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Statistics 300: Elementary Statistics

Understand the Central Limit Theorem, which states that the distribution of sample means approaches a normal distribution under certain conditions. Learn how to calculate probabilities using this theorem with a practical example.

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Statistics 300: Elementary Statistics

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  1. Statistics 300:Elementary Statistics Section 6-5

  2. Central Limit Theorem • Given: X has mean = m and standard deviation = s • For a specified sample size “n” • The number of possible samples of size n is usually very large

  3. Central Limit Theorem • The number of possible samples of size n is usually very large • Example: Population N = 100 and sample size n = 10. • The number of possible samples is 100C10 = 1.73 * 1013

  4. Central Limit Theorem • Each of the possible samples has its own sample mean • The collection (set or population) of possible sample means has a mean and standard deviation • The mean = m and the standard deviation = s/sqrt(n)

  5. Central Limit Theorem • Furthermore, • If n > 30 or if X~N(m,s) then • The distribution of all possible sample means is approximately a normal distribution

  6. The Mean of a Random Sample has the distribution below if n > 30 or the “parent population” is “normal”

  7. Weights of oranges have a mean weight of 34.2 grams and a standard deviation of 6.4 grams. If 12 oranges are selected at random, what is the probability the average weight of the 12 oranges will be greater than 30 g?

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