Ch6 Sampling Distributions

Ch6 Sampling Distributions - Stat1801 Probability and...

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Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 Chapter VI Sampling Distributions § 6.1 Population and Sample Population : a group of individuals about which we wish to make an inference. We usually do not gather information from the entire population. Sample : a subset of the population. We usually have data on the sampled individuals. Random Sample : sample drawn in such a way that every possible sample of n objects will have known chance of being selected Example 6.1 Farmer Jane owns 1,264 sheep. These sheep constitute her entire population of sheep. If 15 sheep are selected to be sheared, then these 15 sheep represent a sample from Jane’s population. Further, if the 15 sheep were selected at random from these 1,264 sheep, then they would constitute a random sample . Parameter : A numerical characteristic of a population, such as a mean or standard deviation. Statistic : Numerical characteristic of a sample. Statistics may be calculated from data in a sample. Statistical Inference : A conclusion about a population based on sampled observations. Example 6.2 Suppose we want to investigate the weights of wool that can be sheared from the sheep. Then the mean and standard deviation of the weights of wool of the entire 1,264 sheep are the population parameters . For a random sample of size 15 from this population, the sample mean and sample standard deviation of the weights of wool of these 15 sheep are the statistics . Statistical inference may involve the estimation of the population parameters by these sample statistics. p. 107
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Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 § 6.2 Sampling Distribution of Sample Statistics Sampling distribution : the distribution of possible values of a sample statistic over all random samples of a given size. Example 6.3 Consider a hypothetical population with size 5 = N : { 2, 4, 6, 8, 10 } If we draw a number randomly from this population and denote it as , then the probabilistic behaviour of can be described by the following pmf : 1 X 1 X () ( ) 5 1 10 8 6 4 2 1 1 1 1 1 = = = = = = = = = = X P X P X P X P X P Hence this pmf completely described the population. Moreover, it can be easily determined that the population mean and population variance are 6 1 = = X E μ , ( ) 1 2 X Var = σ . Suppose we draw one more number from the same population and denote it as (sample with replacement). The pmf of is the same as and are independent. Then 2 X 2 X 1 X 2 1 , X X { } 2 1 , X X forms a random sample with size . From a particular random sample, we can calculate the following sample statistics: 2 = n 2 2 1 X X X + = , 2 1 2 1 2 2 1 2 2 2 1 2 X X X X X X S = + = Since are random, 2 1 , X X X and 2 S are also random. Their probabilistic behaviour can be described by the following table: Sample X X 2 S 2 2 S Probability 2, 2 2 - 4 0 - 8 1/25 2, 4 3 - 3 2 - 6 2/25 2, 6 4 - 2 8 0 2/25 2, 8 5 - 1 18 10 2/25 2, 10 6 0 32 24 2/25 4, 4 4 - 2 0 - 8 1/25 4, 6 5 - 1 2 - 6 2/25 4, 8 6 0 8 0 2/25 4, 10 7 1 18 10 2/25 6, 6 6 0 0 - 8 1/25 p. 108
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Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 6, 8 7 1 2 - 6 2/25 6, 10 8 2 8 0 2/25 8, 8 8 2 0 - 8
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Ch6 Sampling Distributions - Stat1801 Probability and...

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