STA 301 - Ch 08 - pp 85-95

STA 301 - Ch 08 - pp 85-95 - Chapter 8 Fundamental Sampling...

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Chapter 8: Fundamental Sampling Distributions and Data Descriptions 85 CHAPTER 8 — FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS Recall in Chapter 1 we introduced the idea of random sampling . In this chapter, we formalize the concept and introduce the idea of a sampling distribution . Definitions Population – “The totality of the observations with which we are concerned” o The population may be finite or infinite. o Each individual element of a population is a value of a random variable X having some probability distribution f ( x ). Parameter – A numerical characteristic of the population. o Typically, we think of the population mean and population variance, but any numerical characteristic of a population can be thought of as a parameter. Sample – A subset of a population o Generally, the sample is used for inference about some population parameter. o To avoid inadvertent bias in our inference, we prefer using random samples . Random Sample Let n X X X , , , 2 1 K be n independent random variables, each having the same probability distribution f ( x ). Then we define n X X X , , , 2 1 K to be a random sample of size n from the population f ( x ), and write its joint probability distribution as: ( ( ( ( n n x f x f x f x x x f L K 2 1 2 1 , , , = Statistic o A statistic is any function of the random variables that constitute a random sample. Note that since a statistic is a function of random variables, it is also a random variable.
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Chapter 8: Fundamental Sampling Distributions and Data Descriptions 86 Some Common Statistics Sample Mean If n X X X , , , 2 1 K represent a random sample of size n , then the sample mean is defined by the statistic: = = n i i X n X 1 1 . Sample Variance If n X X X , , , 2 1 K represent a random sample of size n , then the sample variance is defined by the statistic: ( 29 = - - = n i i X X n S 1 2 2 1 1 . Of course, this is computationally equivalent to: - - = = 2 1 2 2 1 1 X n X n S n i i . The sample standard deviation , S , is the positive square root of the sample variance.
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