Lecture 16

# Lecture 16 - 1 Random Sampling A population consists of the...

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Unformatted text preview: 1 Random Sampling A population consists of the totality of the observations with which we are concerned. The size of the population can be finite or infinite. Examples of finite populations are the height of the students at Columbia University, the lengths of fish in a lake, and the lifetime of a set of light bulbs. The set of all measurements on the depth of a lake from any conceivable position is an example of a population whose size is infinite. In either case there are population parameters that we want to estimate by observing a sample. For example, we may select a 30 male students at random and use the average of their heights to estimate the average height of all male students at Columbia University. We will start our discussion with the case of sampling without replacement from a finite population. This is often referred as survey sampling. We will then discuss the case where we are either sampling with replacement or sampling from an infinite population. 2 Survey Sampling: The Finite Population Case Let x i be numerical value of interest associated with the i th member of the population i = 1 ,...,N where N is the size of the population. Let μ = 1 N N X i =1 x i τ = Nμ = N X i =1 x i σ 2 = 1 N N X i =1 ( x i- μ ) 2 = 1 n N X i =1 x 2 i- μ 2 . Notice that μ , τ and σ are population parameters that can only be computed if we have access to all the x i . In many cases it is impossible or not practical to observe the entire population and compute the population parameters. 2.1 Random Samples Sampling surveys are used to obtain information about a large population by examining only a small fraction of that population. Examples: Census surveys, agricultural surveys, political surveys. It is important that the samples are selected in a random fashion to avoid introducing bias. In random sampling each member of the population has the same probability of being included in the sample and the actual composition of the sample is random. Random Sampling Advantages: Guard against biases Less costly than complete enumeration May be more accurate than complete count Can calculate error estimates Can control size of error by selecting sample size 1 Definition. A choice of a subset of size n from a larger set of size N is called a random sample if each of the ( N n ) possible subsets of size n taken without replacement is equally likely to be selected. Example: Suppose N = 4 and n = 2 then a simple random sample of size 2 consists of selecting one of the six subsets of size 2: { 1 , 2 } , { 1 , 3 } , { 1 , 4 } , { 2 , 3 } , { 2 , 4 } , { 3 , 4 } where each of the subsets has equal probability of being selected, i.e., 1/6. 2.2 How to do random sampling from a finite population Naive way: List all K subsets of size n from the set of N elements. Generate a uniform [0 , 1] random number and select the i th sample if U ∈ ( i- 1 /K,i/K ] . This is a lot of work: For N = 393 and n = 16 we have K ≥ 10 33 ....
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Lecture 16 - 1 Random Sampling A population consists of the...

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