CC BY
47
Double Sampling Based Parameter Estimation in Big Data and Application in Control Charts
Abdul Alim and Diwakar Shukla
Department of Computer Science and Applications, Dr. Harisingh Gour University, Sagar (MP), India E-mail: abdulaleem1990@gmail.com, diwakarshukla@rediffmail.com
Abstract
Double sampling technique and control charts are used for predicting about unknown parameters of the big population and developing algorithms for imposing control over growth factor. This sampling procedure has two approaches like sub-sample and independent sample. Aim is to estimate mean file-size by both and to find out which approach is better in big data setup. Comparative mathematical tools used herein are mean squared error, confidence interval, relative confidence interval length measure and control charts of digital file-size for monitoring. Estimation strategies are proposed and confidence intervals are computed over multiple points of time. At each time, it was found that confidence intervals are catching the true values. First kind of approach (as case I) of double sampling found better than the second. A new simulation strategy is proposed who is observed efficient for comparison purpose. Single-valued simulated confidence intervals are obtained using the new simulation strategy and found covering the truth in its range. As an application of outcomes, control charts are developed to monitor the parametric growth over long duration. Upper and Lower control limits are drawn for business managers to keep a watch on digital file-size estimates whether their growth under control? Outcomes may be extended for reliability evaluation under discrete time domain. The content herein is a piece of thought, idea and analysis developed by deriving motivation from past references to handle big data using double sampling. Findings of the study can be used for developing software based monitoring system using process control charts for managers.
Keywords: Big Big-Data, Double Sampling, Sub-Sample, Estimation, Control Limits, Control Charts, Process Control, Social Media Portal, Mean squared error, Simulation, Confidence Interval (CI).
I. Introduction
Due to emergence of digital technologies and appearance of social media platforms worldwide, people are habitual for ease and comforts contained therein. While user registration, participation and content-communication through these platforms, the digital data is facing challenges in terms of drastic growth in volume, velocity and variety. Momentum of data over time domain has got immense speed to occupy memory space at servers/data centers. Often users do not remove their long past garbage data from the social media account. Space allocation to users is unlimited due to inherent competition in IT business. No service provider wants reduction in user database. Therefore, forecasting (or prediction) require about possible expansion of digital space over continuous time. A manager of Data center is interested to know how much investment cost needed for enhancing capacity of storing units relating to social media portal. Fig.1 to Fig. 3 reveal scenario of expanding digital space over time t1, t2 and t3 (tx< t2 < t3).
Digital Storage related to big data
User 1 User 2 User 3
e
fCZ c_^
Space Space Space
User N
Space
Figure 1: The digital model of data storage at time t1
The figure 1 is allocation of default memory space at the time of user-registration on a portal at time ti.
Digital Storage related to big data
Figure 2: Digital model of data storage at time t2
After ti and before t2 (t2 > tx) , figure 2 shows increment in default allocated digital space. While at time instant t3 (t3 > t2> tx). the default space demand got extra ordinary longevity.
Digital storage related to big data
Figure 3: The digital model of data storage at time t3
Alert system requires for constant monitoring of the storage space who can convey managers of IT-business for further planning and cost investments. It could be developed by the joint efforts of sampling methodologies available in literature and process control charts over time frame.
Abdul Alim, Diwakar Shukla
DOUBLE SAMPLING BASED PARAMETER ESTIMATION IN BIG RT&A, No 2(62) DATA AND APPLICATION IN CONTROL CHARTS_Volume 16, June 2021
Double sampling scheme is a tool for estimating population parameters where first sample provides guess(low cost) estimate of parameters of the support variable while second sample provides precise estimates of main variable of interest. This scheme has two variants like (a) second sample as sub-sample of first (b) second sample as independent. Fig 4 shows the scheme diagrammatic layout of double sampling.
Figure 4.1: Double sampling Scheme (at jth occasion ti) under case I
Figure 4.2: Double sampling Scheme (at ith occasion ti) under case II
Using scheme of figure 4.1,and 4.2, one can obtain the sample based prediction about average file-size floating in portal of social media communication model described in Figure 5 where two-way communication exist among large voluminous group of registered users.
Figure 5. Portal based Social Media communication model and floating files
Statistical methods are used in process control and size measure, which exhibit the extent of conformity of the situation under specifications determined by the relevant authority. It is one of aspects affecting decision making based on specifications set at early level and continued until the completion [1]. Big data take into account digital streams with observations on file-size generated sequentially over time. Among many different purposes, one common task is to collect and analyze big data and to monitor the longitudinal performance of the related processes. Big data assume different forms of data-streams gathered through complex engineering systems like
Abdul Alim, Diwakar Shukla
DOUBLE SAMPLING BASED PARAMETER ESTIMATION IN BIG RT&A, No 2(62) DATA AND APPLICATION IN CONTROL CHARTS_Volume 16, June 2021
sequences of satellite images, climate data, website transaction logs, credit cards, etc which have
complicated data structure and complex storage mechanism. Statistical process control charts [28]
could be utilized as an important tool a for monitoring and decision making [2].
- Upper Control Limit
Text size Video size Image size
Lower Control Limit
tl t2 t* tj t? tfi - Time-►
Figure 6: Process control chart of file size variable II. Literature Review
Parameter estimation problem in big data setup is an opportunity which enhanced by contributions to the higher education sectors in terms of developing indicators for decision making [3][4]. Broad aspect of big data including special features and characteristics was narrated [6] with consolidated description by integrating definitions from practitioners and academics. It was focused on analytics related to unstructured data, whose share is 95% of the big data. While dealing with big data and sampling methodologies, scientists can derive machine learning algorithms by the implementation of supervised statistical data mining. Analysis techniques can be used for a single machine scheduling problem in light of hidden patterns using optimized scheduling sequence [7]. Analysis depends on the ability of data scientists to make sense and develop insight into huge data volume. Way of developing an actionable idea is known as data exploration who brings out hidden facts and so is challenging task.
Data requires first a small view to have insights [8] for further course of actions. Sampling methodologies can play vital role by creating preliminary insight into big data. Sampling frame is collection of all units of the population that can use in big data for sample selection. Random sample of registered users on portal can be drawn to obtain estimates of individuals and such precisions are the same as of those when frame consisting of a list of individuals is taken into account [9]. Online social network portals generate large population of individuals where unbiased sampling could be used as a tool for prediction. Convergence properties of the random walk were studied using sampling of Facebook data [10]. Estimation of the influence of an event connected through social media is a problem to handle with because social media is widely exploited as communication platform for relevant and irrelevant information. It opens avenue for mathematical formulation and characterization using casual inference [11]. Avenues in web-based large scale social networks are to explore concise and coherent methods for summarization and to draw valid conclusions. Sampling methodologies on social media network have key role as knowledge discovery tools while a suitable methodology can carry out with efficient decision [12]. Big data may suffer due to incompleteness of required values, and so, need to be replaced by neighboring computations. Digital media platforms often have distinctive temporal patterns that can be exploited for computations in situations involving incomplete information. Iterative type methods suggest to estimate the parameters in the underlying point process and assign weights to the unknown events with direct calculable score function [13].
Text size
Video size
Image size
The ratio based chain-type exponential estimator for finite population mean under double sampling with the auxiliary variable support can provide an efficient estimation methodology [14] applicable in big data. Empirical study as a tool could be used for the betterment of outcome. Extension of exponential-type estimator under double sampling was an outcome [15] with comparative efficiency of the past. In practice, one can find multiple variants affecting the main variable of interest. The multivariate exponential type estimators could be used in setup of double sampling [16] who may efficient performer than single support variant. In presence of non-response and with the help of fractional raw moments, estimate of population parameter could be more efficient using such inputs [17].
The Gaussian process is useful for prediction and can be applied over big data. It is useful where the additive model works well and response depends upon a small number of features [18]. International groups of customers relating to business deals constitute big data and, therefore, exist like open problem for business manager to deal with. Data of sale of cars available on the internet can be analyzed [19] just to have an insight for perception, preferences, preparedness and internet experience of potential car customers while making a purchase/sale decision. A coordination among manufacturer, supplier, retailer is essential to manage forward and reverse supply chain in business. It relates to price-fixing, storing capacity, profit sharing and decision making in a closed-loop supply chain in order to maintain the smooth functioning. Responsibility-sharing is prime factor that affects outcome and profit. Such require optimal selling price, optimal time, wholesale price, sharing percentage and optimal return rate in such a manner that objective function be maximize [20]. The stratified double sampling scheme can be used for estimating finite population mean in presence of more than one support variables using regression time estimators [21]. The basic idea used is to use the ranks of two support variables and extension of the same idea in double sampling setup is due to [22].
III. Motivation and Problem Undertaken
Mean estimation strategies exist in literature [23] for estimating average file-size of text, video and image type communication among registered users on asocial media platform. According to sampling theory [25-27],while the population mean of support variable is unknown, the double sampling is used. This motivates for developing generalization of the aspect [23] for real world situation. In big data, it is difficult to find out parametric information of support variable priorly known because of data-volume, velocity and variety. This motivates for use of double sampling in early literature. Moreover, simulation method with this scheme still not explored. This paper considers the scenario of absence of support variable parametric information and presents new estimation strategies along with a new simulation procedure useful in comparative analysis. As an application, control charts are developed to monitor parametric changes in big data over multiple time occasions. Reliability could be examined through discrete-time domain over a long period. The mathematical support is derived from [23], [24-26] and [27].
I. Assumptions
1. Let at time tj, the jth registered user on a web portal generates data values {(Ti)tj} as text, {(^i)t7} as video and {(/i)^.j as image (reveals variety) i = 1,2,3, ...N,j = 1,2,3, ...M, where M is the total time points of observations ( reveals velocity), N is total registered users, large in numbers, on a web portal ( reveals volume).
2. Symbol T is used for text, V for video, and I for image.
IV. Parameters
<Xext= (1)
^X ldeo= N-1№i(V^ (2)
i^J lmaae=N-1№1(1^ (3)
Symbols C(Tn, C(j\ C(n are coefficients of variation shown in (7), (8), (9) and pTV, pvl, plT are correlation coefficients of respective pair of populations. Also pab = pba holds for any two pair of variables a and b. Whatever follows hereunder, the t1 used for time occasion j=1; tj for jth time occasion. Some other symbols in use are:
(ST(j))text =1-iZli[TTdtj-TT.)tj]2 = S2T(j) at tj (4)
(Svi))video=lh^^-1[TV~)^i-TV^:)t;\2 = SVW at tj (5)
] = Sf" at t] (6)
The (4), (5), (6) show the variability factor of T, V and I with respect to their means (1), (2), (3). Moreover equations (7), (8), (9) are derived as ratio of (4), (5), (6) with (1), (2), (3).
(CrXxt= [^/(T-)^ C? at tj (7)
(Cnideo = №/^ = C? at tj (8)
(Cnmage = №})/(J-)tj]= C? at tj (9)
V. Sample Selection using Double Sampling
Consider figure 4.1 and figure 4.2 where a primary large sample n (n < N)is drawn by random sampling without replacement and second sample of size n (n ' < n ) is drawn in either of the following manners:
Case I: second sample as a sub-sample.
Case II: second sample as independent to primary sample.
Mean of primary sample n' at jth point of time are:.
(r. )text = (n T (rf)] = (T. ™ ) at tj (10)
(V-)videg = (n J" [li*. (O] = (T-W ) at tj (11)
(T.)image = (nV [ULi (4°°)] = (T-™ ) at tj (12)
The (10),(11) and (12) help to have an idea about the unknown (1), (2), (3) in rough manner with low cost and time effort. Means based on n are not very accurate because role of primary sample n is just to provide a guess of (1), (2), (3). To get precise estimate, second sample n (n ' ' < n ) is drawn and appropriate methodologies are used on accurate data of n ' ' to obtain
sample means as under :
(T ■(j))aa = (n r[&i(0]
(I tM&iCO]
= (T. ' ' (J)) at tj -(V. ''&) at tj = (T ' at tj
(13)
(14)
(15)
The (13), (14), (15) are used to estimate unknown (1), (2), (3). Moreover, sample mean squares on n ' ' are:
(s.2 n^-^KO-OT '"") Î
2 ' ' (j)
-S,
_ ç2 ' '(])
at L
at t:
-S,
2 ' ' (j)
at L
Sample coefficients of variations are:
(C <j>)]
- (C'')!') at tj = (Cat tj
= (C'at tj
(16)
(17)
(18)
(19)
(20)
(21)
The symbols pTy , pVJ, p, T are used for correlation coefficient in population while p TV , p
V,l'
p j T are used for the same purpose but on data of n called as sample estimate of correlation
coefficient.
VI. Double Sampling based Methods of Estimation
Let Ei, E2, E3 are double sampling based estimation strategies for estimating unknown big-data population parameters (1), (2), (3) respectively. At jth point of time tj , they are as under:
wi«-^. Vv~ "j>)
(£,">) -[I■ 'J>((T. \JT.' ■'")
v ' image \ LJ /
Equations (22), (23), (24) are in accordance with references [24][25][26][27] extended for set-up of big data. These are logically formulated such that in (22) the T is variable of main interest while V is support variable correlated to T since larger T provides increment in V. Therefore, with the help of V, a better estimate of T could be obtained. Similar logical justifications are used for formulation of (23) and (24). The EI ,E2, E3 are biased for estimation of (1), (2), (3) because E[Em] ^ (T. >text or ('V.or (I.>image for m=1,2,3 where E[.] denotes expectation of estimate.
(22)
(23)
(24)
video
The general form of mean squared error (MSE) is describe below for 9 to be an estimator of true value 9.
M(9) - E[B - 8]2 where 9 corresponds to £j0>,i=1,2,3 sequentially while 9 are (1), (2), (3) respectively.
Mean squared error under ,Case I ,are (see references [24] [25] [26] [27] ):
v
[MSE (E(S) ] = [(T. )2}] [(V20)<P + ((VofW - (V0f)r) - 2 {orf - (Vii)^}] (25)
textJj
[MSE (E2j))videol = [<y. )2tj] [(V20W + ((VofW - (vOf)™) - 2 {(ViiW - (Vli)^}] (26)
[MSE (E^)^ = [(T.)2tj] [(Yto)^ + (( Vof)() - (Kf)^) - 2 {(YiD^ - (Vi)^}] (27)
Mean squared errors, under case II, are (see references [24][25] [26][27]):
[MSE (E^^J^ = [(r.)2tj] [orf + ((V02)? + (V02)?) - 2(Vii)^] (28)
fe(j)) ] = [(V.)t] [(V,n)(p + ((Vm)(p +
' video*¡j
[mSE (^'^J^ = [(T^] [(Vto)?' + (Wot)? + (Vot)((J)) - 2(Vii)\»] (30)
Using expectation E[. ]of sample mean, following are expressions up-to first order of approximations (see references [24][25] [26][27]):
[MSE (E*P) ] = [TV:^] [TV20)? + (TV02)V!) + ^ - 2TVU)V)] (29)
(VT^T = E ' ^ ) - TT: It} [(V: ' ^ ) - TV: ), ^ ]
= prrvN~-^[icT}))tJ[(C^^))vldS (31)
V
= E [[(r."T^ ) - TT: )tj}" [(V.yi ) - TV: ),/]
= prrvN~-t[(cT}))tJ[(C^^))vldJ (32)
iTr,„0> im
"V
(yTRX = E [{(№ ) - TV )tj}q [(L'M ) - TL)tj}m]
= pVl Nn [(ClVJ))video] [(CT))image] (33)
(vT^))v = E[[(vrrTn)-TV:)tj}q[(T^:Tn)-T1^.)tj}m]
= pV'NNnr[(CVJ)) video] [(ClJ)\mage] (34)
(vTm)) , = E [[(r" ) - TV ) tj}q [(T:^ ) - TT:
= prTNN-n[(CTn)lmage]T[(CT]))tJm (35) (V Tim)., = E [[(':"Ti) ) - TLX} [(T.™ ) - TT:)tj}m]
= p,T Nn [(C/ )images] [(Ct XeJ (36)
where q = 0,1,2 m = 0,1,2 and r = 1 if q = p = 1 else r = 0.
The pooled estimates, based on sample n" , over M different time points (occasions) are :
[TEi)teXt ] = Uf-1 Wjt (Elj))text ,WiT=± (37)
[TE2)video]l=Y!j-lWjV №) ,WjV=1 (38)
v ' video M
[TE3)image\ = Uf-1 Wj, (e^) , Wj, = 1 (39)
1 v ' image M
The (37), (38), (39) are weighted average over M occasions of E1 ,E2, E3 under case I and same is derived for case II in (40), (41), (42).
[(Ei)text L =Zf=lWj
-■Utext J„ _ ¿-¡j = i'vjT
= Y>Uwiv
[(El)vid
№)image]n = T,Uwj> (F3J))
video
wjT =-
11 M iv M
image
, W„ = -
11 M
(40)
(41)
(42)
The pooled mean squared errors (MSE) on M points of time also have weighted sum shown in (43) to (48) for Case I and II.
[MSE(E-)text]l =Y™=iWfT MSE(E(j))text (43)
[MSE(E2)video]l = Ylf=i W?v MSE (Ei(i))Video (44)
[MSE(E3)image\ = Zf=i Wj2 MSE(E(3j))image (45)
[MSE(E-)text]u = Ylf=i W}2T MSE(E-j))text (46)
[MSE(E2)video]u = Y!f=- W}2V MSE (E2j))video (47)
[MSE(E3)image]u = Zy=iWj MSE(E(3j))image (48)
The 95% confidence interval, in general, is defined for two estimated numbers a , b' in probability sense denoted as P[. ] like P[a < True Value <b ] = 0.95 . It is explained as estimate a, b obtained from sample, there is 95% chance that a , b will catch (predict) the true value. More explicitly, the 95% confidence interval is computed as P[sample mean ± 1.96^standard error] = 0.95 (see [25]).
For Case I, the confidence intervals (CI) are in (49), (50) and (51).
P[(Ei)text - 1.96^[MSE(Ei)text]i , (Ei)text + 1.96^[MSE(Ei)text]i ] P[(E2)video - 1.96 ^[MSE(E2)video]l , (E2)video + 1.96 ^MSE^X^ ]
P
(E3)
3Jimage
-1.96 ^[MSE(E3)image]l, (E3)image + 1.96 MSE fc)^^
=0.95 (49) =0.9 (50) =0.95 (51)
For second case II, CI are expressed in (52), (53) and (54).
P[(Ei)text - 1.96j[MSE(Ei)text]u ,
P[(E2) video - 1.96 ^[MSE(E2)Video]\\ (E3)image - 1.96 J[MSE(E3)image]u -
P
(Ei)text + 1.96j[MSE(Ei)text]u ] (ß2)video + 1.96 J[MSE(E2) video^U ] (E3)image + 1.96 l[MSE(E3)image]u
=0.95 (52) =0.95 (53)
=0.95 (54)
I. Population Description
For calculation and comparison, in order to avoid complexity, a small population of size N=100 is considered whose detail is in annexure A. Descriptive statistics of the population as per (1), (2), (3), (4), (5), (6), (7), (8) are in table 1 calculated at six points of time t± to t6.
Table 1: Descriptive statistics of population at six points of time (users are the same)
tlr N=100 [T.]ti =74.14 [V.]ti =105.3 [T.]ti =145.07 pTV(1) =0.7 Pv,,(1) =0.8 P,,t(1) =0.7
S2(1) =1537.04 S2(1) =3756.03 S2(1) =6784.69
c(1) =0.53 C^ =0.58 C(1) =0.57
t2, N=100 [T.]t7 =67.7 \y.]t, =98.13 [T.]t7 =226.18 Ptv{2) =0.6 Pv/2) =0.7 P,,t(2) =0.5
S2(2) =1365.71 S2(2) =3501.81 S2(2) =16979.73
C(2) =0.55 C(2) =0.60 C,(2) =0.58
t3, N=100 [T.]t3 =125.92 [F.]t, =137.29 [T.]t3 =362.74 pTy(3) =0.5 Pv/3) =0.8 P,,t(3) =0.7
S2(3) =4212.01 S2(3) =7083.59 Sf(3) =42405.57
c(3) =0.52 c(3) =0.61 C!(3) =0.57
t4, N=100 [T.]u =110.79 =144.05 [T.]u =142.45 Pt,v(4) =0.7 Pv/4) =0.8 p,/4) =0.6
S2(4) =2382.75 S2(4) =5670.83 Sf(4) =7309.01
C(4) =0.44 44) =0.52 c(4) =0.60
t5 , N=100 [T.]t5 =148.92 [7.]^ =236.51 [T.]t5 =257.97 pTy(5) =0.5 Pv/5) =0.8 p,/5 =0.5
S2(5) =7393.63 S2(5) =15047.95 Sf(5) =17480.67
C(5) =0.58 C(5) =0.52 C,(S) =0.51
W5T =0.167 W5V =0.167 W5I =0.167
t6 , N=100 [T.]t6 =173.5 =308.78 [T.]t6 =306.78 pTy(6) =0.7 Pv/6) =0.8 pITi6 =0.6
S2(6) =4997.55 S2(6) =29899.47 S?(6) =29761.89
C(6) =0.41 C(6) =0.56 C(6) =0.56
Primary sample of size n = 40 is drawn from N=100 to calculate the mean size of unknown parameters [T.]t. , [V.~]t. and I.t over six points of time. This sample is used to have a guess value of the population parameter to use as supportive information. Calculation of sample means on n =40 is in table 2.
Table 2: Sample-based mean estimates at six occasions (n =40 primary sample)
At time tx (occasion one) n' = 40 [T.']t =83.58 [V. ' ]t =114.85 I. 't =152.85
At time t2 (occasion two)n' = 40 [T.']t7 =75.95 [V. ]t =107.12 T. 't =244.53
At time t3 (occasion third) n' = 40 [T.']t =139.07 [v. \ =1535 II =382.25
At time t4 (occasion four) n' = 40 [T.']t =120.95 \y. \ =150.9 11 't4 =157.97
At time ts (occasion five) n' = 40 [T.']t =174.12 [V. ']t =274.93 II =281.48
At time t6 (occasion six) n' = 40 [T.']t6 =181.38 [V. '] =362.38 V.'t =337.05
A second sample of size n' =10 is taken for estimation of means on variable of main interest over six points of time. Estimates on n are in table 3 for strategy under case I. Similarly, for strategy under case II, the calculations are in table 4. The pooled estimate of text-data, video-data and images-data using equation (22), (23), (24) are in table 5 and table 6 along with MSE calculation using (25) to (30).
Table 3: Result of sample-based calculation at six occasions (n '=10, first sample) under Case I [eq. (13)-
(21) and (25)-(27)]
[T.' ']t —99.30 [V.' ']t —118.30 [I.' ']t —148.90 MSE_Text=265.51 PTV(1) —0.4 P„(1) —0.4 P,T(1) —0.2
tl S2T' '(1) —1220.90 ST' (1) —5572.46 S2' (1) —7921.43 MSE_Video=566.86
n — 1U CT(1) —0.35 cv(1) —0.63 C, (1) —0.60 MSE_Image=768.16
[T. ' ']t —84.80 [VT. ']t —102.40 [I' ']t —238.20 MSE_Text=424.44 „ tT (2) n a pTV —-0.3 Pv,(2) —0.3 P,T(2) —0.1
t2 ST ' (2) —1230.18 ST'(2) —4394.04 S2' '(2) —20243.96 MSE_Video=490.99
n — 1U CT(2) —0.41 Cv(2) —0.65 C, '(2) —0.60 MSE_Image=2348.19
[T.' ']t —144.40 [V.'' ] —165.40 L [1.' ']t —372.40 MSE_Text=1197.09 p'TV(3) —-0.4 P^(3) —0.4 PIT(3) —-0.0
,*3 ST ' (3) —3185.16 ST'(3) —10899.60 S2'T(3) —49400.49 MSE_Video=1106.98
n — 1U c'(3) —0.39 cv(3) —0.63 C, '(3) —0.60 MSE_Image=6106.08
[T. ' ']t —129.80 [V.'' ] —132.20 L K [1.' ']t —147.10 MSE_Text=98.56 ,, (4) pTV —0.6 P^,(4) —0.7 PIT(4) —0.6
> ST ' (4) —1307.51 ST' '(4) —2028.84 S2'T(4) —8529.66 MSE_Video=345.68
n — 1U CT(4) —0.28 Cv(4) —0.34 C[ '(4) —0.63 MSE_Image=523.50
[T.' ']t —193.20 [V.'' ] —307.40 L Jt5 [1.' ']t —323.30 MSE_Text=752.35 Ptv(S) —0.4 Pv,(S) —0.8 pIT(S) —0.0
,*s ST ' (s) —9574.18 ST' '(s) —8915.60 S2' '(s) —13703.34 MSE_Video=520.64
n — 1U CT(s) —0.51 Cv(s) —0.31 C, '(s) —0.36 MSE_Image=3214.32
T.''t —212.00 L6 [V.''] —412.60 L ^6 [1.' ']t —276.10 MSE_Text=478.96 Ptv(6) —0.7 P^,(6) —0.2 pIT(6) —0.6
te ST ' (e) —2686.44 ST' '(e) —38532.04 S2' '(e) —27937.66 MSE_Video=6424.23
CT(e) —0.24 cTT(e) —0.48 CTXe) —0.61 MSE_Image=1875.89
Table 4: Result of sample-based calculation (n '=10, first sample) under Case II [eq. (13)-(21) and (28)-(30)]
ti n'' — 10 [T.' ']t —99.30 [V.' ']t —118.30 [I.' ']t —148.90 MSE_Text=355.53 Ptv(1) —0.4 P^ —0.4 pIT(1) —0.2
ST' (1) —1220.90 ST' (1) —5572.46 S2' (1) —7921.43 MSE_Video=654.93
CT(1) —0.35 Cv(1) —0.63 q (1) —0.60 MSE_Image=820.38
t2 n'' — 10 [Tt ' ']t —84.80 [v.. ']t —102.40 [I.' ']t —238.20 MSE_Text=532.38 „ ' ' (2) na pTV —-0.3 Pv,(2) —0.3 ' ' (2) n 1 pIT —0.1
ST'(2) —1230.18 ST'(2) —4394.04 S2'X2) —20243.96 MSE_Video=566.21
CT(2) —0.41 Cv(2) —0.65 q (2) —0.60 MSE_Image=2599.03
ts n'' — 10 [Tt ' ']t —144.40 [v.. ']t —165.40 [I.' ']t —372.40 MSE_Text=1503.78 PtV3 —-0.4 P^ —0.4 pIT(3) —-0.0
ST'(3) —3185.16 ST'(3) —10899.60 S2'X3) —49400.49 MSE_Video=1278.36
CT(3) —0.39 Cv(3) —0.63 q (3) —0.60 MSE_Image=6755.85
t* n'' — 10 [Tt ' ']t —129.80 [V.' ']t —132.20 [I.. ']t —147.10 MSE_Text=124.07 ,, (4) pTV —0.6 P^,(4) —0.7 ' ' (4) n £ p[T —0.6
ST'(4) —1307.51 ST'(4) —2028.84 S2'X4) —8529.66 MSE_Video=481.63
CT(4) —0.28 Cv(4) —0.34 C[ '(4) —0.63 MSE_Image=499.85
ts n'' — 10 [Tt ' ']t —193.20 [v.. ']t —307.40 [I.' ']t —323.30 MSE_Text=783.31 p'TV(S) —0.4 P^,(S) —0.8 p'/T(s) —0.0
sT''(s) —9574.18 S2''(s) —8915.60 sf'(s) —13703.34 MSE_Video=650. 11
c:;(s) —0.51 C^(s) —0.31 C;'(s) —0.36 MSE_Image=4012.67
te n'' — 10 [T—212.00 [V.'']tk —412.60 [L'']t6 —276.10 MSE_Text=678.99 p^(e) —0.7 pV,(e) —0.2 p'/T(e) —0.6
ST''(6) —2686.44 s2"(e) —38532.04 S2"(e) —27937.66 MSE_Video=7951.35
C!;(e) —0.24 C'(e) —0.48 c; '(e) —0.61 MSE_Image=1816.54
Table 5: Result of sample-based pooled calculation at six occasions under case I using one sample
n'' = 10 [(E1)teXt]l =126.06 [(E2)video ] =195.47 [(Es)image ] =258.18
[MSE(E1)teXt]i =40.93 [MSE(E2)video]i =44.61 [MSE(E3)image\ =304.28
CI (113.52-138.59) (182.38-208.56) (223.99-292.36)
True Value 116.82 171.62 240.19
Length (CI) 25.07 26.18 68.37
Table 6: Result of sample-based pooled calculation at six occasions under case II using one sample
n'' = 10 [(Ei)teXt]u =126.06 [(E2)video ]n =195.47 [(Es)image =258.18
[MSE(Ei)teXt]u =47.85 [MSE(E2)video]n =54.95 [MSE(E3)image]u =339.36
CI (112.50-139.61) (181.21-209.99) (222.04-294.31)
True Value 116.82 171.62 240.19
Length (CI) 27.11 28.78 72.27
Table 5 and table 6 contain one-sample combined estimates, pooled to six occasions, on the variable of main interest (T or V or I). The MSE under case I is smaller than Case II. The length of confidence intervals in case I is lower showing efficiency over case II.
II. Practically Difficulty
The confidence intervals (CI) in table 5 and table 6 are sample dependent therefore difficult to conclude uniquely. Reason behind is that one can draw many samples of size n'' from n' (total n'c ,,) and many from N (total Nc ,,). Each time the average of sample estimate fluctuates and accordingly variation occur in predicted value of confidence intervals. Look at table 6 , [(E2)video ]„ =195.47, CI = (181.21-209.99) where CI does not catch the true value 171.62 which is evidence of difficulty. To cope up this, a new simulation procedure is proposed in section 6.2 based on many samples who ultimately determines the single-value of lower and upper limits.
III. Simulation Procedure Algorithm for Double Sampling
In order to get single-value of limits of 95% confidence interval , a simulation procedure is proposed:
Step 1: Draw a primary random sample of size n'. Step2: Draw second sample as under
Case I: as sub-sample of n' Case II: as independent sample from N Step 3: Compute lower limit (say 'a') and upper limit (say 'b') of confidence interval(CI) using
each sub-sample (or independent sample) , where 95% confidence interval is Prob. [a < true value < b] = 0.95. It is like table 3 and table 4 form t1 to t6 using equations (49) to (54).
Step 4: Repeat step 2 and step 3 for k times (k=200 ).
Step 5: Compute the Less Than Type (LTT) and More Than Type (MTT) cumulative probabilities
by constructing class-intervals for 'a' and 'b' separately for each CI. Step 6: Plot data of step 5 of cumulative probabilities (on y-axis) over class-intervals (on x-axis) and draw two graphs. A perpendicular drawn from point of intersection of two graphs ,on the x-axis, determines single-point of simulated value of lower limit 'a' (and corresponding upper limit 'b') of confidence interval for unknown parameters to be predicted. Express outcomes in tabular presentation like tables 8,9,10 and table 11.
IV. Features of proposed simulation procedure for double sampling:
(a) It is based on k-samples, where K may be as large possible.
(b) It considers cumulative probabilities which is ratio of cumulative frequency to total frequency.
(c) It takes into account the perpendicular drawn from point of intersection of the cumulative probability curves which always remain unique for lower as well as upper limit.
(d) It eliminates problem discussed in section I.
V. Demonstration of Simulation procedure:
Out of k= 200 samples, after calculation of confidence intervals on each sample, let f be frequencies of class intervals — xi+1 relating to lower limit of CI , such that 2fi = k = 200 holds. Probabilities are pt = f/k , i = 1,2,3,...
Table 7. Demonstration of Simulation Procedure
Class Intervals (for lower limit 'a') Frequencies (Occurrence of estimate 'a ') Probabilities LTT (Step 5) MTT (Step 5)
fi Pi = fi/k Ci = Pi c' = i
f2 P2 = f2/k C2 = Pi+P2 C2 = i-Pi
K3-K4 Î3 Ps = fs/k C3 = Pl+P2+P3 C3 = i-Pi-P2
K4-K5 U Pa = f4/k Ca = Pl+P2+P3 c' = i-pi-p2
--- + P4 -Ps
Total ^fi = k = 200 Y*<=1
Plot Ci and C[ over class-interval on a graph to find point of intersection of two curves ( step 6). Draw a perpendicular on X-axis from point of intersection which uniquely determine single-value of 'a'.
VI. Application of proposed Simulation Procedure
Figure 7-78 provide the simulated single-valued lower limit 'a' and simulated single-valued upper limit 'b' of confidence intervals as an application.
Graphs (fig. 7-42) are under case I on ti to t6 . Note that SCI symbolized for simulated confidence interval, TD text dataset, YD video data and ID indicates image dataset in all hereunder:
Lower i_irr»it of Text Dolo at Occasion One under case I Upper Limit of Text Data at Occasion One under case I
60 es TO 75 SO 83 BO SO lOO HO 120 130
Figure 7: At tv Case I, Lower limit of text data at (a=65.30) Figure 8: At tv Case I, Lower limit of text data at (b=101.56)
The figure 7 is calculates value a=65.30 (perpendicular from intersection point) which is the lower limit of simulated confidence interval of text dataset at i^under case I. Similarly, figure 8 has upper limit of simulated confidence interval of text data at ti under case I whose perpendicular from point of intersection is b=101.56.
Figure 9: At t v Case I, Lower limit of video data at (a=96.04) Figure10: At t v Case I, Upper limit of video data at (b=137.79)
Figure 9 provides value a=96.04 (perpendicular from intersection point) as lower limit of simulated confidence interval of video dataset at ^ under case I. Likewise, figure 10 shows upper limit of simulated confidence interval of video data at ti under case I where perpendicular from intersection point is at b=137.79.
Figure 11: At tv Case I, Lower limit of image data at (a=109.36) Figure 12: At t v Case I, Upper limit of image data at (b=197.01 )
Figure 11 reveals the value a=109.36 (perpendicular from intersection point) as lower limit of SCI of image dataset at t± under case I. Figure 12 displays upper limit of SCI of image data at occasion one, under case I which is b=197.01.
Figure 13: At t2, Case I, Lower limit of text data at (a=53.08) Figure 14: At Case I, Upper limit of text data at (b=96.30)
Figure 13 is at time showing value a=53.08 , under case I , and figure 14 is similar for upper limit b=96.30 under case I at
Figure 15: At t2, Case I, Lower limit of video data at (a=89.56) Figure 16: At t2, Case I, Upper limit of video data at (b=128.33)
Figure 15 and 16 reveal value a=89.56 as lower and b=128.33 as upper at t2, for case I.
iper Limit of imaae Data at Occasion Tvwo under
/ 1 s 1
1 1
37 ►a-*
Figure 17: At t2, Case I, Lower limit of image data at (a=164.44) Figure 18: At t 2, Case I, Upper limit of image data at (b=320.34)
Figure 17 and 18 reflect towards value a=164.44 and b=320.34 at t2 case I.
Figure 19: At t3, Case I, Lower limit of text data at (a=97.39) Figure 20: At t3, Case I, Upper limit of text data at (b=176.44)
Figure 19 and 20 reveal for a=97.39 and b=176.44 at t2 case I.
Figure 21: At t3, Case I, Lower limit of video data at (a=125.06) Figure 22: At t3, Case I, Upper limit of video data at (b=187.50)
The figure 21 and 22 have a=125.06, b=187.50.
Figure 23: At t3, Case I, Lower limit of image data at (a=278.48) Figure 24: At t3, Case I, Upper limit of image data at (b=487.83)
Values a=278.48 and b=487.83 are in figure 23 to 24. Similar are in figure 25 to 78 under case I and case II for T, V and I over time t1 to t6. Figure caption from 25-78 are self explanatory and reveal auto interpretation as above
Figure 25: At t4, Case I, Lower limit of text data at (a=93.22) Figure 26: At t4, Case I, Upper limit of text data at (b=146.75)
Figure 27: At t4, Case I, Lower limit of video data at (a=129.03) Figure 28: At t4, Case I, Upper limit of video data at (b=175.43)
Figure 29: At t4, Case I, Lower limit of image data at (a=112.47) Figure 30: At t4, Case I, Upper limit of image data at (b=204.60)
Figure 31: At t5, Case I, Lower limit of text data at (a=130.39) Figure 32: At t5, Case I, Upper limit of text data at (b=214.44)
Figure 33: At t5, Case I, Lower limit of video data at (a=236.75) Figure 34: At t5, Case I, Upper limit of video data at (b=314.93)
Figure 35: At t5, Case I, Lower limit of image data at (a=200.37) Figure 36: At t5, Case I, Upper limit of image data at (b=368.03)
Figure 37: At t6, Case I, Lower limit of text data at (a=142.73) Figure 38: At t6, Case I, Upper limit of text data at (b=216.89)
/1
1 1
1 1
Figure 39: At t6, Case I, Lower limit of video data at (a=285.58) Figure 40: At t6, Case I, Upper limit of video data at (b=461.31)
Figure 41: At t6, Case I, Lower limit of image data at (a=244.35) Figure 42: At t6, Case I, Upper limit of image data at (b=431.30)
The following Fig. 43-78 are showing time point wise ti to t6 the simulated results under
case II.
Figure 43: At t1, Case II, Lower limit of text data at (a=63.72) Figure 44: At t1, Case II, Upper limit of text data at (b=103.49)
Figure 47: At t1/ Case II, Lower limit of image data at (a=106.98) Figure 48: At t1r Case II, Upper limit of image data at (b=199.75)
Figure 49: At t2, Case II, Lower limit of text data at (a=50.86) Figure 50: At t2, Case II, Upper limit of text data at (b=98.45)
Figure 51: At t2, Case II, Lower limit of video data at (a=88.27) Figure 52: At t2, Case II, Upper limit of video data at (b=129.26)
Figure 53: At t2, Case II, Lower limit of image data at (a=158.51) Figure 54: At t2, Case II, Upper limit of image data at (b=324.26)
Abdul Alim, Diwakar Shukla
DOUBLE SAMPLING BASED PARAMETER ESTIMATION IN BIG RT&A, No 2(62) DATA AND APPLICATION IN CONTROL CHARTS_Volume 16, June 2021
Figure 59: At t3, Case II, Lower limit of image data at (a=271.49) Figure 60: At t3, Case II, Upper limit of image data at (b=491.64)
Figure 61: At t4, Case II, Lower limit of text data at (a=90.28) Figure 62: At t4, Case II, Upper limit of text data at (b=150.54)
Figure 63: At t4, Case II, Lower limit of video data at (a=125.51)
„ — - -
/l\
1 ^ - 1
/ / 1 1
o
Figure 64: At t4, Case II, Upper limit of video data at (b=178.96)
so so loo 120 i-io- i«o loo leo zoo aao j*o aoo
Figure 65: At t4, Case II, Lower limit of image data at (a=111.75) Figure 66: At t4, Case II, Upper limit of image data at (b=205.07)
Low*r Limn or Tout Data at Occasion Flw* undar ca»« I ■ U t> r>*r I Limit of T**t Data at Occasion Flva unda-r ca»« H
i»o uo mo too lao xoo lao 200 220 -ao 2«o
Figure 67: At t5, Case II, Lower limit of text data at (a=129.04) Figure 68: At t5, Case II, Upper limit of text data at (b=216.24)
Figure 69: At t5, Case II, Lower limit of video data at (a=235.86) Figure 70: At t5, Case II, Upper limit of video data at (b=318.93)
Figure 71: At t5, Case II, Lower limit of image data at (a=191.57) Figure 72: At t5, Case II, Upper limit of image data at (b=376.60)
Figure 73: At t6, Case II, Lower limit of text data at (a=138.04) Figure 74: At t6, Case II, Upper limit of text data at (b=221.40)
Figure 75: At t6, Case II, Lower limit of video data at (a=278.96) Figure 76: At t6, Case II, Upper limit of video data at (b=469.30)
13C !W »O JOO 350 <00 vljij aa toa «Q soa
Figure 77: At t6, Case II, Lower limit of image data at (a=241.10) Figure 78: At t6, Case II, Upper limit of image data at (b=434.08)
VII. Tabular Presentation for summarization (part of Step 5): After simulation is over, outcomes of all above graphs are summarized in table 8 9, 10 and 11.
Table 8: Summary of simulated CI over t1 to t6 under case I ( based on figures 7-42 )
Time-Occasions Dataset Figures Lower Limit Figures Upper Limit True Value
h T Figure 7 a=65.30 Figure 8 b=101.56 74.14
V Figure 9 a=96.04 Figure 10 b=137.79 105.3
I Figure 11 a=109.36 Figure 12 b=197.01 145.07
t2 T Figure 13 a=53.08 Figure 14 b=96.30 67.7
V Figure 15 a=89.56 Figure 16 b=128.33 98.13
I Figure 17 a=164.44 Figure 18 b=320.34 226.18
ts T Figure 19 a=97.39 Figure 20 b=176.44 125.92
V Figure 21 a=125.06 Figure 22 b=187.50 137.29
I Figure 23 a=278.48 Figure 24 b=487.83 362.74
t4 T Figure 25 a=93.22 Figure 26 b=146.75 110.79
V Figure 27 a=129.03 Figure 28 b=175.43 144.05
I Figure 29 a=112.47 Figure 30 b=204.60 142.45
ts T Figure 31 a=130.39 Figure 32 b=214.44 148.92
V Figure 33 a=236.75 Figure 34 b=314.93 236.51
I Figure 35 a=200.37 Figure 36 b=368.03 257.97
te T Figure 37 a=142.73 Figure 38 b=216.89 173.5
V Figure 39 a=285.58 Figure 40 b=461.31 308.78
I Figure 41 a=244.35 Figure 42 b=431.30 306.78
Table 9: Summary of simulated CI over t1 to t6 under case II (based on figures 43-78 )
Time-Occasions Dataset Figures Lower Limit Figures Upper Limit True Value
h T Figure 43 a=63.70 Figure 44 b=103.49 74.14
V Figure 45 a=94.95 Figure 46 b=139.18 105.3
I Figure 47 a=106.98 Figure 48 b=199.75 145.07
h T Figure 49 a=50.86 Figure 50 b=98.45 67.7
V Figure 51 a=88.27 Figure 52 b=129.26 98.13
I Figure 53 a=158.51 Figure 54 b=324.26 226.18
ts T Figure 55 a=92.06 Figure 56 b=180.96 125.92
V Figure 57 a=124.15 Figure 58 b=188.82 137.29
I Figure 59 a=271.49 Figure 60 b=491.64 362.74
t4 T Figure 61 a=90.28 Figure 62 b=150.54 110.79
V Figure 63 a=125.51 Figure 64 b=178.96 144.05
I Figure 65 a=111.75 Figure 66 b=205.07 142.45
t5 T Figure 67 a=129.04 Figure 68 b=216.24 148.92
V Figure 69 a=235.86 Figure 70 b=318.93 236.51
I Figure 71 a=191.57 Figure 72 b=376.60 257.97
t6 T Figure 73 a=138.04 Figure 74 b=221.40 173.5
V Figure 75 a=278.96 Figure 76 b=469.30 308.78
I Figure 77 a=241.10 Figure 78 b=434.08 306.78
Table 8 and 9 reflect scenario of corresponding true values within the predicted range which is beauty of method.
Table 10: Pooled simulated confidence interval average result over t1 to t6 undercase I [Using eq. (37)-(39)]
Time-Occasions Dataset Lower Limit Upper Limit True Value Length
T a=96.67 b=158.17 116.82 61.5
V a=160.00 b=233.67 171.62 73.67
I a=184.50 b=334.50 240.19 150
Table 11: Pooled simulated confidence interval average result over t1 to t6 under case II [Using eq (40).(41),(42)]
Time-Occasions Dataset Lower Limit Upper Limit True Value Length
T a=93.67 b=161.33 116.82 67.66
t1-t6 V a=157.33 b=236.83 171.62 79.5
I a=179.67 b=338.17 240.19 158.5
VII. Discussion
In view to outcomes of table 8, 9, 10, 11 , one can observe in table 8, at t1, the true values are 74.14 for text-data, 105.3 for video-data and 145.07 for image-data whereas simulated confidence intervals are(65.30 - 101.56)T, (69.04 - 137.79)v and (109.36 - 197.01), respectively. All true values are within the simulated confidence intervals. Similarly, at t2, the true value are 67.7 for text-data, 98.13 for video-data and 226.18 for the image-data while simulated CI are (53.08 — 96.30)T, (89.56 - 128.33)v and (164.44 - 32034),. At t3, true values are 125.92 for T, 137.29 for V and 362.74 for I while corresponding CI are (97.39 - 176.44)T, (125.06 - 187.50)^ and (278.48 -487.83), . All true values found well within the simulated confidence intervals under case I.
Observing t4, in table 8, simulated CI are(93.22 - 146.75)T, (129.03 - 175.43)v and (112.47 -204.60), against true values 110.79, 144.05 and 142.45. These are also catching the truth. At t5, true values are 148.92 for T, 236.51 for V and for I 257.97 while CI are (130.39 - 214.44)T, (236.75 - 314.93)v and (200.37 - 368), At t6, the true are 173.05, 308.78 and 306.78 whereas the SCI are predicting accurately to true values being within the range (142.73 - 216.89)T, (285.58 -461.31)v and (244.35 - 431.30), respectively for the case I.
Looking at estimation by Double sampling strategy under case II, true values are same as earlier but the confidence intervals, at t1, are (63.70 - 103.49)T, (94.95 - 139.18)v and (106.98 -199.75), showing all true values within the confidence intervals. Similarly, at t2, the confidence interval are (50.86 - 98.45)T, (88.27 - 129.26)v and (158.51 - 324.26),, at t3, they are (92.06 -180.96)T, (124.15 - 188.82)v and (271.49 - 491.64),, at t4, we have (90.28 - 150.54)T, (125.51 -178.96)v and (11.75 - 205.07),, at t5, one can find as (129.04 - 216.24)T, (235.86 - 318.93)v
Abdul Alim, Diwakar Shukla
DOUBLE SAMPLING BASED PARAMETER ESTIMATION IN BIG RT&A, No 2(62)
DATA AND APPLICATION IN CONTROL CHARTS_Volume 16, June 2021
and (191.57- 376.60),, and lastly, at t6, their presence are as (138.04 - 221.40)T, (278.96-469.30)v and (241.10 - 434.08), respectively.
In the context to table 10, the single valued pooled simulated confidence intervals, under case I, are (96.67 - 158.17)T, (160.00 - 233.67)v and (184.50 - 334.50), with respect to average true values 116.82, 171.62 and 240.19. The length of simulate confidence intervals, at average level, are 61.5, 73.67, and 150 in sequence for T, V and I.
Likewise, table 11 contains same under case II who are (93.67 - 161.33)T, (157.33 - 236.83)v and (179.67 - 338.17), Lengths of confidence intervals, at average level ,are 67.66, 79.5, and 158.5 respectively.
VIII. Comparison and Efficiency
Define Relative CI Length Measure (RCILM) as
(CI length)case „
RCILM =
x 100
(CI length) case I _
Table 12: Relative CI Length Measure using table 10 and table 11 (under simulation)
Dataset RCILM
T 110.01%
V 107.91%
I 105.66%
Table 12.1: Relative CI Length Measure using table 5 and table 6 (without simulation)
Dataset RCILM
T 108.13%
V 109.93%
I 105.70%
It is observed in table 12, and table12.1, the case I is having a smaller length of confidences than case II consistently in every type T, V and I.
IX. Developing Control Charts using simulated confidence intervals as tools for
managerial decision
Control charts for managerial decision about web-portals, data-centers are one of applications of confidence interval. Consider the theory discussed in section 1 and in figure 6. The graphical trace of CI over t-tto t6 displayed in figure 79 to 90, for the case I and case II , can be used.
Figure 7: CL of Text file-size measures under case I Figure 80: CL of Video file-size measures under case I
Figure 81: CL of Image file-size measures under case I Figure 82: CL of Text file-size measures under case II
600 400 200
0
At
■Video_LL Video UL
n-1-1-1-1-1
2t 3t 4 t 5 t 6
1 l2 l3 l4 l5
Figure 83: CL of Video file-size measures under case II Figure 84: CL of Image file-size measures under case II
Fig. 79-81 reveal Upper Control Limit (UCL or UL) and Lower Control Limit (LCL or LL) of the file-size measures of text-data, video-data and image-data used in communication . Similarly, Fig. 82-84 are showing same application for case II and these are file-size production procss control charts. The simulated value 'a' is LCL(or LL) and simulated value 'b' is UCL( or UL) of the confidence intervals.
Such are helpful for decision making regarding control over size measure of communication files on social media web-portal and ,as a consequence, alert can issue for further infrastructure, resources required to achieve the goal of profit. For example, IT- industry (Servers/Data Center/ hardware/software), if alarmed well before about flowing digital file-size, who is growing fast in big data environment over time, better management can be thought of in timely manner. While file-size measure, if increases exponentially over time then investment in Data Centers urgently needed .
In view to fig. 85-90 , matter of importance is to watch whether same habits of communication of users are maintained ? If at nth point of time tn (n = 1,2,3 the Upper
Control Limit (UCL) or Lower Control Limit (LCL) are crossed in control charts , there is significant evidance exist for change of habits of communication of file-size. At this juncture, the industry owner needs to review decision regarding up-gradation or framing new policy to share memory resources with others in order to maximize profit. Simulated confidence intervals play key role for developing such monitoring.
200
150
100 -Text_ .LCL
50 Text_ UCL
0 1 1 1 1 1 1
At 1t 2t 3t 4 t 5 t 6 l1 l2 l3 l4 l5
300 200 100 0
At
■Video_LCL Video UCL
t1 t2 t3 4t
5t 6t
tt
Figure 85: CL of Text file-size under case I Figure 86: CL of Video file-size under case I
Figure 87: CL of Image file-size under case I Figure 88: CL of Text file-size under case II
300 200 100 0
At
•Video_LCL Video UCL
-1-1—i-1-1—i
t1 t2 t3 4 5t 6
l1 l2 l3 l4 l5
400 300 200 100 0
At
•Image_LCL Image_UCL
~l-1-1-1-1-1
t11 t22 t33 4t4
5t 6t
tt
Figure 89: CL of Video file-size under case II Figure 90: CL of Image file-size under case II
X. Conclusion
On recapitulation, the double sampling approach has been adopted in the content for estimating the population parameter in the setup of big data where volume, variety and velocity characteristics are present simultaneously. The idea of number of registered users on a social networking platform communicating through Text, Video and Image files has been considered over different time span. Estimate of average file size is focused whose growth needs to be monitored over time variations. Estimation strategies have been suggested in the setup of double sampling. When has two approaches as (a) sub-sample and (b) independent sample. Both have been compared and found that the proposed methods capture the true values of the population mean over several occasions (time frame). The merge setup of the average of all occasions also reveals that both strategies (case I and case II) cover the true values. A new simulation algorithm based on double sampling is suggested to obtain a single value estimate of 95% confidence interval whose estimate also predict about the true value. Efficiency comparison of two cases is made through the tool RCILM which shows case I better than case II. The study is useful for a managerial decision since the lower control limit and upper control limit growth can generate an alert for IT-business managers. Control charts predict for the future event to check whether the average values of file sizes at farther occasions are within the UCL or LCL or not. If the control limits violate then re-thinking about IT-business infrastructure may be originated to cope up the future challenges. If the control limits violates then re-thinking about IT-business infrastructure may be originated to cope up the future challenges.
References
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[2] Qiu, Peihua (2017). Statistical process Control Charts as a tool for analyzing Big Data. Big Data and Complex Analysis, Springer Cham, pp. 123-138.
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ANNEXURE A Population N=100
ID T V I T V I T V I T V I T V I T V I
0 10 15 6 13 10 10 18 21 15 20 25 27 31 35 38 41 49 52
1 5 8 7 9 3 11 8 11 17 9 12 13 15 16 18 19 23 24
2 9 12 11 5 7 17 16 17 27 18 23 24 28 32 34 37 44 47
3 8 11 14 16 6 23 14 15 36 15 19 21 24 27 29 32 38 40
4 12 20 18 10 15 29 22 28 45 24 30 32 37 42 46 49 59 62
5 11 16 22 9 11 35 19 22 54 21 27 29 33 37 41 44 52 56
6 13 18 25 11 13 41 23 25 64 26 33 35 40 46 49 53 64 68
7 14 21 29 17 16 47 25 29 73 27 34 37 42 48 52 56 67 71
8 15 23 33 13 18 52 27 32 82 30 38 41 47 53 58 62 74 79
9 17 26 37 15 21 58 30 36 91 33 42 45 52 58 63 68 82 87
10 18 28 40 24 23 64 33 39 101 36 46 49 56 64 69 75 89 95
11 20 31 44 18 26 70 35 43 110 39 49 53 61 69 75 81 97 102
12 21 33 48 19 28 76 38 46 119 42 53 57 66 74 81 87 104 110
13 23 36 51 21 31 82 41 50 128 45 57 61 70 79 86 93 111 118
14 24 38 55 22 33 88 44 53 138 48 61 65 75 85 92 99 119 126
15 26 41 59 24 36 94 46 57 147 51 64 69 80 90 98 105 126 134
16 27 43 62 25 38 100 49 60 156 54 68 73 84 95 103 112 133 141
17 29 46 66 27 41 106 52 64 165 57 72 77 89 100 109 118 141 149
18 30 48 70 28 43 112 54 67 175 60 76 82 94 106 115 124 148 157
19 32 51 74 30 46 118 57 71 184 63 79 86 98 111 120 130 155 165
20 33 53 77 31 48 124 60 74 193 66 83 90 103 116 126 136 163 173
21 35 56 81 33 51 129 62 78 202 69 87 94 108 121 132 142 170 180
22 36 58 85 34 53 135 65 81 212 72 91 98 112 127 138 148 177 188
23 38 61 88 36 56 141 68 85 221 75 94 102 117 132 143 155 185 196
24 39 63 92 37 58 147 71 88 230 78 98 106 122 137 149 161 192 204
25 41 66 96 39 61 153 73 92 239 81 102 110 126 142 155 167 199 212
26 42 68 99 40 63 159 76 95 249 84 106 114 131 148 160 173 207 219
27 44 71 103 42 66 165 79 99 258 87 109 118 135 153 166 179 214 227
28 45 73 107 43 68 171 81 102 267 90 113 122 140 158 172 185 221 235
29 47 76 111 45 71 177 84 106 276 93 117 126 145 163 177 191 229 243
30 48 78 114 46 73 183 87 109 286 96 121 130 149 169 183 198 236 251
31 50 81 118 48 76 189 89 113 295 99 124 134 154 174 189 104 244 258
32 51 83 122 49 78 195 92 116 304 102 128 138 159 179 195 110 251 266
33 53 86 125 51 81 200 95 120 313 105 132 142 163 184 200 116 258 274
34 54 88 129 52 83 206 98 123 323 108 136 146 168 190 206 122 266 282
35 56 91 133 54 86 212 100 127 332 111 139 150 173 195 212 128 273 290
36 57 93 136 55 88 218 103 130 341 114 143 154 177 200 217 135 280 297
37 59 96 140 57 91 224 106 134 350 117 147 158 182 205 223 141 288 305
38 60 98 144 58 93 230 108 137 360 120 151 163 187 211 229 147 295 313
39 62 101 148 60 96 236 111 141 369 123 154 167 191 216 234 153 302 321
40 63 103 151 61 98 242 114 144 378 126 158 171 196 221 240 159 310 329
41 65 106 155 63 101 248 116 148 387 129 162 175 101 226 246 165 317 336
42 66 108 159 64 103 254 119 151 397 132 166 179 105 232 252 171 324 344
43 68 111 162 66 106 260 122 155 406 135 169 183 110 237 257 178 332 352
44 69 113 166 67 108 266 125 158 415 138 173 187 115 242 263 184 339 360
45 71 116 170 69 111 272 127 162 424 141 177 191 119 247 269 190 346 368
46 72 118 173 70 113 277 130 165 434 144 181 195 124 253 274 96 354 375
47 74 121 177 72 116 283 133 169 443 147 184 199 128 258 280 102 361 383
48 75 123 181 73 118 289 135 172 452 150 188 203 133 263 286 208 368 391
49 77 126 185 75 121 295 138 176 461 153 192 207 138 268 291 114 376 399
50 78 128 188 76 123 301 141 179 471 156 196 211 142 274 297 121 383 407
51 80 131 192 78 126 307 143 183 480 159 199 215 147 279 303 127 391 414
52 81 133 196 79 128 313 146 186 489 162 203 219 152 284 309 233 398 422
53 83 136 199 81 131 319 149 190 498 165 207 223 156 289 314 139 405 430
54 84 138 203 82 133 325 152 193 508 168 211 227 123 295 320 145 413 438
55 86 141 207 84 136 331 154 197 517 171 214 231 120 300 326 251 420 446
56 87 143 210 85 138 337 157 200 526 174 218 235 170 305 331 158 427 453
57 89 146 214 87 141 343 160 204 535 177 222 239 175 310 337 264 435 461
58 90 148 218 88 143 348 162 207 545 180 226 244 180 316 343 170 442 469
59 92 151 222 90 146 354 165 211 554 183 229 248 84 321 348 245 449 477
60 93 153 225 91 148 360 168 214 563 186 233 252 28 326 354 180 457 485
61 95 156 229 93 151 366 170 218 572 189 237 256 94 331 360 288 464 492
62 96 158 233 94 153 372 173 221 582 192 241 260 98 337 366 135 471 500
63 98 161 236 96 156 378 176 225 591 195 244 264 102 342 371 202 479 508
64 99 163 240 97 158 384 179 228 600 198 248 268 108 347 377 207 486 516
65 101 166 244 99 161 390 181 232 609 101 252 272 113 352 383 196 493 524
66 102 168 247 80 163 396 184 235 619 104 256 276 114 358 388 199 501 531
67 104 171 251 35 166 402 187 239 628 107 259 280 213 363 394 258 508 539
68 105 173 255 81 168 408 189 242 637 105 263 284 126 368 400 231 515 547
69 107 176 259 102 171 414 192 246 646 113 267 288 131 373 405 237 523 555
70 108 178 262 99 173 420 195 249 656 116 271 292 135 379 411 244 530 563
71 110 181 266 88 176 425 197 253 665 109 274 296 140 384 417 250 538 570
72 111 183 270 106 178 431 200 256 674 85 278 300 145 389 423 256 545 578
73 113 186 273 59 181 437 203 260 683 95 282 304 149 394 428 262 552 586
74 45 188 277 43 183 443 81 263 693 100 113 122 140 158 171 185 221 234
75 116 191 281 114 186 449 208 267 702 132 289 212 159 405 440 274 567 602
76 117 193 284 26 188 455 211 270 711 102 293 123 163 410 445 281 574 609
77 119 196 288 117 191 461 214 274 720 132 297 13 168 415 451 284 582 617
78 120 45 292 118 40 467 216 63 730 140 102 25 173 421 457 293 589 625
79 122 201 70 120 196 112 219 281 175 125 100 30 177 129 462 299 596 633
80 123 170 85 121 165 136 178 238 213 142 108 33 182 331 268 205 604 641
81 125 106 103 123 101 165 159 148 258 127 113 137 187 336 274 211 611 648
82 126 208 107 82 203 171 130 291 268 152 116 241 191 342 340 217 618 156
83 128 211 210 43 206 336 85 295 525 155 119 145 196 347 385 224 626 164
84 129 13 214 55 8 342 133 18 535 150 125 110 101 352 291 230 633 272
85 131 216 215 129 102 344 135 302 538 142 140 270 105 357 497 236 640 180
86 132 218 231 130 103 370 138 305 578 178 179 154 110 363 302 242 148 287
87 134 25 225 132 20 200 142 35 563 167 180 102 114 368 508 248 155 295
88 135 223 229 133 218 205 243 112 573 170 165 78 319 373 375 254 162 203
89 137 226 233 135 221 100 246 116 583 173 141 69 324 378 519 260 170 311
90 138 105 110 136 100 176 249 125 275 176 88 73 328 384 525 267 300 319
91 122 5 15 120 0 24 220 7 38 144 107 29 378 27 464 200 98 234
92 102 152 210 100 147 336 184 102 525 104 102 75 316 57 288 218 300 230
93 113 136 148 111 131 237 203 130 370 126 99 10 350 396 229 300 290 188
94 110 20 35 108 15 56 198 28 88 120 100 97 341 385 218 200 201 172
95 106 120 152 104 115 243 191 68 380 112 142 86 329 371 203 205 219 201
96 102 119 125 100 114 200 184 67 313 102 153 175 316 357 288 203 200 130
97 105 100 101 103 95 162 189 40 253 103 152 184 326 368 299 231 215 220
98 100 85 90 98 80 144 180 19 225 108 50 70 310 350 280 210 290 175
99 120 75 60 118 70 96 216 100 150 130 100 100 372 320 356 292 288 124
100 125 70 25 123 65 40 225 98 63 85 113 12 388 338 375 213 213 150
ANNEXURE B CI of 200 Sample each size n=10 (Case I)
SNo Text Lower Text Upper Video Lower Video Upper Image Lower Image Upper
1 113.52 138.60 182.37 208.56 223.99 292.37
2 134.15 169.82 178.21 208.20 183.23 259.15
3 108.70 125.20 172.51 192.64 278.78 322.86
4 98.32 117.18 172.70 207.84 301.50 346.83
5 119.85 144.72 175.90 211.48 223.60 286.24
6 121.10 150.73 182.54 218.73 205.16 279.25
7 135.22 165.82 162.97 191.85 213.89 276.67
8 107.45 131.03 173.59 190.97 267.18 329.69
9 115.60 145.78 179.46 217.16 223.00 289.18
10 102.87 128.53 177.76 216.92 263.89 328.07
11 119.50 148.09 185.18 227.06 202.44 264.31
12 102.11 119.69 179.10 199.41 275.68 331.41
13 118.51 153.85 180.19 218.56 199.86 279.23
14 106.84 127.94 160.98 185.00 302.64 342.08
15 103.74 129.75 179.12 204.78 245.36 322.51
16 117.55 148.77 186.01 224.57 198.17 272.13
17 105.39 122.01 182.75 223.76 252.77 303.68
18 105.64 131.99 181.69 221.76 242.17 309.79
19 115.26 151.50 184.97 228.85 191.27 272.69
20 139.27 173.37 175.76 208.77 182.90 256.83
21 106.22 135.39 186.06 226.95 225.24 299.26
22 126.17 154.80 165.86 192.99 239.90 295.12
23 119.28 148.90 186.63 228.59 197.25 265.25
24 104.62 129.03 182.03 233.05 244.82 302.58
25 109.70 132.10 180.57 207.67 241.39 301.99
26 109.59 136.66 193.94 250.75 224.91 283.77
27 112.16 134.94 171.35 186.03 265.34 319.65
28 110.89 132.73 172.02 192.35 269.83 321.56
29 117.83 142.03 169.00 195.69 250.72 309.13
30 104.50 128.95 186.04 239.64 229.02 294.26
31 115.42 142.35 178.97 219.60 216.32 284.98
32 109.82 133.50 172.50 192.44 259.31 319.72
33 118.35 148.42 170.85 207.27 220.90 290.64
34 117.23 147.41 168.79 210.57 232.76 302.81
35 106.93 128.39 172.62 192.64 277.41 326.53
36 100.11 124.66 174.54 212.28 276.27 334.68
37 116.23 151.28 170.03 222.55 222.16 297.20
38 115.01 143.91 172.40 209.72 228.85 298.03
39 124.20 151.61 181.42 204.62 209.17 274.60
40 109.92 134.92 168.40 202.58 255.59 315.39
41 96.04 116.00 180.62 229.87 280.33 332.95
42 114.19 135.99 179.51 203.90 241.11 295.96
43 112.50 141.00 186.95 227.50 216.99 286.86
44 105.28 131.40 173.25 212.63 245.62 313.91
45 119.59 152.94 194.83 247.39 182.06 258.71
46 109.18 138.42 178.99 215.70 232.22 306.16
47 114.30 143.11 175.95 213.92 223.81 295.47
48 135.22 163.57 178.67 197.15 196.52 260.81
49 111.77 140.48 181.12 209.78 224.21 295.44
50 110.99 142.28 165.22 211.79 236.64 304.69
51 114.41 139.16 182.11 206.76 225.46 290.48
52 129.25 157.94 193.22 236.13 178.00 237.86
53 106.16 137.59 168.71 222.81 245.27 312.38
54 113.25 135.28 179.40 206.03 240.28 296.19
55 113.81 140.55 174.47 212.01 229.48 297.07
56 102.93 130.27 193.84 245.45 213.22 291.80
57 117.30 143.95 178.53 203.92 225.81 289.36
58 102.00 126.05 162.87 202.67 297.51 344.91
59 102.80 120.83 173.43 208.17 290.48 332.71
60 94.26 113.51 178.32 227.06 279.52 341.99
61 110.06 136.33 177.09 214.88 246.23 312.45
62 98.90 117.90 173.64 219.16 289.73 339.52
63 120.69 152.76 169.95 200.50 221.78 298.05
64 125.12 157.64 183.58 229.75 190.87 265.37
65 122.31 148.69 180.86 204.78 219.87 281.78
66 107.87 139.21 168.18 214.57 231.77 314.20
67 122.91 147.34 176.85 198.15 225.04 282.71
68 108.28 132.06 177.18 200.03 252.48 315.42
69 119.26 149.26 189.96 232.44 194.19 260.22
70 115.23 137.86 172.60 208.24 248.60 299.72
71 130.76 158.31 189.61 225.55 193.67 253.14
72 101.05 117.19 171.90 185.84 311.34 348.68
73 130.26 167.89 186.65 231.77 167.56 244.90
74 113.95 137.82 173.55 191.39 253.35 311.21
75 108.32 132.68 170.30 207.15 259.30 317.84
76 127.24 149.82 176.72 195.96 234.56 280.55
77 108.21 132.38 170.43 187.15 270.10 328.65
78 102.58 131.93 172.38 226.51 249.05 317.04
79 105.12 133.54 192.45 251.23 223.54 293.44
80 119.23 137.94 160.77 182.95 285.97 317.89
81 106.73 134.75 185.04 235.27 223.06 293.84
82 127.90 150.28 173.66 192.43 227.59 277.71
83 118.47 142.52 162.79 190.44 265.12 312.64
84 112.34 139.64 190.83 240.43 206.42 278.53
85 112.02 139.97 166.58 211.26 248.96 306.42
86 113.40 145.49 174.46 219.57 226.59 296.77
87 117.36 153.15 171.03 224.48 216.81 292.28
88 107.48 130.00 177.39 202.16 253.91 314.81
89 116.87 144.85 177.76 197.15 227.76 295.46
90 102.62 129.83 173.64 219.05 256.83 321.29
91 132.26 165.75 186.10 226.84 183.44 248.20
92 116.66 145.53 180.27 227.12 221.99 283.49
93 118.67 147.80 174.80 194.71 222.91 293.50
94 106.96 125.08 168.88 203.92 281.77 324.85
95 121.57 152.26 177.17 201.19 218.88 286.17
96 107.10 132.29 178.30 205.28 242.23 312.23
97 124.98 151.55 161.41 188.17 250.34 301.76
98 144.42 173.99 179.73 205.87 180.77 241.99
99 113.76 142.47 174.36 198.14 236.32 304.03
100 111.82 136.96 185.79 225.34 222.31 289.02
101 118.94 147.54 160.93 188.91 247.23 310.88
102 120.52 150.59 168.16 211.94 230.45 292.34
103 112.27 134.90 172.52 191.48 264.89 316.28
104 110.50 134.47 179.39 217.60 243.27 300.45
105 108.86 131.46 175.06 197.80 264.91 322.23
106 105.38 130.44 175.56 213.76 248.98 313.69
107 100.86 126.16 176.44 224.74 266.79 328.91
108 137.86 168.70 179.01 204.44 188.28 254.12
109 112.96 134.38 179.05 202.83 244.80 300.10
110 146.40 172.92 191.06 215.25 171.53 228.63
111 133.29 161.08 178.94 202.07 195.95 263.39
112 112.13 136.56 176.36 212.66 253.29 306.83
113 112.07 141.27 180.71 220.82 217.99 288.03
114 121.40 151.16 182.64 221.29 199.96 264.56
115 130.97 160.24 184.38 225.38 193.31 246.37
116 101.96 119.74 168.33 203.89 292.71 339.24
117 131.08 157.41 180.32 204.41 206.36 264.20
118 107.00 126.61 183.28 221.06 246.91 307.00
119 114.62 141.94 174.13 206.74 233.25 302.13
120 109.11 135.06 183.31 222.81 223.14 290.25
121 106.97 137.69 171.73 211.77 237.46 313.65
122 138.62 169.74 162.26 192.30 209.02 273.30
123 123.40 152.62 168.72 211.98 227.90 287.55
124 123.95 154.05 185.21 224.92 188.40 259.66
125 100.97 124.92 182.99 235.75 239.76 306.35
126 113.10 139.03 184.98 224.20 220.55 289.13
127 131.00 161.47 188.77 229.57 187.59 242.68
128 104.45 130.78 174.70 215.45 248.93 309.91
129 116.14 145.17 182.67 220.74 217.37 287.61
130 119.59 143.56 177.61 200.25 224.88 288.67
131 139.59 168.84 186.43 212.12 175.49 238.51
132 101.07 127.38 177.86 227.41 260.52 322.35
133 113.49 140.83 187.81 228.50 209.60 279.21
134 123.47 150.70 177.77 214.50 221.05 276.41
135 129.57 156.81 177.44 202.23 211.75 274.06
136 113.90 137.60 185.45 213.09 225.00 289.65
137 151.23 182.37 185.20 210.32 166.18 229.20
138 109.05 137.35 182.31 222.73 241.16 306.61
139 111.25 140.68 182.88 221.79 230.25 294.18
140 119.68 151.28 172.65 204.08 216.32 291.78
141 111.33 135.82 178.23 217.73 234.30 294.04
142 129.24 154.75 176.48 198.12 219.97 276.14
143 115.70 147.14 183.91 233.86 200.61 273.73
144 133.68 169.69 194.57 245.77 166.55 230.67
145 106.15 139.15 180.00 220.91 218.83 300.05
146 105.37 123.99 170.08 205.25 274.75 328.35
147 118.10 155.00 177.60 233.26 193.53 276.06
148 125.58 150.07 188.84 229.81 197.75 251.16
149 135.94 164.45 184.08 210.96 186.99 246.94
150 104.12 124.58 169.63 204.70 274.30 329.46
151 126.64 158.34 186.55 236.42 181.21 251.28
152 111.67 135.75 181.82 222.37 228.22 291.62
153 129.85 146.69 179.01 196.17 240.98 279.24
154 129.12 160.75 179.37 225.28 192.42 257.94
155 118.93 147.14 178.49 203.57 223.14 291.47
156 123.62 154.94 183.44 220.45 194.34 264.52
157 115.46 138.07 178.16 215.99 240.49 287.55
158 142.28 172.84 178.38 201.38 178.61 246.30
159 132.57 170.26 170.69 219.54 182.60 259.57
160 134.76 169.72 176.13 202.65 190.19 260.80
161 110.83 136.12 192.88 243.94 217.56 282.83
162 103.09 118.83 171.20 186.36 307.64 342.80
163 117.30 150.31 169.34 215.27 221.04 290.24
164 134.78 166.99 172.47 203.11 205.06 271.89
165 127.64 158.93 185.65 227.58 186.98 251.48
166 116.50 146.59 191.48 234.36 192.69 263.23
167 112.50 135.44 179.32 203.39 238.32 301.97
168 117.43 139.12 182.66 210.58 227.24 282.80
169 107.71 128.50 162.05 184.25 303.12 340.93
170 126.09 157.91 166.31 194.73 221.03 292.49
171 110.77 127.46 176.64 200.93 262.13 309.77
172 121.91 154.19 190.60 231.74 194.20 267.03
173 111.93 135.28 171.07 186.38 263.67 317.41
174 114.30 140.12 171.24 193.14 243.84 305.55
175 124.56 143.17 173.88 191.05 250.50 291.29
176 117.73 140.49 175.37 198.99 242.57 294.25
177 138.12 164.32 181.89 204.01 195.24 254.53
178 98.23 117.15 172.21 205.04 303.78 347.61
179 102.97 121.83 176.99 224.98 261.30 315.23
180 138.15 174.29 179.45 223.88 185.93 253.26
181 117.81 142.28 184.95 224.19 210.27 272.82
182 111.68 135.20 174.22 196.36 254.10 311.13
183 115.89 135.99 180.74 218.71 242.30 284.06
184 104.81 123.76 175.46 211.93 285.40 329.53
185 112.44 132.76 177.43 211.57 261.46 305.85
186 112.63 133.06 180.89 219.72 239.74 291.32
187 111.58 138.81 169.62 213.62 246.36 304.13
188 100.55 116.56 171.81 185.89 313.20 350.22
189 120.20 145.68 178.03 215.08 213.41 278.58
190 121.14 155.67 167.54 213.37 215.88 292.09
191 123.48 151.46 171.57 193.28 222.67 288.17
192 106.22 128.97 168.46 198.06 269.61 327.15
193 110.94 134.72 180.65 216.55 250.46 303.02
194 103.31 128.20 181.03 221.10 246.16 311.76
195 117.42 147.06 183.95 222.36 216.35 284.42
196 110.70 146.09 166.99 220.61 233.86 308.95
197 108.89 135.16 172.39 209.54 241.38 312.29
198 134.67 161.67 186.68 212.45 184.12 244.42
199 122.52 155.03 163.82 193.10 232.96 303.74
200 123.31 154.67 173.09 211.61 204.41 275.16
ANNEXURE C CI of 200 Sample each size n=10 (Case II)
SNo Text Lower Text Upper Video Lower Video Upper Image Lower Image Upper
1 112.5025 139.6188 180.9357 209.9947 222.0762 294.2889
2 132.1565 171.8114 176.3829 210.0216 181.4341 260.9476
3 107.8549 126.0527 172.3538 192.8011 278.2256 323.4121
4 97.28737 118.2201 171.5393 209.0049 300.528 347.8061
5 119.0032 145.5677 174.4599 212.9213 221.1926 288.6421
6 119.6648 152.1567 180.9075 220.362 202.5174 281.8964
7 133.4725 167.5706 161.5407 193.2813 212.6052 277.958
8 106.3945 132.0859 173.0844 191.4717 264.9768 331.8918
9 114.2793 147.0991 177.7743 218.8427 220.1311 292.0491
10 101.6652 129.7376 175.9887 218.6873 261.7236 330.2351
11 117.9328 149.6625 183.0345 229.202 200.5343 266.216
12 101.3133 120.485 177.9547 200.5548 273.9451 333.1438
13 116.7251 155.6399 178.0277 220.7157 197.357 281.7325
14 105.6572 129.1162 160.3586 185.6207 302.185 342.5367
15 102.6085 130.8796 177.2882 206.6135 243.301 324.5688
16 116.2565 150.0644 183.7047 226.8764 195.5426 274.7563
17 104.1755 123.2273 181.0051 225.5034 251.9731 304.4761
18 104.5469 133.0791 179.7977 223.6472 239.8835 312.0763
19 113.4035 153.364 182.4263 231.3981 189.4269 274.5269
20 137.7428 174.8995 173.485 211.0486 181.0419 258.6833
21 104.8304 136.781 183.4532 229.554 222.7347 301.7651
22 124.682 156.2853 164.5653 194.2808 238.3067 296.7146
23 117.6546 150.5266 184.4309 230.7915 195.521 266.9823
24 103.3036 130.3428 179.9946 235.0848 242.992 304.4074
25 108.5566 133.2433 179.3298 208.9106 240.0204 303.3612
26 108.2785 137.9636 191.7557 252.9432 222.619 286.0615
27 111.4435 135.6584 170.8133 186.5734 262.7295 322.2601
28 109.8958 133.7237 171.5627 192.8094 268.3515 323.0353
29 117.0376 142.8189 167.4221 197.2665 248.2261 311.6299
30 103.1887 130.268 183.2596 242.4131 227.2967 295.9827
31 113.7117 144.0559 177.2317 221.3402 214.475 286.8212
32 108.6669 134.6575 172.1115 192.8282 257.6001 321.4268
33 116.9968 149.7776 169.2943 208.8191 218.1333 293.4088
34 115.7656 148.8779 166.8516 212.5117 230.6757 304.8981
35 105.8899 129.4305 172.0856 193.1804 275.9146 328.0331
36 98.78647 125.9776 173.0555 213.7639 274.4069 336.545
37 114.1647 153.3442 167.345 225.2402 219.7946 299.5632
38 113.3935 145.5338 170.5726 211.5429 226.6755 300.2043
39 123.1525 152.6558 180.143 205.89 206.9668 276.8079
40 108.9226 135.9182 167.0473 203.9361 252.8565 318.1285
41 94.78109 117.2568 178.463 232.0219 279.3064 333.9716
42 113.0547 137.1217 178.6081 204.7991 239.4949 297.5753
43 111.0507 142.4457 184.4639 229.9909 214.6993 289.1541
44 104.0987 132.5776 171.2642 214.6111 243.4342 316.0932
45 117.7712 154.7576 191.8136 250.4098 179.607 261.17
46 108.0273 139.5723 176.8206 217.8769 229.442 308.9327
47 113.0693 144.3429 174.0301 215.8397 221.1914 298.0877
48 134.4277 164.3694 176.9454 198.8787 193.7765 263.5526
49 110.2277 142.0148 179.8179 211.0823 222.3989 297.2529
50 109.4672 143.8032 163.0289 213.979 235.2198 306.1044
51 112.9665 140.5978 181.0044 207.8623 223.8664 292.0724
52 127.6045 159.583 191.2084 238.1448 176.3396 239.5216
53 104.4922 139.2542 165.9372 225.5766 243.4363 314.2153
54 112.1048 136.4315 178.4944 206.9279 239.0231 297.442
55 112.707 141.657 172.8391 213.6413 227.3741 299.1733
56 101.7639 131.4424 190.2886 248.9992 210.958 294.0564
57 116.2937 144.959 177.0841 205.3588 223.2077 291.9614
58 100.6352 127.4147 161.2607 204.2833 296.4775 345.9437
59 101.7613 121.8703 172.3789 209.2262 289.5168 333.6739
60 93.31854 114.4549 175.6672 229.7062 277.6419 343.8712
61 108.4735 137.9135 175.9495 216.0254 244.3709 314.3022
62 97.85368 118.9428 171.6796 221.1267 287.9315 341.3232
63 119.3528 154.0918 167.9264 202.5278 219.4674 300.3561
64 123.5925 159.1735 180.8238 232.5091 188.5559 267.6838
65 121.0287 149.9641 179.6038 206.0375 217.7891 283.8648
66 106.4229 140.6509 165.4635 217.2859 229.691 316.2784
67 121.6842 148.5709 176.536 198.4615 223.3503 284.4004
68 107.1758 133.173 176.0211 201.1831 250.487 317.4138
69 117.8529 150.6682 187.9882 234.4153 192.1432 262.2635
70 114.0986 138.9898 171.4805 209.3636 246.5689 301.7476
71 129.5237 159.5445 188.6509 226.5084 191.1456 255.6659
72 100.1975 118.0451 171.8944 185.8425 310.5186 349.5013
73 128.1539 170.001 183.8726 234.5459 165.4989 246.967
74 112.7568 139.0161 173.1246 191.8106 251.5786 312.9848
75 106.9005 134.0995 168.8298 208.6236 257.5159 319.6272
76 126.3709 150.6879 176.0532 196.6314 232.4573 282.6508
77 107.3033 133.2811 169.9509 187.6264 267.6271 331.1282
78 100.941 133.5688 169.1504 229.743 247.266 318.8223
79 103.3659 135.2866 189.7995 253.8758 221.5567 295.4243
80 118.2461 138.9255 160.0072 183.7118 285.9793 317.8764
81 105.0789 136.4004 182.3798 237.9373 221.1902 295.7084
82 127.3167 150.8695 173.0966 192.9996 225.2877 280.0133
83 117.3727 143.6146 161.816 191.4196 264.0154 313.7467
84 111.1262 140.8496 188.3426 242.918 203.8726 281.0866
85 110.4382 141.5568 164.5645 213.2762 247.9363 307.4482
86 111.5953 147.2998 171.925 222.1018 224.5882 298.7639
87 115.1705 155.3432 168.4218 227.0897 214.7593 294.3302
88 106.2111 131.2661 176.1948 203.3549 252.4467 316.2729
89 115.834 145.8789 176.3625 198.5479 225.2968 297.9281
90 101.4559 130.9953 171.5939 221.0973 253.754 324.3593
91 130.4779 167.5326 184.7005 228.2409 181.154 250.484
92 115.396 146.7969 178.6149 228.7789 218.7634 286.7163
93 117.9274 148.5433 173.338 196.1644 220.197 296.2196
94 105.9154 126.1265 167.785 205.0123 280.7667 325.8502
95 119.9302 153.8944 176.0191 202.3343 216.9271 288.1204
96 105.7466 133.6415 177.3202 206.2592 240.4651 313.994
97 123.4758 153.0476 160.4645 189.1168 249.5679 302.5239
98 143.2534 175.1605 178.4569 207.136 178.6118 244.1425
99 112.9867 143.2411 173.0715 199.4294 233.3486 306.9992
100 110.6316 138.1493 184.2426 226.8832 219.9981 291.3308
101 117.6532 148.8222 159.9025 189.9333 245.2808 312.8312
102 119.0967 152.0117 165.9236 214.1748 228.1585 294.6355
103 110.9365 136.2283 172.0827 191.9236 263.6667 317.5054
104 109.1383 135.8324 177.9972 218.9959 241.5299 302.183
105 108.0657 132.2497 174.1478 198.716 262.8316 324.3086
106 104.0505 131.7665 173.7124 215.6026 247.1579 315.5129
107 99.54875 127.4673 174.2892 226.8904 264.2541 331.4424
108 136.637 169.9269 177.7045 205.7468 185.7354 256.6656
109 111.8287 135.5027 178.0552 203.8318 243.1506 301.7452
110 145.5738 173.7484 189.6491 216.6588 169.0974 231.0597
111 132.2655 162.0995 177.8587 203.1535 193.9084 265.4285
112 110.5288 138.1531 175.1532 213.8681 251.8669 308.2521
113 110.9403 142.4005 178.2127 223.3101 215.333 290.6819
114 120.2631 152.2921 180.8275 223.1041 197.455 267.0683
115 129.6357 161.5753 182.8615 226.9011 190.9967 248.6889
116 101.0557 120.6438 167.1494 205.0728 291.5208 340.4316
117 129.9248 158.558 179.7832 204.9454 204.4899 266.0657
118 105.7336 127.8727 181.5886 222.7499 245.1186 308.7944
119 113.3288 143.2347 172.1463 208.7279 231.0965 304.2803
120 108.093 136.0751 181.241 224.8768 220.9618 292.4217
121 105.2444 139.4157 169.8732 213.6228 235.3795 315.7308
122 136.9557 171.4033 160.8921 193.6687 207.6163 274.7066
123 122.1504 153.8653 166.551 214.1491 225.4448 289.9981
124 122.8419 155.1536 182.8078 227.3192 185.8128 262.2422
125 99.70152 126.1885 179.9936 238.7465 237.9428 308.1689
126 111.5485 140.5842 183.4325 225.7504 218.3515 291.324
127 129.3029 163.1657 187.4728 230.8761 185.528 244.7415
128 103.3323 131.8956 173.1128 217.0381 247.1662 311.6779
129 114.8862 146.4198 180.9907 222.4157 214.7773 290.1951
130 118.8749 144.2761 176.5123 201.3529 223.1046 290.4441
131 138.3963 170.0363 184.5617 213.9877 173.3222 240.6787
132 99.62565 128.8244 175.7908 229.4773 258.4013 324.4739
133 111.9322 142.3953 185.9308 230.3726 207.8235 280.9824
134 122.3692 151.7961 176.5784 215.6949 218.2109 279.252
135 128.7016 157.6722 175.7999 203.8778 209.0222 276.7923
136 112.8307 138.6694 184.0605 214.4809 222.9509 291.697
137 149.7657 183.8338 183.7802 211.7355 164.2749 231.1092
138 107.8751 138.5291 180.4376 224.5936 238.6345 309.1309
139 109.8208 142.1098 181.2026 223.4656 227.8544 296.584
140 118.2247 152.7289 170.3822 206.3409 214.1352 293.9634
141 110.0752 137.0799 176.7099 219.2462 232.6888 295.6495
142 127.9007 156.0903 175.8884 198.7099 218.2205 277.8915
143 114.3328 148.5068 181.4102 236.3515 198.13 276.2131
144 131.7717 171.6004 192.3412 248.0014 164.0839 233.1391
145 104.5398 140.7624 177.6861 223.2159 216.4322 302.4524
146 104.47 124.8887 168.545 206.7871 272.9205 330.1836
147 116.2984 156.8052 174.386 236.4733 191.0548 278.5333
148 124.033 151.6164 187.5675 231.087 196.4229 252.4811
149 134.8634 165.529 182.7169 212.3166 184.3861 249.5506
150 102.8558 125.8455 168.0124 206.3133 272.8902 330.8742
151 125.2486 159.7288 183.7088 239.2591 179.211 253.2814
152 110.5752 136.8422 179.8302 224.3573 225.8394 293.9977
153 129.6543 146.8904 178.4509 196.7308 238.645 281.5744
154 127.8518 162.0119 177.0669 227.5873 190.1802 260.1814
155 117.8019 148.2678 177.4925 204.5629 220.7603 293.8481
156 122.4298 156.1313 181.6233 222.2668 191.6167 267.2493
157 114.07 139.4558 176.7261 217.4209 239.1659 288.8712
158 141.2664 173.853 176.8435 202.9079 176.3973 248.5132
159 130.4362 172.3929 168.117 222.1044 180.8877 261.2883
160 132.8644 171.6159 174.6374 204.1507 188.0881 262.8969
161 109.7245 137.229 190.5085 246.3098 215.027 285.3557
162 102.3437 119.5781 171.2071 186.3528 307.0306 343.4032
163 115.8665 151.7409 166.8963 217.7171 218.7386 292.5458
164 133.2373 168.5281 170.6996 204.8804 203.0442 273.9077
165 125.9633 160.6011 183.2076 230.0186 184.9568 253.5047
166 115.3864 147.7065 189.0317 236.8081 190.4789 265.4336
167 111.2776 136.6608 178.3354 204.3773 236.5786 303.7095
168 116.1472 140.4064 181.7183 211.5185 226.201 283.8409
169 106.7678 129.4426 161.3679 184.9314 301.7355 342.314
170 124.6367 159.3601 164.6671 196.3782 219.115 294.4027
171 109.7975 128.4284 175.9608 201.6023 261.4589 310.4398
172 120.6427 155.4623 188.467 233.8666 191.5926 269.6411
173 111.1079 136.0974 170.5417 186.9067 261.3276 319.7563
174 113.0445 141.3792 170.6827 193.6963 242.1039 307.2931
175 123.6469 144.0884 173.6048 191.3177 249.253 292.5336
176 116.3674 141.8538 174.9314 199.4319 241.5215 295.2986
177 137.1128 165.3278 180.5412 205.3599 193.0907 256.6776
178 97.34554 118.0329 170.8858 206.3621 302.1018 349.2931
179 101.7979 122.9972 174.4955 227.4812 259.8926 316.6379
180 136.2239 176.2208 177.4457 225.8892 183.9656 255.22
181 116.6495 143.4375 183.0356 226.1076 208.4037 274.6845
182 110.3211 136.5583 173.6371 196.9454 252.8043 312.4312
183 114.8908 136.9921 179.6202 219.8316 240.3142 286.0459
184 103.4087 125.1564 174.5705 212.8136 285.218 329.7199
185 111.5307 133.6676 176.3725 212.631 259.0846 308.2217
186 111.5881 134.1007 179.2584 221.352 237.8675 293.1888
187 110.1395 140.2489 167.3017 215.9429 244.8081 305.6851
188 99.8415 117.2698 171.8125 185.8856 312.1827 351.2323
189 119.1446 146.7326 176.3303 216.783 211.1309 280.8579
190 119.3813 157.4293 165.3169 215.5944 213.7207 294.2584
191 122.3186 152.6193 170.6861 194.1627 220.5516 290.2861
192 104.9061 130.2859 167.0852 199.4431 268.4693 328.2909
193 109.9692 135.6914 179.3392 217.8561 247.9055 305.5764
194 102.0151 129.4977 178.9765 223.1482 244.4347 313.4777
195 115.9212 148.5581 182.438 223.87 213.8521 286.9141
196 108.8774 147.9102 164.328 223.2739 231.0375 311.7662
197 107.4867 136.5637 170.6027 211.3267 239.1731 314.5028
198 133.9345 162.4037 185.2204 213.9091 181.8937 246.6481
199 120.6932 156.8539 162.5758 194.3481 231.302 305.4011
200 121.6709 156.3097 171.2976 213.3957 202.1978 277.3722
ANNEXURE D
200 Sample each size n=10 (estimate on value Case I))
S.No. Mean(T) Mean(V) Mean (I) MSE(T) MSE(V) MSE (I)
1 126.06 195.47 258.18 40.93 44.61 304.28
2 151.98 193.20 221.19 82.81 58.51 375.05
3 116.95 182.58 300.82 17.71 26.38 126.48
4 107.75 190.27 324.17 23.15 80.34 133.73
5 132.29 193.69 254.92 40.23 82.36 255.37
6 135.91 200.63 242.21 57.13 85.26 357.26
7 150.52 177.41 245.28 60.96 54.26 256.51
8 119.24 182.28 298.43 36.16 19.66 254.32
9 130.69 198.31 256.09 59.25 92.50 285.11
10 115.70 197.34 295.98 42.87 99.79 268.01
11 133.80 206.12 233.38 53.19 114.16 249.09
12 110.90 189.25 303.54 20.10 26.83 202.10
13 136.18 199.37 239.54 81.25 95.80 410.04
14 117.39 172.99 322.36 28.97 37.52 101.20
15 116.74 191.95 283.93 44.01 42.86 387.44
16 133.16 205.29 235.15 63.44 96.79 355.90
17 113.70 203.25 278.22 17.96 109.47 168.70
18 118.81 201.72 275.98 45.20 104.48 297.62
19 133.38 206.91 231.98 85.47 125.28 431.42
20 156.32 192.27 219.86 75.67 70.89 355.72
21 120.81 206.50 262.25 55.37 108.83 356.51
22 140.48 179.42 267.51 53.33 47.91 198.43
23 134.09 207.61 231.25 57.09 114.56 300.92
24 116.82 207.54 273.70 38.76 169.45 217.10
25 120.90 194.12 271.69 32.67 47.83 238.95
26 123.12 222.35 254.34 47.69 210.03 225.43
27 123.55 178.69 292.49 33.77 14.03 191.94
28 121.81 182.19 295.69 31.03 26.88 174.10
29 129.93 182.34 279.93 38.11 46.38 222.02
30 116.73 212.84 261.64 38.89 186.98 276.96
31 128.88 199.29 250.65 47.19 107.46 306.73
32 121.66 182.47 289.51 36.48 25.87 237.46
33 133.39 189.06 255.77 58.85 86.33 316.47
34 132.32 189.68 267.79 59.27 113.58 319.38
35 117.66 182.63 301.97 29.98 26.08 157.01
36 112.38 193.41 305.48 39.22 92.69 222.06
37 133.75 196.29 259.68 79.95 179.53 366.48
38 129.46 191.06 263.44 54.36 90.63 311.41
39 137.90 193.02 241.89 48.87 35.03 278.62
40 122.42 185.49 285.49 40.68 76.05 232.69
41 106.02 205.24 306.64 25.92 157.88 180.21
42 125.09 191.70 268.54 30.92 38.72 195.83
43 126.75 207.23 251.93 52.88 107.03 317.74
44 118.34 192.94 279.76 44.38 100.91 303.46
45 136.26 221.11 220.39 72.37 179.74 382.32
46 123.80 197.35 269.19 55.64 87.71 355.76
47 128.71 194.93 259.64 54.02 93.83 334.17
48 149.40 187.91 228.66 52.30 22.22 269.04
49 126.12 195.45 259.83 53.64 53.46 330.12
50 126.64 188.50 270.66 63.69 141.16 301.33
51 126.78 194.43 257.97 39.88 39.53 275.14
52 143.59 214.68 207.93 53.55 119.83 233.16
53 121.87 195.76 278.83 64.30 190.49 293.11
54 124.27 192.71 268.23 31.58 46.15 203.44
55 127.18 193.24 263.27 46.53 91.74 297.30
56 116.60 219.64 252.51 48.63 173.33 401.87
57 130.63 191.22 257.58 46.22 41.95 262.80
58 114.02 182.77 321.21 37.67 103.06 146.22
59 111.82 190.80 311.60 21.16 78.52 116.07
60 103.89 202.69 310.76 24.10 154.59 253.94
61 123.19 195.99 279.34 44.90 92.92 285.39
62 108.40 196.40 314.63 23.49 134.83 161.35
63 136.72 185.23 259.91 66.91 60.73 378.53
64 141.38 206.67 228.12 68.83 138.69 361.17
65 135.50 192.82 250.83 45.28 37.26 249.44
66 123.54 191.37 272.98 63.92 140.09 442.15
67 135.13 187.50 253.88 38.83 29.54 216.41
68 120.17 188.60 283.95 36.80 33.97 257.85
69 134.26 211.20 227.20 58.59 117.44 283.75
70 126.54 190.42 274.16 33.33 82.65 170.07
71 144.53 207.58 223.41 49.41 84.03 230.22
72 109.12 178.87 330.01 16.94 12.65 90.75
73 149.08 209.21 206.23 92.17 132.51 389.23
74 125.89 182.47 282.28 37.09 20.72 217.87
75 120.50 188.73 288.57 38.61 88.39 222.98
76 138.53 186.34 257.55 33.20 24.10 137.62
77 120.29 178.79 299.38 38.03 18.19 223.08
78 117.25 199.45 283.04 56.03 190.68 300.84
79 119.33 221.84 258.49 52.56 224.87 317.92
80 128.59 171.86 301.93 22.77 32.04 66.31
81 120.74 210.16 258.45 51.10 164.20 326.00
82 139.09 183.05 252.65 32.61 22.92 163.49
83 130.49 176.62 288.88 37.63 49.75 146.97
84 125.99 215.63 242.48 48.49 160.04 338.39
85 126.00 188.92 277.69 50.85 129.88 214.85
86 129.45 197.01 261.68 67.00 132.42 320.52
87 135.26 197.76 254.54 83.37 185.87 370.72
88 118.74 189.77 284.36 32.99 39.92 241.38
89 130.86 187.46 261.61 50.95 24.47 298.25
90 116.23 196.35 289.06 48.19 134.21 270.41
91 149.01 206.47 215.82 73.00 108.04 272.90
92 131.10 203.70 252.74 54.22 142.83 246.11
93 133.24 184.75 258.21 55.21 25.80 324.25
94 116.02 186.40 303.31 21.37 79.88 120.78
95 136.91 189.18 252.52 61.28 37.55 294.62
96 119.69 191.79 277.23 41.32 47.37 318.86
97 138.26 174.79 276.05 45.95 46.58 172.08
98 159.21 192.80 211.38 56.89 44.46 243.92
99 128.11 186.25 270.17 53.61 36.79 298.34
100 124.39 205.56 255.66 41.16 101.79 289.60
101 133.24 174.92 279.06 53.24 50.93 263.69
102 135.55 190.05 261.40 58.86 124.74 249.30
103 123.58 182.00 290.59 33.34 23.40 171.81
104 122.49 198.50 271.86 37.40 95.00 212.76
105 120.16 186.43 293.57 33.24 33.66 213.80
106 117.91 194.66 281.34 40.85 94.99 272.57
107 113.51 200.59 297.85 41.66 151.78 251.18
108 153.28 191.73 221.20 61.89 42.09 282.16
109 123.67 190.94 272.45 29.86 36.79 199.05
110 159.66 203.15 200.08 45.75 38.10 212.23
111 147.18 190.51 229.67 50.25 34.80 295.99
112 124.34 194.51 280.06 38.84 85.76 186.50
113 126.67 200.76 253.01 55.46 104.70 319.24
114 136.28 201.97 232.26 57.63 97.21 271.59
115 145.61 204.88 219.84 55.79 109.37 183.23
116 110.85 186.11 315.98 20.57 82.26 140.90
117 144.24 192.36 235.28 45.12 37.75 217.72
118 116.80 202.17 276.96 25.02 92.85 234.94
119 128.28 190.44 267.69 48.59 69.21 308.75
120 122.08 203.06 256.69 43.85 101.53 293.06
121 122.33 191.75 275.56 61.44 104.31 377.78
122 154.18 177.28 241.16 63.02 58.69 268.87
123 138.01 190.35 257.72 55.57 121.76 231.56
124 139.00 205.06 224.03 58.96 102.61 330.48
125 112.95 209.37 273.06 37.34 181.13 288.63
126 126.07 204.59 254.84 43.73 100.14 306.05
127 146.23 209.17 215.13 60.43 108.34 197.54
128 117.61 195.08 279.42 45.10 108.05 241.96
129 130.65 201.70 252.49 54.82 94.35 321.07
130 131.58 188.93 256.77 37.36 33.37 264.76
131 154.22 199.27 207.00 55.67 42.95 258.44
132 114.23 202.63 291.44 45.06 159.76 248.81
133 127.16 208.15 244.40 48.64 107.75 315.29
134 137.08 196.14 248.73 48.24 87.80 199.44
135 143.19 189.84 242.91 48.30 39.99 252.65
136 125.75 199.27 257.32 36.58 49.71 272.05
137 166.80 197.76 197.69 63.13 41.05 258.45
138 123.20 202.52 273.88 52.12 106.32 278.73
139 125.97 202.33 262.22 56.36 98.49 265.97
140 135.48 188.36 254.05 64.99 64.29 370.61
141 123.58 197.98 264.17 39.03 101.52 232.21
142 142.00 187.30 248.06 42.33 30.48 205.28
143 131.42 208.88 237.17 64.32 162.36 347.96
144 151.69 220.17 198.61 84.40 170.59 267.51
145 122.65 200.45 259.44 70.87 108.92 429.33
146 114.68 187.67 301.55 22.57 80.52 187.00
147 136.55 205.43 234.79 88.60 201.57 443.32
148 137.82 209.33 224.45 39.05 109.25 185.66
149 150.20 197.52 216.97 52.90 47.02 233.88
150 114.35 187.16 301.88 27.26 80.06 197.97
151 142.49 211.48 216.25 65.40 161.84 319.56
152 123.71 202.09 259.92 37.72 107.00 261.58
153 138.27 187.59 260.11 18.44 19.17 95.27
154 144.93 202.33 225.18 65.09 137.16 279.30
155 133.03 191.03 257.30 51.82 40.93 303.91
156 139.28 201.95 229.43 63.86 89.13 320.53
157 126.76 197.07 264.02 33.28 93.13 144.16
158 157.56 189.88 212.46 60.78 34.43 298.15
159 151.41 195.11 221.09 92.42 155.29 385.56
160 152.24 189.39 225.49 79.56 45.77 324.47
161 123.48 218.41 250.19 41.61 169.62 277.22
162 110.96 178.78 325.22 16.12 14.95 80.46
163 133.80 192.31 255.64 70.93 137.25 311.60
164 150.88 187.79 238.48 67.51 61.10 290.62
165 143.28 206.61 219.23 63.73 114.38 270.67
166 131.55 212.92 227.96 58.93 119.70 323.81
167 123.97 191.36 270.14 34.24 37.71 263.61
168 128.28 196.62 255.02 30.62 50.74 200.94
169 118.11 173.15 322.02 28.11 32.09 93.05
170 142.00 180.52 256.76 65.92 52.58 332.29
171 119.11 188.78 285.95 18.14 38.40 147.74
172 138.05 211.17 230.62 67.79 110.13 345.17
173 123.60 178.72 290.54 35.47 15.27 187.97
174 127.21 182.19 274.70 43.37 31.19 247.81
175 133.87 182.46 270.89 22.53 19.18 108.27
176 129.11 187.18 268.41 33.72 36.31 173.75
177 151.22 192.95 224.88 44.69 31.85 228.76
178 107.69 188.62 325.70 23.29 70.17 125.00
179 112.40 200.99 288.27 23.16 149.86 189.31
180 156.22 201.67 219.59 84.98 128.45 294.98
181 130.04 204.57 241.54 38.96 100.20 254.60
182 123.44 185.29 282.62 36.02 31.90 211.65
183 125.94 199.73 263.18 26.29 93.83 113.45
184 114.28 193.69 307.47 23.37 86.57 126.73
185 122.60 194.50 283.65 26.89 75.84 128.24
186 122.84 200.31 265.53 27.17 98.12 173.16
187 125.19 191.62 275.25 48.27 125.97 217.14
188 108.56 178.85 331.71 16.69 12.90 89.18
189 132.94 196.56 245.99 42.26 89.34 276.46
190 138.41 190.46 253.99 77.60 136.73 377.96
191 137.47 182.42 255.42 50.94 30.67 279.22
192 117.60 183.26 298.38 33.69 57.02 215.45
193 122.83 198.60 276.74 36.79 83.87 179.82
194 115.76 201.06 278.96 40.29 104.49 280.06
195 132.24 203.15 250.38 57.18 96.00 301.57
196 128.39 193.80 271.40 81.50 187.09 366.96
197 122.03 190.96 276.84 44.91 89.82 327.25
198 148.17 199.56 214.27 47.46 43.19 236.56
199 138.77 178.46 268.35 68.76 55.81 326.04
200 138.99 192.35 239.79 63.98 96.56 325.79
ANNEXURE E
200 Samples each of size n=10 (estimates value (Case 11))
S.No. Mean(T) Mean(V) Mean (I) MSE(T) MSE(V) MSE (I)
1 126.06 195.47 258.18 47.85 54.95 339.36
2 151.98 193.20 221.19 102.33 73.64 411.44
3 116.95 182.58 300.82 21.55 27.21 132.88
4 107.75 190.27 324.17 28.52 91.35 145.46
5 132.29 193.69 254.92 45.92 96.27 296.06
6 135.91 200.63 242.21 68.70 101.30 410.05
7 150.52 177.41 245.28 75.66 65.56 277.94
8 119.24 182.28 298.43 42.95 22.00 291.39
9 130.69 198.31 256.09 70.10 109.76 336.59
10 115.70 197.34 295.98 51.28 118.65 305.46
11 133.80 206.12 233.38 65.52 138.71 280.75
12 110.90 189.25 303.54 23.92 33.24 228.06
13 136.18 199.37 239.54 98.55 118.59 463.30
14 117.39 172.99 322.36 35.81 41.53 105.96
15 116.74 191.95 283.93 52.01 55.96 429.80
16 133.16 205.29 235.15 74.38 121.29 408.35
17 113.70 203.25 278.22 23.62 128.86 179.39
18 118.81 201.72 275.98 52.98 125.13 339.17
19 133.38 206.91 231.98 103.92 156.07 471.29
20 156.32 192.27 219.86 89.85 91.83 392.30
21 120.81 206.50 262.25 66.43 138.31 406.46
22 140.48 179.42 267.51 65.00 57.46 222.01
23 134.09 207.61 231.25 70.32 139.87 332.33
24 116.82 207.54 273.70 47.58 197.50 245.46
25 120.90 194.12 271.69 39.66 56.94 261.09
26 123.12 222.35 254.34 57.35 243.64 261.93
27 123.55 178.69 292.49 38.16 16.16 230.63
28 121.81 182.19 295.69 36.95 29.38 194.60
29 129.93 182.34 279.93 43.25 57.96 261.61
30 116.73 212.84 261.64 47.72 227.71 307.02
31 128.88 199.29 250.65 59.92 126.61 340.61
32 121.66 182.47 289.51 43.96 27.93 265.11
33 133.39 189.06 255.77 69.93 101.66 368.75
34 132.32 189.68 267.79 71.35 135.68 358.51
35 117.66 182.63 301.97 36.06 28.96 176.77
36 112.38 193.41 305.48 48.12 107.84 251.27
37 133.75 196.29 259.68 99.90 218.13 414.09
38 129.46 191.06 263.44 67.22 109.24 351.84
39 137.90 193.02 241.89 56.65 43.14 317.43
40 122.42 185.49 285.49 47.43 88.56 277.26
41 106.02 205.24 306.64 32.87 186.68 194.47
42 125.09 191.70 268.54 37.69 44.64 219.53
43 126.75 207.23 251.93 64.14 134.89 360.76
44 118.34 192.94 279.76 52.78 122.28 343.56
45 136.26 221.11 220.39 89.02 223.44 432.93
46 123.80 197.35 269.19 64.76 109.70 411.21
47 128.71 194.93 259.64 63.65 113.76 384.80
48 149.40 187.91 228.66 58.34 31.31 316.84
49 126.12 195.45 259.83 65.76 63.61 364.63
50 126.64 188.50 270.66 76.72 168.93 326.99
51 126.78 194.43 257.97 49.69 46.94 302.74
52 143.59 214.68 207.93 66.55 143.37 259.79
53 121.87 195.76 278.83 78.64 231.47 326.01
54 124.27 192.71 268.23 38.51 52.61 222.09
55 127.18 193.24 263.27 54.54 108.34 335.48
56 116.60 219.64 252.51 57.32 224.32 449.38
57 130.63 191.22 257.58 53.47 52.03 307.62
58 114.02 182.77 321.21 46.67 120.45 159.24
59 111.82 190.80 311.60 26.32 88.36 126.89
60 103.89 202.69 310.76 29.07 190.04 285.45
61 123.19 195.99 279.34 56.40 104.52 318.25
62 108.40 196.40 314.63 28.94 159.11 185.51
63 136.72 185.23 259.91 78.53 77.91 425.80
64 141.38 206.67 228.12 82.39 173.85 407.46
65 135.50 192.82 250.83 54.49 45.47 284.13
66 123.54 191.37 272.98 76.24 174.77 487.91
67 135.13 187.50 253.88 47.04 31.28 242.55
68 120.17 188.60 283.95 43.98 41.20 291.49
69 134.26 211.20 227.20 70.08 140.27 319.97
70 126.54 190.42 274.16 40.32 93.39 198.14
71 144.53 207.58 223.41 58.65 93.27 270.91
72 109.12 178.87 330.01 20.73 12.66 98.89
73 149.08 209.21 206.23 113.96 167.10 431.92
74 125.89 182.47 282.28 44.87 22.72 245.39
75 120.50 188.73 288.57 48.14 103.05 251.06
76 138.53 186.34 257.55 38.48 27.56 163.95
77 120.29 178.79 299.38 43.92 20.33 262.42
78 117.25 199.45 283.04 69.28 238.93 333.21
79 119.33 221.84 258.49 66.31 267.19 355.09
80 128.59 171.86 301.93 27.83 36.57 66.21
81 120.74 210.16 258.45 63.84 200.87 361.37
82 139.09 183.05 252.65 36.10 25.78 194.90
83 130.49 176.62 288.88 44.81 57.03 160.95
84 125.99 215.63 242.48 57.49 193.83 387.99
85 126.00 188.92 277.69 63.02 154.42 230.48
86 129.45 197.01 261.68 82.96 163.85 358.06
87 135.26 197.76 254.54 105.02 223.99 412.04
88 118.74 189.77 284.36 40.85 48.01 265.11
89 130.86 187.46 261.61 58.75 32.03 343.30
90 116.23 196.35 289.06 56.78 159.48 324.42
91 149.01 206.47 215.82 89.35 123.37 312.80
92 131.10 203.70 252.74 64.17 163.76 300.50
93 133.24 184.75 258.21 61.00 33.91 376.11
94 116.02 186.40 303.31 26.58 90.19 132.27
95 136.91 189.18 252.52 75.07 45.07 329.84
96 119.69 191.79 277.23 50.64 54.50 351.84
97 138.26 174.79 276.05 56.91 53.43 182.50
98 159.21 192.80 211.38 66.25 53.53 279.46
99 128.11 186.25 270.17 59.57 45.21 353.00
100 124.39 205.56 255.66 49.28 118.32 331.13
101 133.24 174.92 279.06 63.22 58.69 296.95
102 135.55 190.05 261.40 70.50 151.51 287.59
103 123.58 182.00 290.59 41.63 25.62 188.63
104 122.49 198.50 271.86 46.37 109.39 239.41
105 120.16 186.43 293.57 38.06 39.28 245.95
106 117.91 194.66 281.34 49.99 114.20 304.07
107 113.51 200.59 297.85 50.72 180.06 293.78
108 153.28 191.73 221.20 72.12 51.17 327.41
109 123.67 190.94 272.45 36.47 43.24 223.43
110 159.66 203.15 200.08 51.66 47.48 249.85
111 147.18 190.51 229.67 57.92 41.64 332.88
112 124.34 194.51 280.06 49.66 97.54 206.90
113 126.67 200.76 253.01 64.41 132.35 369.47
114 136.28 201.97 232.26 66.76 116.31 315.36
115 145.61 204.88 219.84 66.39 126.22 216.60
116 110.85 186.11 315.98 24.97 93.59 155.68
117 144.24 192.36 235.28 53.35 41.20 246.74
118 116.80 202.17 276.96 31.90 110.26 263.86
119 128.28 190.44 267.69 58.20 87.09 348.54
120 122.08 203.06 256.69 50.96 123.91 332.32
121 122.33 191.75 275.56 75.99 124.56 420.16
122 154.18 177.28 241.16 77.22 69.91 292.92
123 138.01 190.35 257.72 65.46 147.44 271.18
124 139.00 205.06 224.03 67.94 128.93 380.14
125 112.95 209.37 273.06 45.66 224.64 320.94
126 126.07 204.59 254.84 54.86 116.54 346.53
127 146.23 209.17 215.13 74.62 122.60 228.18
128 117.61 195.08 279.42 53.09 125.56 270.84
129 130.65 201.70 252.49 64.71 111.67 370.15
130 131.58 188.93 256.77 41.99 40.16 295.10
131 154.22 199.27 207.00 65.15 56.35 295.25
132 114.23 202.63 291.44 55.48 187.57 284.10
133 127.16 208.15 244.40 60.39 128.53 348.31
134 137.08 196.14 248.73 56.35 99.57 242.48
135 143.19 189.84 242.91 54.62 51.30 298.89
136 125.75 199.27 257.32 43.45 60.22 307.56
137 166.80 197.76 197.69 75.53 50.86 290.69
138 123.20 202.52 273.88 61.15 126.88 323.42
139 125.97 202.33 262.22 67.85 116.24 307.41
140 135.48 188.36 254.05 77.48 84.15 414.71
141 123.58 197.98 264.17 47.46 117.75 257.97
142 142.00 187.30 248.06 51.71 33.89 231.72
143 131.42 208.88 237.17 76.00 196.44 396.77
144 151.69 220.17 198.61 103.23 201.61 310.33
145 122.65 200.45 259.44 85.39 134.90 481.54
146 114.68 187.67 301.55 27.13 95.17 213.39
147 136.55 205.43 234.79 106.78 250.86 498.00
148 137.82 209.33 224.45 49.51 123.25 204.51
149 150.20 197.52 216.97 61.20 57.02 276.34
150 114.35 187.16 301.88 34.39 95.46 218.80
151 142.49 211.48 216.25 77.37 200.82 357.04
152 123.71 202.09 259.92 44.90 129.03 302.32
153 138.27 187.59 260.11 19.33 21.75 119.93
154 144.93 202.33 225.18 75.94 166.10 318.89
155 133.03 191.03 257.30 60.40 47.69 347.63
156 139.28 201.95 229.43 73.91 107.50 372.26
157 126.76 197.07 264.02 41.94 107.77 160.78
158 157.56 189.88 212.46 69.10 44.21 338.45
159 151.41 195.11 221.09 114.56 189.68 420.67
160 152.24 189.39 225.49 97.73 56.68 364.19
161 123.48 218.41 250.19 49.23 202.64 321.88
162 110.96 178.78 325.22 19.33 14.93 86.09
163 133.80 192.31 255.64 83.75 168.08 354.51
164 150.88 187.79 238.48 81.05 76.03 326.79
165 143.28 206.61 219.23 78.08 142.60 305.79
166 131.55 212.92 227.96 67.98 148.54 365.62
167 123.97 191.36 270.14 41.93 44.13 293.27
168 128.28 196.62 255.02 38.30 57.79 216.21
169 118.11 173.15 322.02 33.46 36.13 107.16
170 142.00 180.52 256.76 78.46 65.44 368.87
171 119.11 188.78 285.95 22.59 42.79 156.13
172 138.05 211.17 230.62 78.90 134.13 396.42
173 123.60 178.72 290.54 40.64 17.43 222.17
174 127.21 182.19 274.70 52.25 34.47 276.55
175 133.87 182.46 270.89 27.19 20.42 121.90
176 129.11 187.18 268.41 42.27 39.06 188.20
177 151.22 192.95 224.88 51.81 40.09 263.13
178 107.69 188.62 325.70 27.85 81.90 144.93
179 112.40 200.99 288.27 29.25 182.70 209.55
180 156.22 201.67 219.59 104.11 152.72 330.41
181 130.04 204.57 241.54 46.70 120.73 285.89
182 123.44 185.29 282.62 44.80 35.35 231.37
183 125.94 199.73 263.18 31.79 105.23 136.10
184 114.28 193.69 307.47 30.78 95.18 128.88
185 122.60 194.50 283.65 31.89 85.56 157.13
186 122.84 200.31 265.53 32.98 115.31 199.16
187 125.19 191.62 275.25 59.00 153.97 241.18
188 108.56 178.85 331.71 19.77 12.89 99.23
189 132.94 196.56 245.99 49.53 106.49 316.40
190 138.41 190.46 253.99 94.21 164.50 422.11
191 137.47 182.42 255.42 59.75 35.87 316.46
192 117.60 183.26 298.38 41.92 68.14 232.89
193 122.83 198.60 276.74 43.06 96.55 216.44
194 115.76 201.06 278.96 49.15 126.97 310.22
195 132.24 203.15 250.38 69.32 111.71 347.38
196 128.39 193.80 271.40 99.15 226.12 424.12
197 122.03 190.96 276.84 55.02 107.93 369.28
198 148.17 199.56 214.27 52.74 53.56 272.88
199 138.77 178.46 268.35 85.09 65.69 357.32
200 138.99 192.35 239.79 78.08 115.33 367.76
The population dataset and Python programming code which we have used in this paper for calculate the results of each occasion is available at: https://abdulalim90.blogspot.com/
