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This chapter explores the concept of sampling distributions, specifically the distribution of the sample mean. A sampling distribution is a probability distribution of all possible values of the mean computed from a sample of size n from a population with mean µ and standard deviation σ. The properties of the sampling distribution of the mean, including its mean and standard deviation, and how it approaches normality as the sample size increases due to the central limit theorem.
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A Sampling Distribution of a statistic is a probability distribution for all possible values of the statistic computed from a sample of size n. The sampling distribution of the mean is a probability distribution of all possible values of the random variable ‘x bar’ computed from a sample of size n from a population with mean μ and standard deviation σ. Example 1: pg. 378- 380 (Pay very close attention!)
Suppose that a simple random sample of size n is drawn from a large population with mean μ and standard deviation σ. The sampling distribution of x bar will have mean μ of x bar = μ and standard deviation σ of x bar = σ / sq. root of n. (see formulas on pg
The standard deviation of the sampling distribution of x bar, σ of x bar, is called the standard error of the mean. Example: pg. 389, #
If a random variable X is normally distributed with mean μ and standard deviation σ, then the distribution of the sample mean, x bar, is normally distributed with mean μxbar = μ and standard deviation σ (^) xbar = σ/√n
Suppose a random variable X has population mean μ and standard deviation σ and that a random sample size n is taken from this population. Then the sampling distribution of x bar becomes approximately normal as the sample size n increases. The mean of the distribution is μxbar = μ and the standard deviation is σ (^) xbar = σ/√n.
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Central Limit Theorem Characteristics