVA table. (Given : F ,(0-05) = 6-44) 25 Define the analysis of covariance model of CRD with one concomitant variable and explain its statistical analysis. 25 13 SNCS-D-STS @) 13. (a) (b) (©) (d) (e) In designing a battery for use in a device three possible plate materials are tested at three temperature levels. The following table gives the life (in hours) of the plate materials : Temperature (°F) Plate material type 15 70 125 1 130, 155 | 34, 40 70, 58 74, 180 80, 75 2 159, 126 | 136, 115 45 3 138, 160 | 150, 139 96 Considering the given data is the proportional data, assess the effect of temperature on the plate material type and present the ANOVA table. (Given : F4 1,(0-05) = 3-36, F, (0-05) = 3-98). 25 SECTION ‘G’ (Computing with C and R) Answer all of the following : 10x5=50 Summarise the rules for type conversion in C when neither operand is unsigned. Explain conditional operator with a suitable example. 10 Given a matrix A,,,., write a C function to find the product of A,,,., and its transpose and to print the result. 10 Describe two different approaches to updating a data file. Write illustrative programs one each. 10 Write R code to create a data frame with name, age and gender of 10 individuals. (i) Extract age and gender of the 4", 8 and 1%t individual in this order. (ii) Add three new records to the data frame created above. 10 Write R code to find the median of the observations x; (i= 1, 2, ..., 10) without using the median function. 10 SNCS-D-STS 14 14. (a) (b) (©) (d) Answer any two from the following : 25x2=50 Given the observed frequency distribution with the variate values x; (i = 0, 1, 2, ..., 8) and the corresponding frequencies /; (i= 0, 1, 2, ..., 8) write a C program to fit a binomial distribution and test for its goodness of fit and to print the result. 25 Given class, section, name, date of birth, roll number and marks secured in four subjects for a group of 30 students write a C program to find the average mark secured by each student and to print roll number and average mark of each student. Make use of structure variables within the program. 25 Write a C program to test for the independence of attributes in a given mxn contingency table and to print the result. 25 (i) For a_ given observed frequency distribution with variate values x; (i=0, 1, 2, ..., 10) and the corresponding frequencies /; (i= 0, 1, 2, ..., 10), write R code to find mean and variance without using R functions and compare them. Also print your comment. 15 (ii) Write R code to generate a random sample of size 15 from N(22, o=.1-5) and another random sample of the same size from N (13, o = 2) and to test for the equality of the means of the two populations at 1% level of significance and to print the model values of the samples. 10 15 SNCS-D-STS P(O<Z < 1-96) =0-4750 Degrees of Freedom for the Denominator (v2) $0:4750 ZZ 1-96 = The Normal Distribution Z. 2.090 ©o.O1 2.02 9.03 0.04 2.08 pomet=) 9.07 0.08 2.09 0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359 o.2 0.0398 0.0438 0.0478 0.0S17 0.O0SS7 C.0S96 0.0636 0.067S 0.0714 0.0753 °.2 0.0793 | 0.0832 | 0.0871 | 0.0910 | 0.0948 | 0.0987 | 0.1026 | 0.1064 | 0.1103 | 0.1142 o.3 O.1179 0.1217 0.1258 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517 0.4 o0.1SSs4 0.1591 0.1628 0.1664 o.1700 0.1736 O.1772 0.1808 0.1844 0.1879 0.5 0.1915 0.1950 0.1985 0.2019 0.2088 0.2123 0.2187 0.2190 o.2224 0.6 0.2287 O.2291 0.2324 0.2357 0.2422 0.2454 0.2486 0.2517 0.2549 o.7 0.2580 o.2611 °o.2642 0.2673 0.2734 0.2764 0.2794 0.2823 0.2852 o.8 0.2881 | 0.2910 | 0.2939 | 0.2967 0.3023 | 0.3051 | 0.3078 | 0.3106 | 0.3133 o.9 0.3159 0.3186 0.3212 0.3238 0.3289 0.3315 0.3340 0.3365 0.3389 1-0 0.3413 | 0.3438 | 0.3461 | 0.3485 0.3531 | 0.3554 | 0.3577 | 0.3599 | 0.3621 1.21 0.3643 0.3665 0.3686 0.3708 0.3749 O.3770 0.3790 0.3810 O.3830 1-2 0.3849 0.3869 0.3888 0.3907 0.3944 0.3962 0.3997 o.4015 1-3 0.4032 0.4049 0.4066 0.4082 0.4115 0.4131 0.4162 0.4177 1.4 0.4192 0.4207 0.4222 0.4236 0.4265 0.4279 0.4306 0.4319 1.5 0.4332 0.4345 0.4387 0.4370 0.4394 0.4406 0.4429 0.4441 1.6 0.4452 0.4463 0.4474 0.4484 0.4505 o0.451S 0.4535 0.4545 1.7 0.4554 0.4573 0.4582 0.4599 0.4608 0.4625 0.4633 1.8 0.4641 0.4656 0.4664 0.4678 0.4686 0.4699 0.4706 1.9 0.4713 0.4726 0.4732 0.4744 0.4750 0.4761 0.4767 2.0 C.4772 0.4783 0.4788 0.4798 0.4803 0.4812 0.4817 2.1 0.4821 0.4830 | 0.4834 0.4842 | 0.4846 0.4854 | 0.4857 2.2 0.4861 0.4868 0.4871 0.4878 0.4881 0.4887 0.4890 2.3 0.4893 0.4898 °o.4901 0.4906 0.4909 0.4913 0.4916 2.4 0.4918 0.4922 | 0.4925 0.4929 | 0.4931 0.4934 | 0.4936 2.5 0.4938 0.4941 0.4943 0.4946 0.4948 0.495851 0.4952 2.6 0.4953 0.4956 0.4957 0.4960 0.4961 0.4963 0.4964 2.7 0.4965 = 0.4967 0.4968 0.4970 0.4971 0.4973 0.4974 2.8 0.4974 0.4975 0.4976 0.4977 0.4978 0.4979 0.4980 0.4981 2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4985 0.4986 0.4986 3.0 0.4987 0.4987 0.4987 0.4988 0.4989 0.4989 0.4990 90.4990 3-1 0.4990 0.4991 0.4991 0.4991 0.4992 0.4992 0.4993 0.4993 3.2 0.4993 | 0.4993 | 0.2994 | 0.4994 0.4994 | 0.4994 0.4995 | 0.4995 3-3 0.4995 0.4995 0.4995 0.4996 0.4996 0.4996 0.4996 0.4997 3.4 0.4997 | 0.4997 | 0.4997 | 0.4997 0.4997 | 0.4997 0.4997 | 0.4998 3.5 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 3.6 0.4998 | 0.4998 | 0.4999 | 0.4999 0.4999 | 0.4999 0.4999 | 0.4999 3.7 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 3.8 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 3.9 9.5000 | 0.5000 | o-s000 | 0.5000 ©.5000 | 0.5000 °.5000 | 0.5000 Percentage Points of the F Distribution Fo.05, 1, vp y Degrees of Freedom for the Numerator (v,) ¥2 1 2 3 4 5 6 7. 8 9 10 12 18-2. 90). 9e 300 SAD 0" 1S ies 1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 241.9 243.9 245.9 248.0 249.1 250.1 251.1 252.2 253.3 254.3 2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19,37 19.38 19.40 19.41 19.43 19.45 19.45 19.46 19.47 19.48 19.49 19,50 3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.74 8.70 866 8.64 862 859 857 8.55 8.53 4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.69 566 5.63 os 661 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.62 4.56 4.53 4.50 4.46 4.43 4.40 4.36 6 |5.99 5.14 4.76 4.53 4.39 4.28 421 4.15 4.10 406 4.00 3.94 3.87 3.84 3.81 3.77 3.74 3.70 3.67 cs 5.59 4.74 435 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.57 3.51 344 3.41 3.38 3.34 3.30 3.27 3.23 8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.28 3.22 3.15 3.12 3.08 3.04 3.01 2.97 2.93 9 |5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.07 3.01 2.94 2.90 2.86 2.83 2.79 2.75 2.71 10 | 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.91 2.85 2.77 2.74 2.70 2.66 2.62 2.58 2.54 11 | 4.84 3.98 3.59 3.36 3.20 3.09 3.01 295 290 285 2.79 2.72 265 261 257 253 249 2.45 2.40 12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 269 262 2.54 2.51 2.47 243 2.38 2.34 2.30 13 467 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67 2.60 2.53 2.46 2.42 2.38 2.34 2.30 2.25 2.21 14 |460 3.74 3.34 3.11 296 2.85 2.76 2.70 2.65 260 253 246 2.39 235 2.31 227 222 218 2.13 15 | 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 248 240 2.33 2.29 2.25 2.20 2.16 2.11 2.07 16 | 4.49 3.63 3.24 3.01 2.85 2.74 266 2.59 2.54 2.49 2.42 2.35 2.28 2.24 2.19 2.15 2.11 2.06 2.01 17 14.45 3.59 3.20 2,96 2.81 2.70 261 255 2.49 245 2.38 231 2.23 2.19 2.15 2.10 2.06 2.01 1.96 18 | 4.41 3.55 3.16 2.93 2.77 266 2.58 2.51 2.46 2.41 2.34 2.27 2.19 2.15 2.11 2.06 2.02 1,97 1.92 19 | 4.38 3.52 3.13 2.90 2.74 263 2.54 248 2.42 2.38 2.31 2.23 2.16 2.11 2.07 2.03 1.98 1.93 1.88 20 | 4.35 3.49 3.10 2.87 2.71 2.60 2.51 245 2.39 2.35 2.28 2.20 2.12 2.08 2.04 1.99 1.95 1.90 1.84 21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 2.25 2.18 2.10 2.05 2.01 1.96 1.92 1.87 1.81 22 | 430 3.44 3.05 2.82 266 2.55 246 2.40 2.34 2.30 2.23 2.15 2.07 2.03 1.98 1.94 1.89 1.84 1.78 23 | 4.28 3.42 3.03 2.80 264 2.53 244 237 2.32 2.27 2.20 2.13 2.05 2.01 1.96 1.91 1.86 1.81 1.76 24 4.26 3.40 3.01 2.78 2.62 2.51 242 2.36 2.30 2.25 2.18 2.11 2.03 1.98 1.94 1.89 1.84 1.79 1.73 25 | 4.24 3.39 2.99 2.76 260 2.49 240 2.34 2.28 2.24 2.16 2.09 2.01 1.96 1.92 1.87 1.82 1.77 1.71 26 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 2.22 2.15 2.07 1.99 1.95 1.90 1.85 1.80 1.75 1.69 27 | 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25 2.20 2.13 2.06 1.97 1.93 1.88 1.84 1.79 1.73 1.67 28 1420 3.34 295 2.71 256 245 2.36 2.29 2.24 2.19 2.12 2.04 1.96 1.91 1.87 1.82 1.77 1.71 1.65 29 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22 2.18 2.10 2.03 1.94 1.90 1.85 1.81 1.75 1.70 1.64 30 4.17 3.32 2.92 269 2.53 2.42 2.33 2.27 2.21 2.16 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.62 40 | 4.08 3.23 2.84 261 245 2.34 2.25 2.18 2.12 2.08 2.00 1.92 1.84 1.79 1.74 1.69 1.64 1.58 1.51 60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 1.92 1.84 1.75 1.70 1.65 1.59 1.53 1.47 1.39 120 | 3.92 3.07 2.68 2.45 2.29 2.17 2.09 2.02 1.96 1.91 1.83 1.75 1.66 1.62 155 155 1.43 1.35 1.25 ca 3.84 3.00 260 2.37 2.21 2.10 2.01 1.94 1.88 1.83 1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1.00 SNCS-D-STS