Any unit 12 to be selected with probability 1 The sampled unit is then replaced

Any unit 12 to be selected with probability 1 the

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Any unit 1,2, … , ? to be selected with probability 1/? The sampled unit is then replaced in the population, and a second unit is randomly selected with again probability 1/? In sampling no additional information is obtained from selecting a unit several times. Thus we prefer to sample without replacement.
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A simple random sample without replacement (SRS): n unites are selected so that every possible subset of n distinct units in has the same probability of being selected as the sample. Each sample ? of size n is selected with probability Pr ? = 1/ ? 𝑛 Each unit 𝑖 is in the sample with probability π 𝑖 = 𝑛/?
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A parameter ϑ : Any real valued function of the population values Examples: ? ? = ? 1 + ? 2 +⋯+? 𝑁 ? the population mean . ? 2 = 1 ?−1 𝑖=1 ? ? 𝑖 ? ? 2 the population variance . S = 1 ?−1 𝑖=1 ? ? 𝑖 ? ? 2 the population standard deviation . A statistic T : any real valued function of the values ? 1 , … , ? ? taken from the sample S. For sample S we write ? 𝑆 to denote the values of the statistic calculated from the sample. A estimator ϑ : just a statistic used to estimate the parameter ϑ . The sampling distribution of a statistic T : A probability distribution that determines Pr ? = ? 𝑖 .
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Illustration map ? 1 ? 𝑖 1 ? 𝑖 2 ? 4 ? ? ? ? .. -2 - 1 0 1 2 … ϑ ? ?−1 ? 2 ? 3 Population U with their measured values ϑ Sample S Parameter ϑ : a real number ϑ 𝑆
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Example Suppose the value of ? 𝑖 for each of the N = 8 units in the whole population is given by Then Pr 2, 4, 6, 7 = 1/70 . Suppose the total t = ? 1 + ? 2 + ⋯ ? ? , is the parameter of interest. We may choose ? = ? ? 𝑆 = 2(? 1 + ? 2 + ? 3 +? 4 ) as an estimator of ? . 𝒊 1 2 3 4 5 6 7 8 ? 𝑖 1 2 4 4 7 7 7 8 k ?
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Sampling distribution of the statistic ?
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More definitions related to an estimator The expected value of an estimator is 𝐸 ϑ = 𝑘=1 ? kPr( ϑ = k) = 𝑆 ϑ 𝑆 𝑃?(?) The bias of ϑ is B 𝑖?? ϑ = 𝐸 ϑ − ϑ The estimator is said to be unbiased if B 𝑖?? ϑ = 0 AN IMPORTANT NOTE: the bias of an estimator is not the same as the selection bias or the measurement bias.
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How to measure the quality of an estimator? The mean squared error of ϑ is ??𝐸 ϑ = 𝐸(( 𝜗 − ϑ) 2 ) An estimator ϑ is accurate if ??𝐸 ϑ is small. An estimator ϑ is precise if 𝑉( ϑ) is small. Note that ??𝐸 ϑ = 𝑉 ϑ + [Bias( ϑ)] 2 Also note that 𝑉 ϑ = 𝑆 Pr(?)[ ϑ 𝑠 − 𝐸 ϑ ] 2 For the previous example 𝑉 ? = 1 70 22 − 40 2 + ⋯ + 1 70 58 − 40 2 = 54.86
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Illustration A: Unbiased but not precise B: Biased but precise C: Accurate and precise
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The Proportion Suppose we are interested in the proportion of the population that the has characteristic of the interest. If we assign the value ? 𝑖 = 1 when 𝑖 has the characteristic and 0 otherwise then the proportion of the population that has the characteristic is 𝑝 = ? 1 + ? 2 + ⋯ + ? ? ? = ? ? We can take 𝑝 = ?
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Example The U.S. government conducts a Census of Agriculture every five years.
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