Chapter -2 Simple Random

Simple random sampling (SRS) is a method of selection of a comprising of n a number of sampling units out of the population having N number of sampling units such that every sampling unit has an equal chance of being chosen.

The samples can be drawn in two possible ways.  The sampling units are chosen without replacement because the units once are chosen are not placed back in the population.  The sampling units are chosen with replacement because the selected units are placed back in the population.

1. Simple random sampling without replacement (SRSWOR): SRSWOR is a method of selection of n units out of the N units one by one such that at any stage of selection, any one of the remaining units have the same chance of being selected, i.e. 1/N .

2. Simple random sampling with replacement (SRSWR): SRSWR is a method of selection of n units out of the N units one by one such that at each stage of selection, each unit has an equal chance of being selected, i.e., 1/N .

Procedure of selection of a random sample: The procedure of selection of a random sample follows the following steps: 1. Identify the N units in the population with the numbers 1 to N. 2. Choose any random number arbitrarily in the random number table and start reading numbers. 3. Choose the sampling unit whose serial number corresponds to the random number drawn from the table of random numbers. 4. In the case of SRSWR, all the random numbers are accepted ever if repeated more than once. In the case of SRSWOR, if any random number is repeated, then it is ignored, and more numbers are drawn.

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Such a process can be implemented through programming and using the discrete uniform distribution. Any number between 1 and N can be generated from this distribution, and the corresponding unit can be selected in the sample by associating an index with each sampling unit. Many statistical software like R, SAS, etc. have inbuilt functions for drawing a sample using SRSWOR or SRSWR.

Notations: The following notations will be used in further notes:

N : Number of sampling units in the population (Population size). n : Number of sampling units in the sample (sample size) Y : The characteristic under consideration

th Yi : Value of the characteristic for the i unit of the population 1 n yy  i : sample mean n i1 1 N Yy  i : population mean N i1

NN 222211 SYYYNY()ii ( ) NN11ii11 11NN  2222()YY   ( Y  NY ) NNii ii11 nn 222211 syyyny()ii ( ) nn11ii11

Probability of drawing a sample : 1.SRSWOR: N If n units are selected by SRSWOR, the total number of possible samples are . n 1 So the probability of selecting any one of these samples is . N  n

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th Note that a unit can be selected at any one of the n draws. Let ui be the i unit selected in the sample. This unit can be selected in the sample either at first draw, second draw, …, or nth draw.

th Let Pij () denotes the probability of selection of ui at the j draw, j = 1,2,...,n. Then

Pijn() Pi12 () Pi () ... Pi () 11 1  ... (n times ) NN N n  . N

Now if uu12, ,..., un are the n units selected in the sample, then the probability of their selection is

Pu(12 , u ,..., unn ) Pu ( 1 ). Pu ( 2 ),..., Pu ( ). Note that when the second unit is to be selected, then there are (n – 1) units left to be selected in the sample from the population of (N – 1) units. Similarly, when the third unit is to be selected, there are (n – 2) units left to be selected in the sample from the population of (N – 2) units and so on. n If Pu() , then 1 N n 11 Pu( ) ,..., Pu ( ) . 2 NNn11n Thus nn12 n 1 1 Pu( , u ,.., u ) ...... 12 n NN12 N N  n 1N  n

Alternative approach: The probability of drawing a sample in SRSWOR can alternatively be found as follows:

th th th Let uik() denotes the i unit drawn at the k draw. Note that the i unit can be any unit out of the N units. Then suuoii ((1) , (2) ,..., u in ( ) ) is an ordered sample in which the order of the units in which they are drawn, i.e., ui(1) drawn at the first draw, ui(2) drawn at the second draw and so on, is also considered. The probability of selection of such an ordered sample is

Ps(oiiiiiiiniiin ) Pu ((1) ) Pu ( (2) | u (1) ) Pu ( (3) | u (1) u (2) )... Pu ( ( ) | u (1) u (2) ... u ( 1) ).

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th Here Pu(ik() | u i (1)(2) u i ... u ik ( 1) ) is the probability of drawing uik() at the k draw given that

uuii(1), (2) ,..., u ik ( 1) have already been drawn in the first (k – 1) draws. Such a probability is obtained as 1 Pu( | u u ... u ) . ik() i (1)(2) i ik ( 1) Nk1 So n 1()!Nn Ps()o  . k 1 Nk1! N The number of ways in which a sample of size nn can be drawn  ! ()!Nn Probability of drawing a sample in a given order  N ! So the probability of drawing a sample in which the order of units in which they are drawn is

()!1Nn irrelevantn ! . N ! N  n

2. SRSWR When n units are selected with SRSWR, the total number of possible samples are N n . 1 The Probability of drawing a sample is . N n

th Alternatively, let ui be the i unit selected in the sample. This unit can be selected in the sample either at first draw, second draw, …, or nth draw. At any stage, there are always N units in the population in case of SRSWR, so the probability of selection of ui at any stage is 1/N for all i =

1,2,…,n. Then the probability of selection of n units uu12, ,..., un in the sample is

Pu(12 , u ,.., unn ) Pu ( 1 ). Pu ( 2 )... Pu ( ) 11 1  . ... NN N 1  N n

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Probability of drawing a unit 1. SRSWOR

th Let Ae denotes an event that a particular unit u j is not selected at the  draw. The probability of selecting, say, jth unit at k th draw is

th P (selection of u j at k draw) = PA(12 A .... Akk 1  A )

 PA(121312 ) PA ( A ) PA ( AA )..... PA (kkkk 1122 A , A ...... A ) PA ( A 121 , A ...... A  ) 11 1 1  1 11  1  ...1   NN12 N  NkNk  21   NN12 Nk  11  . ... . N N121 Nk Nk 1  N

2. SRSWR 1 P[ selection of u at kth draw] = . j N

Estimation of population mean and population One of the main objectives after the selection of a sample is to know about the tendency of the data to cluster around the central value and the scatteredness of the data around the central value. Among various measures of central tendency and dispersion, the popular choices are arithmetic mean and variance. So the population mean and population variability are generally measured by the arithmetic mean (or weighted arithmetic mean) and variance, respectively. There are various popular estimators for estimating the population mean and population variance. Among them, sample arithmetic mean and sample variance is more popular than other estimators. One of the reasons to use these estimators is that they possess nice statistical properties. Moreover, they are also obtained through well- established statistical estimation procedures like maximum likelihood estimation, least squares estimation, method of moments etc., under several standard statistical distributions. One may also consider other measures like median, mode, geometric mean, harmonic mean for measuring the central tendency and mean deviation, absolute deviation, Pitman nearness etc. for measuring the dispersion. Numerical procedures like bootstrapping can study the properties of such estimators.

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1. Estimation of population mean 1 n Let us consider the sample arithmetic mean yy  i as an estimator of the population mean n i1 1 N YY  i and verify y is an unbiased estimator of Y under the two cases. N i1 SRSWOR

n Let tyii  . Then i1 1 n Ey() E ( yi ) n i1 1  Et n i

N  11n   ti n N i1  n N  11n n  yi . n N ii11  n When n units are sampled from N units without replacement, each unit of the population can occur with other units selected out of the remaining  N 1 units in the population, and each unit occurs in

N 1 N  the  possible samples. So n 1 n

N  n nNN 1 So yyii   . ii11n 1 i  1 Now (NnNn 1)! !( )! N E()yy  i (1)!()!nNnnN ! i1 1 N   yi N i1  Y .

Thus y is an unbiased estimator of Y .

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Alternatively, the following approach can also be adapted to show the unbiasedness property. Let 1 Pi() denotes the probability of selection of ith unit at jth stage. Then j N 1 n Ey()  Ey (j ) n j1 1 nN  YPij() i n ji11 11nN  Yi . nNji11 1 n  Y n j1 Y

SRSWR 1 n Ey() E ( yi ) n i1 1 n   Ey()i n i1 1 n (YP11 YP 2 2 ... YNN P ) n i1 111n 1 (YY12 ... YN ) nNNi1 N 1 n  Y n i1  Y . 1 where P  for all iN1,2,..., is the probability of selection of a unit. Thus y is an unbiased i N estimator of the population mean under SRSWR also.

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Variance of the estimate Assume that each observation has some variance  2 . Then Vy() Ey ( Y )2 2 1 n EyY()i n i1 nnn 112 EyYyYyY22()iij ()() nniij1

11nnn Ey() Y2 Ey ()() Y y Y 22iij nniij1 1 n K  2 22 nni1 NK1 S 2 Nn n2

nn 2 where K Ey()()ij Y y Y assuming that each observation has variance  . Now we find ij K under the setups of SRSWR and SRSWOR.

SRSWOR

nn KEyYyY ()()ij. ij Consider 1 NN Ey()()ij Y y Y ( y kl  Y )() y Y NN(1) k  Since

NNNN2  2 ()yYkk () yY  ()() yYyY k   kik11  NN 2 0(NS 1)  ( yYyYk  )()  k  NN 112 ()()yYyYk  [(1)] N S NN(1)k  NN (1) S 2  . N

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S 2 Thus Knn(1)  and so substituting the value of K , the variance of y under SRSWOR is N

2 NS112 Vy()WOR  S2 nn (1) Nn n N Nn  S 2. Nn

SRSWR

NN KEyYyY ()()ij ij NN E()()yYEyYij ij  0 because the ith and jth draws ()ij are independent. Thus the variance of y under SRSWR is N 1 Vy() S2 . WR Nn It is to be noted that if N is infinite (large enough), then S 2 Vy() n Nn is both the cases of SRSWOR and SRSWR. So the factor is responsible for changing the N variance of y when the sample is drawn from a finite population in comparison to an infinite Nn population. This is why is called a finite population correction (fpc) . It may be noted that N Nn n Nn n 1, so is close to 1 if the ratio of a sample size to population , is very small or NNN N n negligible. The term is called the sampling fraction. In practice, fpc can be ignored whenever N n  5% and for many purposes, even if it is as high as 10%. Ignoring fpc will result in the N overestimation of the variance of y .

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Efficiency of y under SRSWOR over SRSWR

Nn Vy() S2 WOR Nn N 1 Vy() S2 WR Nn Nn n1 SS22 Nn Nn

V() yWOR a positive quantity Thus

Vy()WR Vy ( WOR ) and so, SRSWOR is more efficient than SRSWR.

Estimation of variance from a sample Since the expressions of of the sample mean, involve S 2 which is based on population values, so these expressions can not be used in real-life applications. In order to estimate the variance of y on the basis of a sample, an estimator of S 2 (or equivalently  2 ) is needed. Consider s2 as an estimator of S 2 (or  2 ) and we investigate its biasedness for s2 in the cases of SRSWOR and SRSWR,

Consider

n 221 syy()i n 1 i1 1 n 2  ()()yYi yY n 1 i1 n 1 22 ()()yYi nyY n 1 i1

n 2221  Es() Ey (i Y ) nEy ( Y ) n 1 i1 11n  2  Var( yi ) nVar () y n nVar () y  nn11i1

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In the case of SRSWOR Nn Vy() S2 WOR Nn and so nNn Es()222 S nNn1  nN1 Nn SS22 nN1  Nn  S 2 In the case of SRSWR N 1 Vy() S2 WR Nn and so

nN1 Es()222 S nNn1  nN11 N SS22 nN1  Nn N 1  S 2 N   2 Hence

SSRSWOR2 in Es()2   2  in SRSWR

An unbiased estimate of Var() y is Nn Vyˆ() s2 in case of SRSWOR and WOR Nn NN1 Vyˆ() . s2 WR Nn N 1

s2  in case of SRSWR. n

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Standard errors

The of y is defined as Var() y . In order to estimate the standard error, one simple option is to consider the square root of the estimate of the variance of the sample mean.

Nn • under SRSWOR, a possible estimator is ˆ()y  s . Nn

N 1 • under SRSWR, a possible estimator is ˆ()ys . Nn

It is to be noted that this estimator does not possess the same properties as of Var() y .  The reason being if ˆ is an estimator of  , then  is not necessarily an estimator of  . In fact, the ˆ()y is a negatively biased estimator under SRSWOR.

The approximate expressions for large N case are as follows: (Reference: Sampling Theory of Surveys with Applications, P.V. Sukhatme, B.V. Sukhatme, S. Sukhatme, C. Asok, Iowa State University Press and Indian Society of Agricultural Statistics, 1984, India)

Consider sS as an estimator of . Let sS22 with E ( )  0, E ( 2 )  S 2 . Write sS (21/2 )

1/2  S  1 2 S 2 S  124 ... 28SS assuming  will be small as compared to S 2 and as n becomes large, the probability of such an event approaches one. Neglecting the powers of  higher than two and taking expectation, we have

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Var() s2 Es() 1 4 S 8S where

4 2 21Sn  Var s13)forlarge.2  N (1)nn  2

1 N j  jiYY N i1    4 : coefficient of kurtosis. 2 S 4 Thus

1 2  3 Es S1  4(nn 1) 8 2 1()Var s2 Var() s S22 S 1  8 S 4  Var() s2  4S 2 Sn2 1  13.2  21nn  2

Note that for a normal distribution, 2  3 and we obtain

S 2 Var() s  . 21n 

Both Var()and s Var ( s2 ) are inflated due to nonnormality to the same extent, by the inflation factor

n 1  132 2n and this does not depend on the coefficient of skewness.

This is an important result to be kept in mind while determining the sample size in which it is assumed that S 2 is known. If inflation factor is ignored and the population is non-normal, then the reliability on s2 may be misleading.

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Alternative approach: The results for the unbiasedness property and the variance of the sample mean can also be proved in an alternative way as follows:

(i) SRSWOR

th With the i unit of the population, we associate a random variable ai defined as follows:

1, if the ith unit occurs in the sample a  i  th 0, if the i unit does not occurs in the sample (iN 1,2,..., ) Then,

th Ea(i ) 1 Probability that the i unit is included in the sample n  ,iN 1,2,..., . N 2 th Ea(i ) 1 Probability that the i unit is included in the sample n  ,iN 1,2,..., N th th Eaa(ij ) 1 Probability that the i and j units are included in the sample nn(1)  ,ij 1,2,..., N . NN(1) From these results, we can obtain

2 nN() n Vara( ) Ea (2 ) Ea ( )  , i  1,2,..., N ii i N 2 nN() n Cov( a , a ) E ( a a ) E ( a ) E ( a )  , i  j 1,2,..., N . ij ij i j NN2 (1) We can rewrite the sample mean as 1 N yay  ii n i1 Then 1 N Ey() Ea (ii ) y Y n i1 and 1 N 1 NN Var() y Var a y Var() a y2 Cov (, a a ) y y . 2  ii 2 ii i jij n  i1  n iij1

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Substituting the values of Var()and aiij Cov (, a a ) in the expression of Var() y and simplifying, we get Nn Var() y S 2 . Nn To show that E()sS22 , consider

nN 2222211 s yiii ny a y ny . (1)nnii11 (1) Hence, taking, expectation, we get

N 2221  Es ( )()() Eaii y nVary Y (1)n  i1

22 Substituting the values of Ea()andi Vary () in this expression and simplifying, we get E()sS .

(ii) SRSWR

th Let a random variable ai associated with the i unit of the population denotes the number of times

th the i unit occurs in the sample iN 1,2,..., . So ai assumes values 0, 1, 2,…,n. The joint distribution of aa12, ,..., aN is the multinomial distribution given by n!1 Pa( , a ,..., a ) . 12 N N N n  ai ! i1

N where  ani  . For this multinomial distribution, we have i1 n Ea() , i N nN(1) Var( a ) , i 1,2,..., N . i N 2 n Cov( a , a ) , i  j  1,2,..., N . ij N 2 We rewrite the sample mean as 1 N yay  ii. n i1

Hence, taking the expectation of y and substituting the value of Ea (i ) n / N we obtain that Ey() Y .

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Further, 1 NN Var() y Var ( a ) y2 Cov (, a a ) y y 2 ii ijij n ii11

22 Substituting, the values of Var() aiij n ( N 1)/ N and Cov (, a a )  n / N and simplifying, we get N 1 Var() y S 2 . Nn N 1 To prove that Es()222 S  in SRSWR, consider N

nN 22222 (1)n s ynyaynyiii   , ii11 N 22 2 (1)()nEs EaynVaryY ()ii  ()  i1 N nN222(1)  yi nSnY.  NnNi1 (1)(1)nN  S 2 N N 1 Es()222 S  N

Estimator of population total: Sometimes, it is also of interest to estimate the population total, e.g., total household income, total expenditures, etc. Let denotes the population total

N YYNYTi i1 which can be estimated by

YNYˆ  ˆ T  Ny. Obviously EYˆ  NEy T  NY

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ˆ 2 Var YT  N Var y

22Nn NNn()  2 N S S for SRSWOR Nn  n    22NNN1(1) 2 N S S for SRSWOR  Nn n ˆ and the estimates of variance of YT are

 NN() n 2  s for SRSWOR  ˆ  n Var() YT   N  s2 for SRSWOR  n

Confidence limits for the population mean Now we construct the 100 (1 ) % confidence interval for the population mean. Assume that the

y Y population is normally distributed N(,  2 ) with mean  and variance  2. then follows Var() y

y Y N(0,1) when  2 is known. If  2 is unknown and is estimated from the sample, then Var() y follows a t -distribution with (n  1) degrees of freedom. When  2 is known, then the 100(1) % confidence interval is given by

yY PZ  Z 1  22Var() y   or P y Z Var() y Y y Z Var () y 1 22 and the confidence limits are  y Z Vary(), y Z Vary () 22

 2 where Z denotes the upper % points on N(0,1) distribution. Similarly, when  is unknown, 2 2 then the 100 (1 ) % confidence interval is

Sampling Theory| Chapter 2 | Simple Random Sampling | Shalabh, IIT Kanpur Page17 17

yY Pt  t 1   22Var() y

  or Py t Vary()  Y y t Vary ()  1 22 and the confidence limits are

  y t Var(), y y t Var () y 22  where t denotes the upper % points on t -distribution with (n  1) degrees of freedom. 2 2

Determination of sample size The size of the sample is needed before the survey starts and goes into operation. One point to be kept ins mind is that when the sample size increases, the variance of estimators decreases but the cost of survey increases and vice versa. So there has to be a balance between the two aspects. The sample size can be determined on the basis of prescribed values of the standard error of the sample mean, the error of estimation, the width of the confidence interval, coefficient of variation of the sample mean, the relative error of sample mean or total cost among several others.

An important constraint or need to determine the sample size is that the information regarding the population standard derivation S should be known for these criteria. The reason and need for this will be clear when we derive the sample size in the next section. A question arises about how to have information about S beforehand? The possible solutions to this issue are to conduct a pilot survey and collect a preliminary sample of small size, estimate S and use it as a known value of S it. Alternatively, such information can also be collected from past data, past experience, the long association of the experimenter with the experiment, prior information, etc.

Now we find the sample size under different criteria assuming that the samples have been drawn using SRSWOR. The case for SRSWR can be derived similarly.

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1. Pre-specified variance The sample size is to be determined such that the variance of y should not exceed a given value, say V. In this case, find n such that Var() y V Nn or ()yV Nn Nn or SV2  Nn 11 V or  nN S2 11 1 or  nN ne n n  e n 1 e N S 2 where n  . e v

2 It may be noted here that ne can be known only when S is known. This reason compels to assume that S should be known. The same reasoning will also be seen in other cases. The smallest sample size needed in this case is n n  e . smallest n 1 e N It N is large, then the required n is

nn e and nnsmallest e .

2. Pre-specified estimation error It may be possible to have some prior knowledge of population mean Y . It may be required that the sample mean y should not differ from it by more than a specified amount of absolute estimation error, i.e., which is a small quantity. Such a requirement can be satisfied by associating a probability (1 ) with it and can be expressed as

 PyY e (1 ).

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Nn 2 Since y follows NY, S assuming the normal distribution for the population, we can write Nn

yY e P 1  Var() y Var () y which implies that e  Z Var() y 2

22 or Z Var() y e 2

222Nn or Z Se 2 Nn

2 ZS  2  e  or n  2 ZS  1 2 1  Ne  which is the required sample size. If N is large then

2 ZS  n  2 . e  

3. Pre-specified width of the confidence interval If the requirement is that the width of the confidence interval of y with confidence coefficient (1 ) should not exceed a pre-specified amount W , then the sample size n is determined such that

2()Z Var y W 2 assuming  2 is known and population is normally distributed. This can be expressed as

Nn 2Z SW 2 Nn

22211 or 4Z SW 2 nN Sampling Theory| Chapter 2 | Simple Random Sampling | Shalabh, IIT Kanpur Page20 20

11 W 2 or 22 nN4 ZS 2

22 4ZS 2 W 2 or n  22. 4Z S 1 2 NW 2

The minimum sample size required is

22 4Z S 2 W 2 nsmallest  22 4Z S 1 2 NW 2 If N is large then

22 4Z S n  2 W 2 and the minimum sample size needed is

22 4Z S n  2 . smallest W 2

4. Pre-specified coefficient of variation The coefficient of variation (CV) is defined as the ratio of standard error (or standard deviation) and mean. The knowledge of the coefficient of variation has played an important role in the sampling theory as this information has helped in deriving efficient estimators.

If it is desired that the coefficient of variation of y should not exceed a given or pre-specified value of the coefficient of variation, say C0 , then the required sample size n is to be determined such that

CV() y C0

Var() y or  C Y 0 Sampling Theory| Chapter 2 | Simple Random Sampling | Shalabh, IIT Kanpur Page21 21

Nn S 2 or Nn  C 2 Y 2 0 11C 2 or 0 nN C2 C 2 C 2 or n  o C 2 1 2 NC0 S is the required sample size where C  is the population coefficient of variation. Y The smallest sample size needed in this case is C 2 C 2 n  0 . smallest C 2 1 2 NC0 If N is large, then C 2 n  C 2 0 C 2 and nsmalest  2 C0

5. Pre-specified relative error When y is used for estimating the population mean Y , then the relative estimation error is defined

yY as . If it is required that such relative estimation error should not exceed a pre-specified value Y R with probability (1 ) , then such requirement can be satisfied by expressing it like such requirement can be satisfied by expressing it like

yY RY P 1. Vary() Vary ()

Nn 2 Assuming the population to be normally distributed, y follows NY,. S Nn

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So it can be written that RY  Z . Var() y 2

2222Nn or Z SRY 2 Nn

11 R2 or 22 nN CZ 2

2 ZC 2  R  or n  2 ZC 1 2 1  NR  S where C  is the population coefficient of variation and should be known. Y If N is large, then

2 zC n  2 .  R 

6. Pre-specified cost

Let an amount of money C be designated for sample survey to called n observations, C0 be the overhead cost and C1 be the cost of collection of one unit in the sample. Then the total cost C can be expressed as

CC01 nC CC Or n  0 C1 is the required sample size.

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