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**Chapter 7: Variation in repeated samples – Sampling distributions**

Recall that a major objective of statistics is to make inferences about a population from an analysis of information contained in sample data. Typically, we are interested in learning about some numerical feature of the population, such as the proportion possessing a stated characteristic; the mean and the standard deviation. A numerical feature of a population is called a parameter. The true value of a parameter is unknown. An appropriate sample-based quantity is our source about the value of a parameter. A statistic is a numerical valued function of the sample observations. Sample mean is an example of a statistic.

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**The sampling distribution of a statistic**

Three important points about a statistic: the numerical value of a statistic cannot be expected to give us the exact value of the parameter; the observed value of a statistic depends on the particular sample that happens to be selected; there will be some variability in the values of a statistic over different occasions of sampling. Because any statistic varies from sample to sample, it is a random variable and has its own probability distribution. The probability distribution of a statistic is called its sampling distribution. Often we simply say the distribution of a statistic.

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**Distribution of the sample mean**

Statistical inference about the population mean is of prime practical importance. Inferences about this parameter are based on the sample mean and its sampling distribution.

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**An example illustrating the central limit theorem**

Figure 7.4 (p. 275) Distributions of for n = 3 and n = 10 in sampling from an asymmetric population.

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**Example on probability calculations for the sample mean**

Consider a population with mean 82 and standard deviation 12. If a random sample of size 64 is selected, what is the probability that the sample mean will lie between 80.8 and 83.2? Solution: We have μ = 82 and σ = 12. Since n = 64 is large, the central limit theorem tells us that the distribution of the sample mean is approximately normal with Converting to the standard normal variable: Thus,

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