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Chapter 7 : Introduction to Sampling Distributions

Chapter 7 : Introduction to Sampling Distributions. Section 2 : The Central Limit Theorem. Theorem 7.1.

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Chapter 7 : Introduction to Sampling Distributions

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  1. Chapter 7: Introduction to Sampling Distributions Section 2: The Central Limit Theorem

  2. Theorem 7.1 • Let x be a random variable with a normal distribution whose mean is and standard deviation is . Let be the sample mean corresponding to random samples of size n taken from the x distribution. 1. The distribution is a normal distribution. 2. The mean of the distribution is . • The standard deviation of the distribution is .

  3. To find the z value: z =

  4. Example • Let x represent the length of a single trout taken at random from the pond. This is normal distribution with a mean of 10.2 inches and standard deviation of 1.4 inches. a.) What is the probability that a single trout taken at random from the pond is between 8 and 12 inches long? b.) What is the probability that the mean length of five trout taken at random is between 8 and 12 inches?

  5. Central Limit Theorem • If x possesses any distribution with mean and standard deviation , then the sample mean based on a random sample of size n will approach a normal distribution when n is sufficiently large (n > 30).

  6. Suppose we know that the x distribution has a mean = 30 and a standard deviation = 8, but we have no information as to whether or not the x distribution is normal. If we draw samples of size 35 from the x distribution and represents the sample mean, what can you say about the distribution?

  7. Suppose x has a normal distribution with mean = 20 and = 5. If we draw random samples of size 10 from the x distribution, and represents the sample mean, what can you say about the distribution? • Suppose you did not know that x had a normal distribution. Would you be justified in saying that the distribution is approximately normal if the sample size was n = 8?

  8. 1. Suppose we have no information as to whether or not the x distribution is normal. If we draw samples of n = 5 from the x distribution and represents the sample mean, will the distribution be approximately normal? Explain why or why not.

  9. 2. Suppose x has a normal distribution with mean = 12 and = 3. If we draw random samples of size 10 from the x distribution, and represents the sample mean, what can you say about the distribution? 3. Suppose you do not know if x has a normal distribution. If we draw random samples of size 40, will the distribution be approximately normal? Explain why or why not.

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