Conseguences of soil crude oil pollution on some wood properties of olive trees Mathematics |177 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 On Double Stage Shrinkage Estimator For the Variance of Normal Distribution With Unknown Mean Muna D. Salman Dept. of Mathematics/College of Education for Pure Science(Ibn-Al-Haitham)/ University of Baghdad Received in :19/ May /2016 , Accepted in : 22 /February/ 2017 Abstract This paper is concerned with preliminary test double stage shrinkage estimators to estimate the variance ( 2 ) of normal distribution when a prior estimate 2 0 ( ) of the actual value ( 2 ) is a available when the mean is unknown , using specifying shrinkage weight factors () in addition to pre-test region (R). Expressions for the Bias, Mean squared error [MSE ()], Relative Efficiency [R.EFF ()], Expected sample size [E(n/ 2 )] and percentage of overall sample saved of proposed estimator were derived. Numerical results (using MathCAD program) and conclusions are drawn about selection of different constants including in the mentioned expressions. Comparisons between the suggested estimator with the classical estimator in the sense of Bias and Relative Efficiency are given. Furthermore, comparisons with the earlier existing works are drawn. Keywords: Normal Distribution, Double Stage Shrinkage Estimator, Bias Ratio, Mean Squared Error , Relative Efficiency , Expected sample size. Percentage of overall sample saved and probability of avoiding the second sample . Mathematics |178 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Introduction " The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a standard of reference for many probability problems. The question, Why is the normal distribution useful?, and the answer is many things actually are normally distributed, or very close to it. For example, height and intelligence are approximately normally distributed; measurement errors also often have a normal distribution , the normal distribution is easy to work with mathematically. In many practical cases, the methods developed using normal theory work quite well, even when the distribution is not normal . The normal distribution plays a vital role in many applied problems of biology, economics, engineering, financial risk management, genetics, hydrology, mechanics, medicine, number theory, statistics, physics, psychology, reliability, etc., and has been extensively studied, both from theoretical and applications point of view, by many researchers, since its inception. There is a very strong connection between the size of a sample N and the extent to which a sampling distribution approaches the normal form. Many sampling distributions based on large sample of size can be approximated by the normal distribution even though the population distribution itself is definitely not normal"; [1],[12] . The probability density of the normal distribution is: 2 2 ( x ) 21 f (x | , ) e , x 2             …(1) Here, μ is the mean or expectation of the distribution (and also its median and mode). The parameter σ is its standard deviation with its variance then σ 2 . If μ = 0 and σ = 1, the distribution is called the standard normal distribution or the unit normal distribution denoted by N(0,1) and a random variable with that distribution is a standard normal deviate. Assume that x1, x2, …, xn be a random sample of size (n) from a normal population with unknown mean () and unknown variance ( 2 ). In conventional notation, we write XN(, 2 ). In this work, we suggest the problem of estimating the variance ( 2 ) when some prior information 2 0 ( ) regarding the variance (2) is available. More specifically, we assume that the prior information regarding is due the following reasons, [10]: 1. We believe that 2 0 ( ) is close to the true value of 2, or 2. We fear that, 2 0 ( ) may be near the true value of 2, i.e.; something bad happens if ( 2 ) approximately equals to 2 0 ( ) and we do not known about it. In such a situation it is natural to start with the MVUE 2 ˆ( ) of 2 and modify it by moving it closure to 2 0 ( ) using shrinkage weight factor [()], so that the resulting estimator though perhaps biased, has a smaller mean squared error [MSE] than that of 2 ˆ( ) in some interval around 2 0 ( ) . This method of constructing an estimator that incorporates the prior value leads to what is known as a shrinkage estimator i.e.; 2 2 2 2 1 1 0 ˆ ˆ ˆ( ) [1 ( )]        ; 0  1 2 ˆ( )  1. (2) https://en.wikipedia.org/wiki/Probability_density https://en.wikipedia.org/wiki/Mean https://en.wikipedia.org/wiki/Expected_value https://en.wikipedia.org/wiki/Median https://en.wikipedia.org/wiki/Mode_%28statistics%29 https://en.wikipedia.org/wiki/Standard_deviation https://en.wikipedia.org/wiki/Variance Mathematics |179 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 where i 2 ˆ( ) , 0  i 2 ˆ( )  1 is shrinkage weight factor specifying the belief in 2 ̂ and 2 i ˆ(1 ( ))   specifying the belief in 2 0 ( ) and 2 i ˆ( )  may be a function of 2 ̂ or a constant. Preliminary test double stage shrinkage estimator for the variance ( 2 ) that utilize a prior estimate 2 0 ( ) is represented as following steps:- 1. Select two positive integers (n1) and (n2). 2. Obtain a random sample of size (n1) on x [first stage sample]. Compute sample variance 2 2 1 1 ˆ (s ) [MVUE], where 1n 2 i 1 2 i 1 1 1 ˆ(x ) s n 1      and 1 ̂ is the first sample mean. 3. Choose a suitable region (R) around 2 0  . Noted that, this work concerns with pre-test region. 4. If 2 1 ̂  R, take the shrinkage estimator of the form defined in (1). Here, we put forward 1() = k= 4  However, if 2 1 ̂  R, obtain a second stage random sample of size n2 on x and advise the estimator 2 p ̂ of  2 as polling estimator of two samples variance. i.e.; 2 2 2 p 1 1 2 2 ˆ ˆ ˆ(n n ) / n     (3) where 2 2 ̂ is the variance of the second random sample and n = n1 + n2. Thus, the suggested form of preliminary test double stage shrunken estimator (PDSE) has the following form: 2 2 2 2 1 0 0 12 DS 2 2 p 1 ˆ ˆ ˆk( ) , if R, ˆ ˆ, if R,             (4) where R is the pre-test region for testing the hypothesis H0 : 2 ̂ = 2 0  against HA : 2 ̂  2 0  with level of significance () using test statistic ̂ ̂ . i.e.; 2 2 2 20 0 1 / 2,n 1 / 2,n 1 R X , X n 1 n 1              (5) 2 i ̂ (i=1,2) is the sample variance of stage (i), 2 1 / 2,n 1 X   and 2 / 2,n 1 X   are the lower and upper 100(/2) percentile point of chi-square distribution with degree of freedom (n – 1) and n = n1 + n2. And for simplest we shall assume that R=[a,b]. The aim of this paper is to show that the proposed estimators defined in (4) is better than the classical estimators and some estimators introduced by some authors in the sense of mean squared error and relative efficiency. Several authors have studied (PDSE) for estimating the mean and variance of Normal distribution and also for estimating the parameters of different distributions, for example, Katti [7],Thompson [10],Al-Joboori [3,4], Al-Joboori et al[5,6],Waikar, Schuurmann and Raghunathan [11] , Handa, Kambo and Al-Hemyari [8] and Maha [9]. Mathematics |180 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Preliminary Test Double Stage Shrunken Estimator (PDSE) In this section, recall (PDSE) defined in (3) for estimating the variance of normal distribution . The expressions for Bias and Mean Squared Error [MSE()] of 2 DS  are respectively as follows: 2 2 ˆBias( , R)   DS 2 2 2 1 2 2 2 2 2 2 2 2 1 0 0 1 2 1 2 ˆ ˆ0 ˆ ˆ ˆ ˆ ˆ ˆ[ ( ) ( )] ( ) ( ) d d        R k f f           + 2 2 2 1 2 2 2 2 2 2 2 21 1 2 2 0 1 2 1 2 1 2ˆ ˆ0 ˆ ˆ1 ˆ ˆ ˆ ˆ[ ( ) ] ( ) ( ) d d 2                    R n n f f n n Where , 1 3 2 2 22 2 2 2 2 12 2 2 ( 1) ˆ ˆ( 1) / exp 2 ˆ, 0 , 0 , 1, 2 ˆ( ) ( 1) 2 2 0 ,                                i i i n n i i i i n i i n n for i f n otherwise         …(6) and by simple calculations, we get   2 2 2 DS 1 i i 1 0 i i 1 ˆBias( , R) k j (a , b ) (n 1) J (a , b ) 1         n 1 0 i i 1 i i 1 0 i i 1 (n 1)( 1)J (a , b ) [J (a , b ) (n 1)J (a , b ) 1 u             …(7) The Bias Ratio of 2 DS  = 2 2 DS ˆBias( , R)  / ( 2 1 1  n ) …(8) And, 2 2 2 2 2 DS DS 2 2 2 2 2 2 2 2 2 2 2 1 0 0 1 2 1 2 Rˆ 0 2 2 2 2 2 2 2 2 2 p 1 2 1 2 Rˆ 0 MSE ( , R) E[ ˆ ˆ ˆ ˆk( ) ( )] f (s / )f (s / )d d ˆ ˆs f (s / )f (s / )d d                                            We conclude,  4 2 2 DS 2 i i 1 1 i i2 1 MSE ( , R) K J (a , b ) 2(n 1) J (a , b ) ( 1)           n   )]b,a(J)1n()b,a(J)[1)(1n(k2)b,a(J)1n( ii01ii11ii0 22 1  Mathematics |181 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 )1un( )1n( . u1 u 2)1n( u1 1 2)b,a(J)1()1n( 1 2 1 2 1 2 ii0 22 1                  )]b,a(J)1n()b,a(J)1n(2)b,a(J[ u1 1 ii0 2 1ii11ii2 2              2 1 0 i i 1 u (n 1) 2 J (a , b ) 1 u (un 1)           …(9) where 1 1 b 1 1 j ( , ) y f (y)dy, 0,1, 2.  a a b (10) i 2 2 20 i i n 1 (n 1)s , y ~ X          , (11) a1 =  1 2 1 / 2,n 1 X   , b1 =  1 2 / 2,n 1 X  , …(12) also u = n2/n1, and n = n1 + n2, The Expected sample size can be obtained as: E(n 2 ,R) = n1 [1 + u(1 – j0(a1,b1))] …(13) The Efficiency of 2 DS  relative to 2 2 ˆ (s ) is given by : 2 DS 2 DS ˆMSE( R.Eff ( , R) [MSE( , R)][E(n , R) / n]               …(14) Where [E (n/  , R)/n] is the Expected sample size proportion. The probability of avoiding the second sample computing by p( 2 1 ˆ R  ). Finally, the percentage of overall sample saved can be obtained by (n2/n) p( 2 1 ˆ R  )*100 …(15) See for example [3],[4], [9] ,[10] and [11]. Numerical Results and Discussion The computations (using MathCAD program) of Relative Efficiency [R.Eff()], Bias ratio [B()], Expected sample size [E(n,R)], Expected sample size proportion, Percentage of the overall sample saved and probability of avoiding the second sample were used for the estimator 2 DS  . These computations were performed for n1 = 5,7,9,11,13; u =,2,6,8,10;  = 0.0(0.1)1,2;  = 0.01,0.05,0.1. Some of these computations are given in annex tables (1)-(17) . The observations mentioned in the tables lead to the following results: i. R.Eff( 2 DS  ) are adversely proportional with small value of . ii. R.Eff( 2 DS  ) are maximum when 2 2 0  and decreasing otherwise Mathematics |182 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 iii. The Bias Ratio of 2 2 1 DS DS DS2 n 1 [B( ) Bias( )]      are reasonably small when  2  2 0  , otherwise B( 2 DS  ) will be maximum. iv. B( 2 DS  ) are reasonably small with small value of . v. R.Eff( 2 DS  ) and B( 2 DS  ) are decreasing function with respect to first sample size n1. vi. The Expected values of sample size of 2 DS  is closer to n1 specially when  2  2 0  and start faraway slowly with increasing of . vii. Percentage of the overall sample saved [ 2n n j0(a1,b1)100] is a decreasing function of , and has a maximum value when  2  2 0  . viii. R.Eff( 2 DS  ) is an increasing function with respect to u (u = n2/n1). iv. The suggested estimator 2 DS  is more efficient than the estimators introduced by Al- Bermani [2] and AL-Joboori[ 4 ] . Conclusions From the above discussions, it is obvious that by using guess point value one can improve the usual estimator by using shrinkage technique. It can be noted that if the guess point is very close to the actual value of the parameter, the proposed estimator perform is better than the usual estimator. If one has no confidence in the guessed value, then proposed preliminary test shrunken estimators can be suggested. We can safely use the proposed estimators for small sample size at the usual level of significance and moderate value of shrunken weight factor . The difficulty of obtaining samples is because of the scarcity and high cost led researchers to use the double stage shrinkage estimators to reduce the sample size that we need and for achieving savings of the items in the sample and obtaining high efficiency estimators. References 1. Ahsanullah,M. et al.,(2014)," Normal and Student’s t Distributions and Their Applications", Atlantis Press and the authors. DOI: 10.2991/978-94-6239-061-4_2. 2. Al-Bermani,M.H.,(2008), "comparison between Bayesian Shrinkage Estimators and Shrinkage Estimators for the variance of normal distribution by using simulation",ph.D. thesis, college of Administration and economics, university of Baghdad. 3. Al-Joboori,A.N.(2010) , “Pre-Test Single and Double Stage Shrunken Estimators for the Mean of Normal Distribution With Known Variance”, Baghdad J. for Science, 7 (4), 1432- 1442. 4. Al-Joboori,A.N.,(2011), Single and Double Stage Shrunken Estimators for the Variance of Normal Distribution". Journal of Education, AL-Mustansiryia. University, 2 , 597-608. 5. Al-Joboori, A.N., et. al , (2014), Single and Double Stage Shrinkage Estimators for the Normal Mean with the Variance Cases , International Journal of Statistics ,.(38),2,.1127 - 1134. Mathematics |183 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 6. Al-Joboori, A.N. and Ameen, M. M.(2016), On Double Stage Minimax-Shrinkage Estimator For Generalized Rayleigh Model, International Journal of Applied Mathematical Research, 5 (1 ), 39-47. 7. Katti, S.K., (1962), Use of Some a Prior Knowledge in the Estimation of Means from Double Samples, Biometrics,.18,.139-147. 8. Handa, B.K., Kambo, N.S. and Al-Hemyari; Z.A., (1988), "On Double Stage Shrunken Estimator of the Mean of Exponential Distribution", IAPQR Trans,.( 13),1,.19-33. 9. Maha.A. M.(2011) , Double Stage Shrinkage Estimator of Two Parameters Generalized Rayleigh Distribution, Education college journal AL-Mustansiriya University (2), 566-573 10. Thompson, J.R., (1968), Some Shrinkage Techniques for Estimating the Mean, J. Amer. Statist. Assoc,.63.113-122. 11. Waikar, V.B., Schuurmann, F.J. and Raghunathar, T.E., (1984), On a Two-Stage Shrinkage Testimator of the Mean of a Normal Distribution, Commum. Statist-Theor. Meth., A,.13 (15),.1901-1913. 12. https://www3.nd.edu/~rwilliam/stats1/x21.pdf. Table (1): Shows Bias ratio [B()] and R.E.ff w.r.t. α, n1 and λ when u =2 λ α n 1 R.Eff(-) B(-) 0.25 0.50 0.75 1 1.25 1.50 1.75 2 0 .0 1 5 R.Eff(-) B(-) 3.4225875 -1.3026559 6.921051 -1.5453646 17.5040876 -0.8971937 34.055869 0.0153905 19.9017239 0.9868869 8.5142224 1.9653436 4.335779 2.9382656 2.5633057 3.902073 7 R.Eff(-) B(-) 2.5218328 -1.4307066 4.5657778 -2.20102 12.7316908 -1.3479203 28.0764855 0.0157813 14.2199712 1.4734652 5.5730906 2.9296484 2.7507472 4.3617489 1.6013574 5.76015 9 R.Eff(-) B(-) 2.2344079 -1.3689823 3.4244664 -2.7706791 10.2563163 -1.7927305 25.4964189 0.0159786 11.3593618 1.9578157 4.1642427 3.8804139 2.0076996 5.7438821 1.1539179 7.5241258 1 1 R.Eff(-) B(-) 2.1910011 -1.2082129 2.7529803 -3.2566881 8.6693885 -2.2313379 24.0612878 0.0160965 9.5571863 2.4399322 3.326823 4.8161439 1.574551 7.0770414 0.8951149 9.1721113 1 3 R.Eff(-) B(-) 2.2811736 -1.0097075 2.3156926 -3.6626852 7.5393297 -2.6635711 23.1480114 0.0161745 8.2890208 2.9197667 2.7683441 5.7353787 1.2902951 8.3540981 0.7267732 10.684783 0 .0 5 5 R.Eff(-) B(-) 3.6194909 -0.8929279 6.1808134 -1.208987 14.5712829 -0.7460543 27.1849786 0.0480284 17.6374379 0.9402995 7.933749 1.8412993 4.0761615 2.7191497 2.406342 3.5619491 7 R.Eff(-) B(-) 2.9246212 -0.8955581 4.2077426 -1.6696883 10.8292699 -1.1277614 22.9410048 0.0490329 12.8134551 1.3835582 5.1866676 2.6961091 2.5615348 3.9248409 1.4850268 5.0444625 9 R.Eff(-) B(-) 2.7826467 -0.7871207 3.25824 -2.0348087 8.8296177 -1.4992136 21.0500992 0.0495215 10.3076319 1.8205121 3.8608721 3.5130704 1.8563702 5.0225485 1.0656958 6.3026994 1 1 R.Eff(-) B(-) 2.8635758 -0.6412647 2.7085502 -2.3148738 7.5250407 -1.8601538 19.9818656 0.0498112 8.7026587 2.2514014 3.0727202 4.2914965 1.4509899 6.008853 0.8312489 7.3316557 1 3 R.Eff(-) B(-) 3.0579131 -0.4968093 2.3588939 -2.5203237 6.5853355 -2.2105946 19.2959357 0.049997 7.561666 2.6762825 2.5489063 5.0307925 1.1900405 6.88241 0.6854149 8.1350713 0 .1 5 R.Eff(-) B(-) 3.809139 -0.7040529 5.8752143 -1.0102327 12.8310536 -0.6359713 22.523826 0.0714824 15.5288307 0.8853387 7.3579544 1.7015398 3.8501684 2.4774536 2.2911515 3.1963249 7 R.Eff(-) B(-) 3.2170698 -0.6720805 4.1163008 -1.3703967 9.6938813 -0.967212 19.242412 0.0727881 11.4827459 1.2837426 4.8503737 2.4483068 2.4332978 3.4831665 1.4289644 4.3586444 9 R.Eff(-) B(-) 3.1448576 -0.5642185 3.2715374 -1.6389512 7.9930075 -1.2865157 17.7562484 0.0733923 9.3307197 1.6732909 3.630191 3.1433387 1.7788781 4.3485916 1.0457584 5.2481266 1 1 R.Eff(-) B(-) 3.2748593 -0.4403664 2.7871275 -1.8297872 6.8728943 -1.5937468 16.9099419 0.0737264 7.9327245 2.054479 2.9040967 3.7879236 1.4075725 5.0795818 0.8379534 5.886642 1 3 R.Eff(-) B(-) 3.4937754 -0.3276427 2.4844107 -1.9555015 6.0604952 -1.8890279 16.3637492 0.0739318 6.9291647 2.4275048 2.4226542 4.3832542 1.1723785 5.6833953 0.7136349 6.3014258 https://www3.nd.edu/~rwilliam/stats1/x21.pdf Mathematics |184 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Table (2): shows Expected Sample Size of w.r.t. α, u, and λ when u=2 Table (3): shows Expected Sample Size Proportion w.r.t. α, u, n1 and λ λ u n1 α 0.25 0.50 0.75 1 1.25 1.50 1.75 2 2 5 0.01 0.05 0.1 0.6308451 0.4107491 0.3518939 0.34 0.3390883 0.3406987 0.3430539 0.3457927 0.7306957 0.4935137 0.3959724 0.3666667 0.3633845 0.3694604 0.3791265 0.3904612 0.7809504 0.5524243 0.4397922 0.4 0.3948049 0.4046508 0.4209772 0.4402237 7 0.01 0.05 0.1 0.727498 0.4396426 0.3552634 0.34 0.339943 0.3433398 0.3481827 0.3541712 0.8196781 0.5363242 0.4035673 0.3666667 0.3664974 0.3792894 0.3976147 0.4191827 0.8608605 0.5995286 0.4497106 0.4 0.3997576 0.4204508 0.4501507 0.4840935 9 0.01 0.05 0.1 0.8028776 0.4689237 0.3586518 0.34 0.3408684 0.3463768 0.3545074 0.3651045 0.8807625 0.576462 0.4110165 0.3666667 0.369759 0.3897431 0.4178015 0.4511351 0.9123294 0.6421232 0.4593403 0.4 0.4048729 0.4366387 0.4800529 0.5288369 11 0.01 0.05 0.1 0.8598196 0.4983351 0.3620964 0.34 0.3418434 0.3497779 0.3619909 0.3785392 0.9220211 0.6140456 0.4184097 0.3666673 0.3731012 0.4006598 0.4392468 0.4852084 0.945175 0.6806899 0.468797 0.4 0.4100512 0.4529807 0.5100832 0.5730448 13 0.01 0.05 0.1 0.9017526 0.5275974 0.3656082 0.34 0.3428607 0.3535288 0.3706006 0.3943621 0.9494897 0.6490968 0.4257681 0.366667 0.3765001 0.4119551 0.4616244 0.5204888 0.9659467 0.7155566 0.4781183 0.4000006 0.4152593 0.4693671 0.5398603 0.6157779 Table (4): shows Percentage of Overall Sample Saved w.r.t. α, u, n1 and λ λ u n1 α 0.25 0.50 0.75 1 1.25 1.50 1.75 2 2 5 0.01 0.05 0.1 36.9154908 58.9250912 64.8106099 66 66.0911727 65.9301318 65.6946101 65.4207316 26.93043 50.6486313 60.402762 63.3333333 63.6615452 63.0539602 62.0873511 60.9538781 21.9049616 44.7575669 56.0207761 60 60.5195074 59.5349218 57.9022763 55.9776293 7 0.01 0.05 0.1 27.2502024 56.035744 64.4736623 66 66.0056967 65.6660242 65.1817254 64.5828769 18.0321895 46.3675819 59.6432699 63.3333333 63.350258 62.0710574 60.2385331 58.0817348 13.9139454 40.0471431 55.0289363 60 60.0242367 57.9549245 54.9849309 51.5906472 9 0.01 0.05 0.1 19.7122408 53.1076264 64.1348246 66 65.9131643 65.3623175 64.5492614 63.4895536 11.9237526 42.3537951 58.898351 63.3333333 63.0241004 61.025691 58.2198476 54.8864896 8.7670569 35.7876824 54.0659732 60 59.5127109 56.3361273 51.994714 47.1163077 11 0.01 0.05 0.1 14.0180372 50.1664879 63.7903596 66 65.8156573 65.0222124 63.8009135 62.1460797 7.797889 38.5954388 58.1590257 63.3332681 62.6898751 59.9340168 56.0753164 51.4791632 5.4824996 31.9310052 53.1202997 60 58.9948844 54.7019278 48.9916837 42.6955182 13 0.01 0.05 0.1 9.824736 47.2402616 63.4391808 66 65.7139302 64.6471179 62.9399354 60.5637872 5.0510283 35.0903195 57.4231882 63.3332994 62.3499858 58.8044877 53.8375641 47.9511213 3.4053325 28.4443414 52.1881723 59.9999423 58.4740712 53.0632869 46.0139709 38.4222114 λ u n1 α 0.25 0.50 0.75 1 1.50 1.25 1.75 2 2 5 0.01 0.05 0.1 9.4626764 6.1612363 5.2784085 5.1 5.0863241 5.1104802 5.1458085 5.1868903 10.9604355 7.4027053 5.9395857 5.5 5.4507682 5.541906 5.6868973 5.8569183 11.7142558 8.286365 6.5968836 6 5.9220739 6.0697617 6.3146586 6.6033556 7 0.01 0.05 0.1 15.2774575 9.2324937 7.4605309 7.14 7.1388037 7.2101349 7.3118377 7.4375958 17.2132402 11.2628078 8.4749133 7.7 7.6964458 7.9650779 8.349908 8.8028357 18.0780715 12.5900999 9.4439234 8.4 8.3949103 8.8294659 9.4531645 10.1659641 9 0.01 0.05 0.1 21.677695 12.6609409 9.6835974 9.18 9.2034457 9.3521743 9.5716994 9.8578205 23.7805868 15.5644753 11.0974452 9.9 9.9834929 10.5230634 11.2806411 12.1806478 24.6328946 17.3373257 12.4021872 10.8 10.9315681 11.7892456 12.9614272 14.2785969 11 0.01 0.05 0.1 28.3740477 16.445059 11.9491813 11.22 11.2808331 11.5426699 11.9456985 12.4917937 30.4266966 20.2635052 13.8075215 12.1000215 12.3123412 13.2217745 14.4951456 16.0118761 31.1907751 22.4627683 15.4703011 13.2 13.5316881 0.4529807 0.5100832 0.5730448 13 0.01 0.05 0.1 35.168353 20.576298 14.2587195 13.26 13.3715672 13.787624 14.4534252 15.380123 37.030099 25.314775 16.604957 14.300013 14.6835055 16.0662498 18.00335 20.2990627 37.6719203 27.9067068 18.6466128 15.6000225 16.1951122 18.3053181 21.0545513 24.0153376 Mathematics |185 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Table (5): shows Bias ratio [B()] and R.E.ff w.r.t. α, n1 and λ when u =6 λ α n 1 R.Eff(-) B(-) 0.25 0.50 0.75 1 1.25 1.50 1.75 2 0 .0 1 5 R.Eff(-) B(-) 11.5006501 -1.5075366 33.8896946 -1.6724435 128.4109443 -0.9400311 367.1232704 6.5959247E-3 144.7705775 0.9894473 50.2690886 1.9725209 23.7539585 2.9485466 13.5001075 3.9153134 7 R.Eff(-) B(-) 7.824044 -1.6642392 19.8793331 -2.3842134 85.9709005 -1.4066272 293.141886 6.7634277E-3 97.6234315 1.4801297 31.6149811 2.9441184 14.4828724 4.3834731 8.0291786 5.7900406 9 R.Eff(-) B(-) 6.7130198 -1.600479 13.6628666 -3.0082667 65.1865558 -1.8677671 261.9441267 6.8479789E-3 74.7424065 1.9690037 22.8642487 3.9040281 10.1646016 5.7813398 5.482838 7.5781944 1 1 R.Eff(-) B(-) 6.5571466 -1.418965 10.2569075 -3.5457157 52.5095898 -2.3232238 244.7941858 6.8984803E-3 60.8410979 2.4560206 17.7495447 4.8507276 7.6624893 7.1345046 4.0189104 9.2577111 1 3 R.Eff(-) B(-) 6.9381658 -1.1906399 8.1644261 -3.9992212 43.8607296 -2.7728235 233.9596269 6.9319302E-3 51.3673362 2.9411151 14.3824044 5.782728 6.0326161 8.4356936 3.0809526 10.8086101 0 .0 5 5 R.Eff(-) B(-) 11.9571477 -1.0751756 26.3678738 -1.3863891 96.7325926 -0.8374679 279.9319496 0.020581 120.1103596 0.9486396 41.7774658 1.8700222 19.2261266 2.7618359 10.5928915 3.6163367 7 R.Eff(-) B(-) 9.2710531 -1.0793316 16.0497764 -1.9078691 64.8867595 -1.2501509 226.6866491 0.0210104 80.7341533 1.4074243 25.5180318 2.7515257 11.1806818 4.0054699 5.9511695 5.1488562 9 R.Eff(-) B(-) 8.8439706 -0.9505521 11.5218048 -2.3241581 49.1298125 -1.6521752 203.8020931 0.0212187 61.3853871 1.8605786 17.9695621 3.5978076 7.5716586 5.1465369 3.9374258 6.4646175 1 1 R.Eff(-) B(-) 9.371124 -0.7761139 9.0678239 -2.6461336 39.493593 -2.0434257 191.1080902 0.0213418 49.5700178 2.3081601 13.6334173 4.4076609 5.5762673 6.1798519 2.8671892 7.5543804 1 3 R.Eff(-) B(-) 10.4950917 -0.6025631 7.579985 -2.8847259 32.9274671 -2.423926 183.0477624 0.0214203 41.5061007 2.7501513 10.8394335 5.1800754 4.3402899 7.102504 2.2319648 8.4182427 0 .1 5 R.Eff(-) B(-) 12.7626413 -0.8649671 23.7335622 -1.2001307 78.9331919 -0.7526722 215.928713 0.030574 99.0740976 0.897929 35.1681173 1.7493915 16.1459181 2.5500459 8.8558915 3.2882648 7 R.Eff(-) B(-) 10.5600862 -0.82466 15.0114678 -1.6168933 53.6450635 -1.1218638 176.6596949 0.0311074 66.9926153 1.3214546 21.3915861 3.6124198 5.0146261 4.5197047 5.0146261 4.5197047 9 R.Eff(-) B(-) 10.5459862 -0.692626 11.1664975 -1.929093 41.008249 -1.4778917 159.5533794 0.0313403 51.0635638 1.7365953 6.3899673 3.2772928 3.4077472 4.5357948 3.4077472 5.4777418 1 1 R.Eff(-) B(-) 11.4988003 9.0868879 33.2356514 -2.1522249 33.2356514 -1.8208237 150.0039513 0.0314575 41.2888982 2.1435705 11.4400968 3.9657723 4.782322 5.3237704 2.578555 6.179403 1 3 R.Eff(-) B(-) 13.0387903 -0.403068 7.8389809 -2.3003672 27.9187493 -2.1508414 143.9156124 0.0315197 34.6026821 2.5424719 9.1401254 4.604823 3.8043577 5.9819398 2.0999618 6.6488272 Mathematics |186 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Table (6): shows Expected Sample Size of w.r.t. α, u, and λ Table (7): shows Expected Sample Size Proportion w.r.t. α, u, n1 and λ λ u n1 α 0.25 0.50 0.75 1 1.25 1.50 1.75 2 6 5 0.01 0.05 0.1 0.5253723 0.2423917 0.1667207 0.1514286 0.1502564 0.1523269 0.155355 0.1588763 0.6537516 0.3488033 0.2233931 0.1857143 0.1814944 0.1893062 0.2017341 0.2163073 0.7183648 0.4245456 0.2797329 0.2285714 0.221892 0.234551 0.2555422 0.2802876 7 0.01 0.05 0.1 0.6496403 0.2795404 0.1710529 0.1514286 0.1513553 0.1557225 0.1619492 0.1696487 0.7681576 0.4038454 0.233158 0.1857143 0.1854967 0.2019435 0.2255046 0.2532348 0.8211064 0.4851082 0.2924851 0.2285714 0.2282598 0.2548653 0.2930509 0.3366917 9 0.01 0.05 0.1 0.7465569 0.3171877 0.1754094 0.1514286 0.152545 0.1596273 0.1700809 0.1837057 0.8466946 0.4554512 0.2427355 0.1857143 0.1896901 0.215384 0.2514591 0.2943166 0.8872807 0.5398727 0.3048661 0.2285714 0.2348366 0.2756784 0.3314965 0.3942189 11 0.01 0.05 0.1 0.8197681 0.3550023 0.1798382 0.1514286 0.1537987 0.1640001 0.1797025 0.200979 0.8997414 0.5037729 0.2522411 0.1857151 0.1939873 0.2294198 0.2790316 0.338125 0.9295107 0.5894585 0.3170247 0.2285714 0.2414943 0.2966895 0.3701069 0.4510576 13 0.01 0.05 0.1 0.873682 0.3926252 0.1843534 0.1514286 0.1551066 0.1688228 0.1907723 0.2213227 0.9350582 0.5488387 0.2617019 0.1857147 0.1983573 0.2439423 0.3078027 0.3834856 0.9562172 0.634287 0.3290092 0.2285722 0.2481905 0.3177577 0.4083918 0.5060001 Table (8): shows Percentage of Overall Sample Saved w.r.t. α, u, n1 and λ λ u n1 α 0.25 0.50 0.75 1 1.25 1.50 1.75 2 6 5 0.01 0.05 0.1 47.4627739 75.7608316 83.327927 84.8571429 84.9743648 84.7673123 84.4644987 84.1123692 34.6248385 65.1196688 77.660694 81.4285714 81.8505581 81.0693774 79.8265943 78.3692718 28.1635221 57.5454432 72.0267121 77.1428571 77.8107952 76.5448994 74.4457838 71.9712376 7 0.01 0.05 0.1 35.0359746 72.0459566 82.8947087 84.8571429 84.8644672 84.4277453 83.8050755 83.0351275 23.1842436 59.6154625 76.6842041 81.4285714 81.4503317 79.8056453 77.4495426 74.6765161 17.8893584 51.489184 70.7514895 77.1428571 77.1740186 74.5134743 70.6949112 66.3308321 9 0.01 0.05 0.1 25.3443096 68.281234 82.4590602 84.8571429 84.7454969 84.0372654 82.9919076 81.6294261 15.330539 54.4548794 75.7264513 81.4285714 81.0309862 78.4616028 74.8540898 70.5683438 0032202111 46.0127345 69.5133941 77.1428571 76.5163426 72.4321637 66.8503466 60.5781099 11 0.01 0.05 0.1 18.0231907 64.4997701 82.0161766 84.8571429 84.6201308 83.5999874 82.029746 79.9021024 10.0258572 49.622707 74.7758902 81.4284876 80.601268 77.0580216 72.0968354 66.1874956 7.0489281 41.0541495 68.2975282 77.1428571 75.8505657 70.3310501 62.9893076 54.8942377 13 0.01 0.05 0.1 12.6318034 60.7374792 81.564661 84.8571429 84.4893389 83.1177231 80.9227741 77.8677263 6.4941792 45.1161251 73.8298134 81.4285277 80.1642675 75.6057699 69.2197253 61.6514417 4.3782846 36.5712961 67.0990787 77.1427829 75.1809487 68.224226 59.1608198 49.3999861 λ u n 1 α 0.25 0.50 0.75 1 1.50 1.25 1.75 2 6 5 0.01 0.05 0.1 18.3880292 8.4837089 5.8352256 5.3 5.2589723 5.3314407 5.4374254 5.5606708 22.8813065 12.2081159 7.8187571 6.5 6.3523047 6.6257179 7.060692 7.5707549 25.1427673 14.8590949 9.7906508 8 7.7662217 8.2092852 8.9439757 9.8100668 7 0.01 0.05 0.1 31.8323725 13.6974812 8.3815927 7.42 7.4164111 7.6304048 7.935513 8.3127875 37.6397206 19.7884234 11.42474 9.1 9.0893375 9.8952338 11.0497241 12.4085071 40.2342144 23.7702998 14.3317701 11.2 11.1847309 12.4883976 14.3594935 16.4978923 9 0.01 0.05 0.1 47.033085 19.9828226 11.0507921 9.54 9.610337 10.0565228 10.7150982 11.5734615 53.3417604 28.693426 15.2923357 11.7 11.9504787 13.5691903 15.8419234 18.5419434 55.8986839 34.0119772 19.2065617 14.4 14.7947042 17.3677369 20.8842816 24.8357908 1 1 0.01 0.05 0.1 63.1221431 27.335177 13.847544 11.66 11.8424993 12.6280097 13.8370956 15.4753811 69.2800899 38.7905156 19.4225645 14.3000646 14.9370236 17.6653234 21.4854368 26.0356284 71.5723254 45.3883048 24.4109033 17.6 18.5950644 22.8450915 28.4982332 34.731437 1 3 0.01 0.05 0.1 79.5050589 35.7288939 16.7761585 13.78 14.1147016 15.362872 17.3602755 20.140369 85.0902969 49.9443262 23.8148698 16.9000397 18.0505166 22.1987494 28.01005 34.8971881 87.015761 57.7201205 29.9398384 20.8000675 22.5853367 28.9159543 37.163654 46.0460127 Mathematics |187 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Table (9): shows Bias ratio [B()] and R.E.ff w.r.t. α, n1 and λ when u =8 λ α n1 R.Eff(-) B(-) 0.25 0.50 0.75 1 1.25 1.50 1.75 2 0 .0 1 5 R.Eff(-) B(-) 15.7992081 -1.5416833 51.243284 -1.6936233 221.9193302 -0.9471706 750.5055682 5.1301631E-3 255.3246925 0.989874 83.6829607 1.9737171 38.7486865 2.9502601 21.7570903 3.9175201 7 R.Eff(-) B(-) 10.5963984 -1.7031613 29.171536 -2.4147456 145.3474248 -1.4164116 596.7733005 5.260443E-3 169.5630469 1.4812404 51.9979693 2.9465301 23.2798893 4.3870937 12.68689 5.7950224 9 R.Eff(-) B(-) 9.045174 -1.6390618 19.616913 -3.0478647 108.3130188 -1.8802732 532.1530326 5.3262048E-3 128.2543243 1.9708684 37.1936914 3.9079638 16.0918181 5.7875827 8.4817049 7.5872058 11 R.Eff(-) B(-) 8.8373127 -1.4540904 14.486276 -3.5938869 86.0098203 -2.3385382 496.6870436 5.3654835E-3 103.3795428 2.458702 28.5774649 4.8564916 11.945746 7.1440818 6.0866627 9.2719778 13 R.Eff(-) B(-) 9.3905281 -1.2207954 11.3847566 -4.0553105 70.9699266 -2.7910323 474.3030401 5.3914999E-3 86.5662921 2.9446732 22.9288293 5.7906195 9.2631688 8.4492929 4.5730995 10.8292479 0 .0 5 5 R.Eff(-) B(-) 16.3784606 -1.1055502 38.1751751 -1.4159561 158.3777524 -0.8527035 548.2465274 0.0160065 202.6710004 0.9500296 65.8214265 1.8748094 29.3966741 2.7689502 15.8692206 3.6254012 7 R.Eff(-) B(-) 12.6272074 -1.1099605 22.6975465 -1.9475659 103.5790628 -1.2705491 441.2897603 0.01634 133.5242593 1.411402 39.3949258 2.7607618 16.6710742 4.0189081 8.6530577 5.1662551 9 R.Eff(-) B(-) 12.074578 -0.9777906 16.0400529 -2.372383 76.9340319 -1.6776688 395.5866532 0.0165016 99.9286385 1.8672564 27.2668572 3.6119305 11.0550078 5.1672016 5.5915848 6.4916038 11 R.Eff(-) B(-) 12.8963957 -0.7985888 12.4842686 -2.7013436 60.9012672 -2.0739711 370.3070947 0.0165969 79.6540237 2.3176199 20.38264 4.4270216 8.0009247 6.2083517 3.9977397 7.5915012 13 R.Eff(-) B(-) 14.6253932 -0.6201888 10.3513612 -2.9454596 50.1327912 -2.4594812 354.283013 0.0166575 65.9664258 2.7624628 15.9974695 5.2049559 6.1379857 7.1391863 3.0679499 8.465438 0 .1 5 R.Eff(-) B(-) 17.5479115 -0.8917862 33.8859019 -1.2317803 125.6270148 -0.7721223 408.0886038 0.0237559 161.466766 0.9000274 53.2351603 1.7573668 23.6664404 2.5621447 12.711558 3.3035882 7 R.Eff(-) B(-) 14.5043071 -0.8500899 21.0108329 -1.6579761 83.2637693 -1.1476391 331.625787 0.0241606 106.8234348 1.3277399 31.7118932 2.5537544 13.4014449 3.633962 7.0154928 4.5465481 9 R.Eff(-) B(-) 14.5867575 -0.7140273 15.4296581 -1.97745 62.481054 -1.5097877 298.5410309 0.0243316 80.0743853 1.747146 21.9405279 3.2996185 8.9865301 4.5669953 4.682014 5.5160111 11 R.Eff(-) B(-) 16.1093432 -0.5579119 12.4465036 -2.2059645 49.9081068 -1.8586699 280.1330665 0.0244127 63.8832813 2.1584191 16.4581892 3.9954137 6.6336712 5.3644685 3.4968988 6.2281965 13 R.Eff(-) B(-) 18.5877299 -0.4156389 10.6714167 -2.3578448 41.4304458 -2.194477 268.4201894 0.024451 52.94306 2.561633 13.0032137 4.6417512 5.2203146 6.0316972 2.8206194 6.7067275 Mathematics |188 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Table (10): shows Expected Sample Size w.r.t. α, u, and λ when u=8 Table (11): shows Expected Sample Size Proportion w.r.t. α, u, n1 and λ λ u n1 α 0.25 0.50 0.75 1 1.25 1.50 1.75 2 8 5 0.01 0.05 0.1 0.5077935 0.2143321 0.1358585 0.12 0.1187844 0.1209316 0.1240719 0.1277236 0.6409276 0.3246849 0.1946298 0.1555556 0.1511794 0.1592805 0.1721687 0.1872816 0.7079338 0.4032324 0.2530563 0.2 0.1930732 0.206201 0.2279696 0.2536316 7 0.01 0.05 0.1 0.636664 0.2528567 0.1403512 0.12 0.119924 0.124453 0.1309103 0.138895 0.7595708 0.3817656 0.2047564 0.1555556 0.1553299 0.1723859 0.1968196 0.2255769 0.8144807 0.4660381 0.2662808 0.2 0.1996768 0.2272677 0.2668676 0.3121247 9 0.01 0.05 0.1 0.636664 0.2528567 0.1403512 0.12 0.119924 0.124453 0.1309103 0.138895 0.8410166 0.4352827 0.2146887 0.1555556 0.1596787 0.1863241 0.2237354 0.2681801 0.8831059 0.5228309 0.2791204 0.2 0.2064972 0.2488516 0.3067371 0.3717826 11 0.01 0.05 0.1 0.7371701 0.2918983 0.144869 0.12 0.1211578 0.1285024 0.1393432 0.1534726 0.8960281 0.4853941 0.2245463 0.1555564 0.164135 0.2008798 0.2523291 0.3136112 0.9269 0.5742533 0.2917293 0.2 0.2134015 0.270641 0.3467776 0.4307264 13 0.01 0.05 0.1 0.8690035 0.3701298 0.1541443 0.12 0.1238143 0.1380384 0.1608009 0.1924828 0.932653 0.5321291 0.2343575 0.155556 0.1686669 0.2159402 0.2821658 0.3606517 0.9545956 0.6207421 0.3041577 0.2000008 0.2203457 0.2924895 0.3864804 0.4877038 Table (12): shows Percentage of Overall Sample Saved w.r.t. α, u, n1 and λ λ u n1 α 0.25 0.50 0.75 1 1.25 1.50 1.75 2 8 5 0.01 0.05 0.1 49.2206544 78.5667883 86.4141465 88 88.1215635 87.9068424 87.5928135 87.2276421 35.9072399 67.5315084 80.537016 84.4444444 84.8820603 84.0719469 82.7831348 81.2718374 29.2066155 59.6767559 74.6943681 80 80.6926765 79.3798957 77.2030351 74.636839 7 0.01 0.05 0.1 36.3336033 74.7143254 85.9648831 88 88.0075957 87.5546989 86.9089672 86.1105026 24.0429193 61.8234426 79.5243598 84.4444444 84.4670106 82.7614099 80.3180442 77.442313 18.5519272 53.3961908 73.3719151 80 80.0323156 77.2732327 73.3132412 68.7875296 9 0.01 0.05 0.1 36.3336033 74.7143254 85.9648831 88 88.0075957 87.5546989 86.9089672 86.1105026 15.8983367 56.4717268 78.5311347 84.4444444 84.0321339 81.3675881 77.6264635 73.1819862 11.6894092 47.7169099 72.0879642 80 79.3502812 75.1148365 69.3262854 62.8217436 11 0.01 0.05 0.1 26.2829877 70.8101686 85.5130995 88 87.884219 87.1497567 86.0656819 84.6527382 10.3971853 51.460585 77.5453677 84.4443575 83.5865002 79.9120224 74.7670885 68.6388843 7.3099995 42.5746736 70.8270663 80 78.6598459 72.9359038 65.3222449 56.9273576 13 0.01 0.05 0.1 13.0996479 62.9870155 84.5855744 88 87.6185737 86.1961572 83.9199139 80.7517162 6.7347044 46.7870927 76.5642509 84.4443991 83.1333144 78.4059836 71.7834189 63.9348284 4.5404433 37.9257886 69.5842298 79.999923 77.9654283 70.7510492 61.3519612 51.2296152 λ u n1 α 0.25 0.50 0.75 1 1.50 1.25 1.75 2 8 5 0.01 0.05 0.1 22.8507055 9.6449453 6.1136341 5.4 5.3452964 5.4419209 5.5832339 5.7475611 28.841742 14.6108212 8.7583428 7 6.8030729 7.1676239 7.7475893 8.4276732 31.857023 18.1454599 11.3875344 9 8.6882956 9.2790469 10.2586342 11.4134224 7 0.01 0.05 0.1 40.1098299 15.929975 8.8421237 7.56 7.5552147 7.8405397 8.2473507 8.7503834 47.8529608 24.0512312 12.8996533 9.8 9.7857833 10.8603118 12.3996322 14.2113428 51.3122859 29.3603998 16.7756935 12.6 12.5796412 14.3178634 16.812658 19.6638564 9 0.01 0.05 0.1 40.1098299 15.929975 8.8421237 7.56 7.5552147 7.8405397 8.2473507 8.7503834 68.1223472 35.2579013 17.3897809 12.6 12.9339716 15.0922537 18.1225646 21.7225912 71.5315786 42.349303 22.608749 16.2 16.7262722 20.1569825 24.8457089 30.1143877 11 0.01 0.05 0.1 59.7107799 23.6437634 11.7343894 9.72 9.8137826 10.4086971 11.2867976 12.4312821 88.7067866 48.0540208 22.230086 15.4000861 16.2493648 19.8870978 24.9805823 31.0475046 91.7631005 56.8510731 28.8812044 19.8 21.1267526 26.7934553 34.3309775 42.641916 13 0.01 0.05 0.1 101.6734119 43.3051919 18.034878 14.04 14.4862688 16.150496 18.8137007 22.520492 109.1203959 62.2591015 27.4198264 18.200053 19.7340221 25.2649992 33.0133999 42.1962508 111.6876814 72.6268274 35.5864512 23.4000901 25.7804489 34.2212724 45.2182054 57.0613502 Mathematics |189 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Table (13): shows Bias ratio [B()] and R.E.ff w.r.t. α, n1 and λ when u =10 λ α n1 R.Eff(-) B(-) 0.25 0.50 0.75 1 1.25 1.50 1.75 2 0 .0 1 5 R.Eff(-) B(-) 20.1608911 -1.5634131 70.0901591 -1.7071014 337.8975179 -0.951714 1.3283519E+3 4.1974058E-3 397.6598069 0.9901455 124.8418315 1.9744783 56.9138232 2.9513505 31.6400086 3.9189244 7 R.Eff(-) B(-) 13.3973709 -1.7279299 39.0274664 -2.4341752 217.5781913 -1.4226381 1.0535918E+3 4.3039982E-3 261.0147471 1.4819473 76.7975902 2.9480647 33.7481223 4.3893978 18.1241341 5.7981926 9 R.Eff(-) B(-) 11.3994496 -1.6636145 25.8338403 -3.0730633 159.9465795 -1.8882317 938.3261324 4.3578032E-3 195.5715793 1.972055 54.4183679 3.9104683 23.0147832 5.7915555 11.8933723 7.5929403 11 R.Eff(-) B(-) 11.1412572 -1.4764429 18.8554033 -3.6245414 125.5839342 -2.3482836 875.1259032 4.38994E-3 156.4276203 2.4604083 41.440367 4.8601595 16.8569689 7.1501764 8.382714 9.2810565 13 R.Eff(-) B(-) 11.8744639 -1.2399852 14.6867545 -4.0910038 102.6244565 -2.8026196 835.2613431 4.411226E-3 130.1326415 2.9469375 32.965176 5.7956414 12.9025777 8.4579469 6.1950229 10.8423811 0 .0 5 5 R.Eff(-) B(-) 20.8613574 -1.1248795 50.5030161 -1.4347714 229.6000232 -0.8623989 929.6749514 0.0130954 302.3386783 0.9509142 93.3190785 1.8778557 40.7379243 2.7734775 21.6438516 3.6311696 7 R.Eff(-) B(-) 16.029387 -1.1294517 29.5492404 -1.9728274 147.3846837 -1.2835298 745.5066524 0.0133679 196.1790652 1.4139333 54.9541643 2.7666393 22.6409719 4.0274596 11.5270072 5.1773272 9 R.Eff(-) B(-) 15.3578356 -0.9951243 20.6607021 -2.4030716 107.9248367 -1.6938919 667.0983423 0.0134998 145.0392399 1.8715058 37.5186889 3.6209178 14.7696354 5.1803519 7.3170393 6.508777 11 R.Eff(-) B(-) 16.497595 -0.812891 15.9609391 -2.7364772 84.4687135 -2.093409 623.805499 0.0135774 114.4561662 2.3236398 27.7215424 4.4393421 10.5487783 6.226488 5.1616786 7.6151236 13 R.Eff(-) B(-) 18.8799178 -0.6314051 13.1624716 -2.9841083 68.8833075 -2.4821073 596.3929771 0.0136267 93.9822353 2.7702973 21.5408041 5.2207889 8.0061526 7.1625296 3.9211192 8.4954713 0 .1 5 R.Eff(-) B(-) 22.4114451 -0.9088528 44.3832194 -1.251921 178.0034482 -0.7844997 670.6915364 0.0194171 233.5318898 0.9013627 72.9627158 1.7624419 31.6895138 2.5698439 16.7617537 3.3133394 7 R.Eff(-) B(-) 18.521388 -0.8662726 27.157685 -1.6841197 115.868162 -1.1640416 542.716628 0.0197399 152.0394491 1.3317397 42.7926667 2.5633405 17.6322489 3.6476707 9.0783545 4.5636303 9 R.Eff(-) B(-) 18.7244342 -0.7276463 19.7741578 -2.0082226 85.7923384 -1.5300851 487.5814411 0.0198716 112.5621374 1.7538601 29.2462296 3.3138258 11.6709842 4.5868502 5.9814853 5.5403642 11 R.Eff(-) B(-) 20.8738002 -0.5685978 15.8588915 -2.2401624 67.8167002 -1.8827538 456.9695181 0.0199297 88.9104098 2.1678682 21.7230208 4.0142765 8.5311762 5.3903673 4.4272972 6.2592469 13 R.Eff(-) B(-) 24.4004725 -0.4236386 13.5423265 -2.3944215 55.8224208 -2.2222451 437.5157543 0.0199527 73.0752811 2.5738265 17.0242182 4.6652509 6.6628313 6.063361 3.5477961 6.7435731 Mathematics |190 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Table (14): shows Expected Sample Size of w.r.t. α, u, and λ Table (15): shows Expected Sample Size Proportion w.r.t. α, u, n1 and λ λ u n1 α 0.25 0.50 0.75 1 1.25 1.50 1.75 2 10 5 0.01 0.05 0.1 0.4966069 0.196476 0.116219 0.1 0.0987567 0.1009527 0.1041644 0.1078991 0.6327669 0.3093368 0.176326 0.1363636 0.131888 0.1401733 0.1533543 0.1688108 0.701296 0.3896695 0.2360803 0.1818182 0.174734 0.1881602 0.2104235 0.2366687 7 0.01 0.05 0.1 0.6284063 0.2358762 0.1208137 0.1 0.0999223 0.1045542 0.1111583 0.1193244 0.7541065 0.3677148 0.1866827 0.1363636 0.1361328 0.1535765 0.1785655 0.2079763 0.8102644 0.4539026 0.2496054 0.1818182 0.1814877 0.2097056 0.2502055 0.2964912 9 0.01 0.05 0.1 0.7311967 0.2758051 0.1254342 0.1 0.1011841 0.1086957 0.1197828 0.1342334 0.8374034 0.4224482 0.1968407 0.1363636 0.1405804 0.1678315 0.206093 0.2515479 0.8804492 0.5119861 0.2627367 0.1818182 0.188463 0.2317801 0.2909812 0.3575049 11 0.01 0.05 0.1 0.8088449 0.3159115 0.1301315 0.1 0.1025138 0.1133335 0.1299875 0.1525535 0.8936652 0.4736986 0.2069224 0.1363645 0.1451381 0.182718 0.2353366 0.2980114 0.9252386 0.5645772 0.2756323 0.1818182 0.1955243 0.2540646 0.3319316 0.4177884 13 0.01 0.05 0.1 0.8660263 0.3558146 0.1349203 0.1 0.103901 0.1184484 0.1417282 0.1741302 0.9311223 0.5214956 0.2169565 0.1363641 0.1497729 0.1981206 0.2658514 0.3461211 0.9535636 0.6121226 0.2883431 0.181819 0.2026263 0.2764097 0.3725368 0.4760608 Table (16) :shows Percentage of Overall Sample Saved w.r.t. α, u, n1 and λ λ u n1 α 0.25 0.50 0.75 1 1.25 1.50 1.75 2 10 5 0.01 0.05 0.1 50.3393056 80.3523972 88.3781044 90 90.1243263 89.9047252 89.5835592 89.2100885 36.7233136 69.0663154 82.3674027 86.3636364 86.811198 85.982673 84.6645697 83.1189246 29.8704022 61.0330458 76.3919674 81.8181818 82.5266009 81.1839842 78.9576495 76.3331308 7 0.01 0.05 0.1 37.159367 76.4123782 87.9186304 90 90.0077683 89.5445784 88.884171 88.0675594 24.5893493 63.2285208 81.3317316 86.3636364 86.3867154 84.642351 82.1434543 79.2023656 18.9735619 54.6097406 75.0394586 81.8181818 81.8512318 79.0294425 74.9794512 70.3508825 9 0.01 0.05 0.1 26.8803284 72.4194906 87.456579 90 89.8815876 89.130433 88.0217202 86.5766641 16.2596626 57.7551751 80.3159332 86.3636364 85.9419551 83.2168514 79.3907013 74.8452131 11.9550776 48.8013851 73.7263271 81.8181818 81.1536967 76.8219918 70.9018828 64.2495105 11 0.01 0.05 0.1 19.1155053 68.4088471 86.986854 90 89.7486236 88.6666533 87.0012457 84.7446541 10.633485 52.6301438 79.3077624 86.3635474 85.4861934 81.7282047 76.4663405 70.1988589 7.4761358 43.5422798 72.4367724 81.8181818 80.4475696 74.5935379 66.8068414 58.2211612 13 0.01 0.05 0.1 13.3973672 64.4185385 86.5079738 90 89.6099049 88.1551608 85.8271847 82.5869825 6.8877658 47.8504357 78.3043475 86.36359 85.0227079 80.1879377 73.4148602 65.3878927 4.6436352 38.7877383 71.1656895 81.8181031 79.7373699 72.3590276 62.746324 52.3939246 λ u n 1 α 0.25 0.50 0.75 1 1.50 1.25 1.75 2 10 5 0.01 0.05 0.1 27.3133819 10.8061816 6.3920426 5.5 5.4316205 5.5524012 5.7290424 5.9344513 34.8021775 17.0135265 9.6979285 7.5 7.2538411 7.7095298 8.4344867 9.2845914 38.5712788 21.4318248 12.984418 10 9.6103695 10.3488087 11.5732928 13.016778 7 0.01 0.05 0.1 48.3872874 18.1624687 9.3026546 7.7 7.6940184 8.0506746 8.5591883 9.1879792 58.0662011 28.314039 14.3745667 10.5 10.4822291 11.8253897 13.7495402 16.0141785 62.3903573 34.9504997 19.2196169 14 13.9745515 16.1473293 19.2658226 22.8298205 9 0.01 0.05 0.1 72.3884749 27.3047043 12.4179868 9.9 10.0172283 10.7608713 11.8584971 13.2891026 82.902934 41.8223766 19.4872261 13.5 13.9174644 16.6153171 20.4032057 24.903239 87.1644732 50.6866287 26.0109362 18 18.6578403 22.9462281 28.8071361 35.3929846 1 1 0.01 0.05 0.1 97.8702386 38.225295 15.7459067 12.1 12.4041654 13.7133495 15.7284927 18.4589686 108.1334832 57.3175261 25.0376075 16.500107 6 17.561706 22.1088723 28.4757279 36.0593807 111.9538756 68.3138414 33.3515054 22 23.6584407 30.7418191 40.1637219 50.5523949 1 3 0.01 0.05 0.1 123.8417649 50.8814899 19.2935975 14.3 14.857836 16.93812 20.2671259 24.900615 133.1504949 74.5738769 31.024783 19.500066 2 21.4175277 28.3312491 38.0167499 49.4953135 136.3596017 87.5335342 41.2330639 26.000112 6 28.9755611 39.5265905 53.2727567 68.0766878 Mathematics |191 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Table (17): shows Probability of a Voiding Second Sample w.r.t. α, n1 and λ ζ n1 α 0.25 0.50 0.75 1 1.25 1.50 1.75 2 5 0.01 0.05 0.1 0.5537324 0.8838764 0.9721591 0.99 0.9913676 0.988952 0.9854192 0.981311 0.4039564 0.7597295 0.9060414 0.95 0.9549232 0.9458094 0.9313103 0.9143082 0.3285744 0.6713635 0.8403116 0.9 0.9077926 0.8930238 0.8685341 0.8396644 7 0.01 0.05 0.1 0.408753 0.8405362 0.9671049 0.99 0.9900855 0.9849904 0.9777259 0.9687432 0.2704828 0.6955137 0.894649 0.95 0.9502539 0.9310659 0.903578 0.871226 0.2087092 0.6007071 0.825434 0.9 0.9003636 0.8693239 0.824774 0.7738597 9 0.01 0.05 0.1 0.2956836 0.7966144 0.9620224 0.99 0.9886975 0.9804348 0.9682389 0.9523433 0.1788563 0.6353069 0.8834753 0.95 0.9453615 0.9153854 0.8732977 0.8232973 0.1315059 0.5368152 0.8109896 0.9 0.8926907 0.8450419 0.7799207 0.7067446 11 0.01 0.05 0.1 0.2102706 0.7524973 0.9568554 0.99 0.9872349 0.9753332 0.9570137 0.9321912 0.1169683 0.5789316 0.8723854 0.949999 0.9403481 0.8990103 0.8411297 0.7721874 0.0822375 0.4789651 0.7968045 0.9 0.8849233 0.8205289 0.7348753 0.6404328 13 0.01 0.05 0.1 0.147371 0.7086039 0.9515877 0.99 0.985709 0.9697068 0.944099 0.9084568 0.0757654 0.5263548 0.8613478 0.9499995 0.9352498 0.8820673 0.8075635 0.7192668 0.05108 0.4266651 0.7828226 0.8999991 0.8771111 0.7959493 0.6902096 0.5763332 Figure (1): Graph for PDF of Normal Distribution