ST3239: Survey Methodology - The Department of Statistics and ...

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Chapter 2 Simple random sampling Definition: If a sample of size n is drawn from a population of size N in such a way that every possible sample of size n has the same probability of being selected, the sampling procedure is called simple random sampling. The sample thus obtained is called a simple random sample. Simple random sampling is often written as s.r.s. for short and is the simplest sampling procedure. 2.1 How to draw a simple random sample Suppose that the population of size N has values {u 1 , u 2 , · · · , u N }. If ( we ) draw n (distinct) items without replacement from the population, ( ) there are altogether N N different ways of doing it. So if we assign probability 1/ to each of the different n n samples, then each sample thus obtained is a simple random sample. We denote this sample by {y 1 , y 2 , · · · , y n }. Remark: In our previous statistics course, we always use upper-case letters like X, Y etc. to denote random variables and lower-case letters like x, y etc. to represent fixed values. However, in sample survey course, by convention, we use lower-case letters like y 1 , y 2 etc. to denote random variables. Theorem 2.1.1 For simple random sampling, we have P (y 1 = u i1 , y 2 = u i2 , · · · , y n = u in ) = 1 N where i 1 , i 2 , · · · , i n are mutually different. 1 (N − 1) · · · 1 (N − n + 1) (N − n)! = . N! 3
Proof. By the definition of s.r.s, the probability ( ) of obtaining the sample {u i1 , u i2 , · · · , u in } N (where the order is not important) is 1/ . There are n! number of ways of ordering n {u i1 , u i2 , · · · , u in }. Therefore, P (y 1 = u i1 , y 2 = u i2 , · · · , y n = u in ) = ( N n 1 ) n! = (N − n)!n! N!n! = (N − n)! . N! Remark: Recall that the total number of all possible samples is ( ) N n , which could be very large if N and n are large. Therefore, getting a simple random sample by first listing all possible samples and then drawing one at random would not be practical. An easier way to get a simple random sample is simply to draw n values at random without replacement from the N population values. That is, we first draw one value at random from the N population values, and then draw another value at random from the remaining N − 1 population values and so on, until we get a sample of n (different) values. Theorem 2.1.2 A sample obtained by drawing n values successively without replacement from the N population values is a simple random sample. Proof. Suppose that our sample obtained by drawing n values without replacement from the N population values is {a 1 , a 2 , · · · , a n }, where the order is not important. Let {a i1 , a i2 , · · · , a in } be any permutation of {a 1 , a 2 , · · · , a n }. Since the sample is drawn without replacement, we have P (y 1 = a i1 , · · · , y n = a in ) = 1 N 1 (N − 1) · · · 1 (N − n + 1) (N − n)! = . N! Hence, the probability of obtaining the sample {a 1 , · · · , a n } (where the order is not important) is ∑ ∑ (N − n)! (N − n)! 1 P (y 1 = a i1 , · · · , y n = a in ) = = n! × = ( ). N! N! all (i 1 ,···,i n ) all (i 1 ,···,i n ) N n The theorem is thus proved by the definition of the simple random sampling. 4
Page 1 and 2: ST3239: Survey Methodology by Wang
Page 3: 1.3 Why sample? If a sample is equa
Page 7 and 8: 2.2 Estimation of population mean a
Page 9 and 10: Summary 1. How to draw a simple ran
Page 11 and 12: Central limit theorem: If n → ∞
Page 13 and 14: e.g. Suppose that a total of 1500 s
Page 15 and 16: 2.3.2 Estimation of population tota
Page 17 and 18: 2.4 Estimation of population propor
Page 19 and 20: Solution 2.5 Comparing estimates Su
Page 21 and 22: Proof. Note that ˆp 1 = X/n and ˆ

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