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Sampling distribution of the sample mean, central limit theorem, constructing CIs for u

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STAT 301
Brett Hunter

26 September Sampling Distribution of the Sample Mean The sampling distribution of the sample mean is a probability distribution consisting of all the sample means of a given sample size selected from a population. Consider two scenarios You know the population distribution is normal x Sampling distribution of is normal We don’t know the distribution of the population, or we know it’s not normal x Sampling distribution of is normal if n is “large” Properties of the Sampling Distribution of x x is the mean of the observations in a random sample of size n, from a population with mean μ and standard deviation σ. The mean of the distribution of x is μx and the standard distribution is σx μ x = μ 2 σ σ x σ σ x = n → = √n When n is significantly large, the sampling distribution of is approximately normal (by Central Limit Theorem) Large enough if n ≥ 30 Central Limit Theorem Regardless of the shape of the population distribution, the shape of the sampling distribution of the sample mean approaches a normal distribution as n increases. As n increases, the normal approximation of the sample distribution of x gets better Probabilities for x x μx σx x Since ~ N( , ), we can standardize our values using x−μ x−μ x Z = = σ
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