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4-4_CLT - THE CENTRAL LIMIT THEOREM The World is Normal...

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THE CENTRAL LIMIT THEOREM The World is Normal Theorem
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Sampling Distribution of x- normally distributed population n=10 σ / 10 σ Population distribution: N( μ , σ ) Sampling distribution of x: N ( μ , σ / 10)
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Normal Populations Important Fact: If the population is normally distributed, then the sampling distribution of x is normally distributed for any sample size n. Previous slide
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Non-normal Populations What can we say about the shape of the sampling distribution of x when the population from which the sample is selected is not normal? 53 490 102 72 35 21 26 17 8 10 2 3 1 0 0 1 0 100 200 300 400 500 600 F req u en cy Salary ($1,000's) Baseball Salaries
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The Central Limit Theorem (for the sample mean x) If a random sample of n observations is selected from a population ( any population), then when n is sufficiently large, the sampling distribution of x will be approximately normal. (The larger the sample size, the better will be the normal approximation to the sampling distribution of x.)
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The Importance of the Central Limit Theorem When we select simple random samples of size n, the sample means we find will vary from sample to sample. We can model the distribution of these sample means with a probability model that is , N n μ σ
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How Large Should n Be? For the purpose of applying the central limit theorem, we will consider a sample size to be large when n > 30.
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Summary Population : mean μ ; stand dev. σ ; shape of population dist. is unknown; value of μ is unknown ; select random sample of size n ;
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