# ch 4 - Stat 0302B Business Statistics Spring 2010-2011...

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Stat 0302B Business Statistics Spring 2010-2011 Chapter IV Population and Sample § 4.1 Introduction 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 subgroup of the population. We usually have data on the sampled individuals. Random Sample : sample drawn randomly by a method involving unpredictable components in such a way that each possible sample of size n have known chance to be selected, e.g. lucky draw on each individual in the population. Example 4.1 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 the population mean or the population 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 4.2 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. 84

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Stat 0302B Business Statistics Spring 2010-2011 § 4.2 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 4.3 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 distribution: 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 This is called the population distribution . Moreover, it can be easily determined that the population mean and population variance are 6 1 = = X E μ , ( ) 8 1 2 = = X Var σ . Suppose we draw one more number from the same population and denote it as (sample with replacement). The distribution of is the same as and are independent. Then 2 X 1 , X 2 X 1 X 2 X { } 2 1 , X X forms a random sample with size 2 = n . From a particular random sample, we can calculate the following sample statistics: 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 probability distributions can be determined 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. 85
Stat 0302B Business Statistics Spring 2010-2011 6, 8 7 1 2 - 6 2/25 6, 10 8

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ch 4 - Stat 0302B Business Statistics Spring 2010-2011...

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