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Section 6-5

Section 6-5. The Central Limit Theorem. THE CENTRAL LIMIT THEOREM. Given : 1. The random variable x has a distribution (which may or may not be normal) with mean µ and standard deviation σ . 2. Samples all of the same size n are randomly selected from the population of x values.

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Section 6-5

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  1. Section 6-5 The Central Limit Theorem

  2. THE CENTRAL LIMIT THEOREM Given: 1. The random variable x has a distribution (which may or may not be normal) with mean µ and standard deviation σ. 2. Samples all of the same size n are randomly selected from the population of x values.

  3. THE CENTRAL LIMIT THEOREM Conclusions: • The distribution of sample means will, as the sample size increases, approach a normal distribution. • The mean of the sample means will be the population mean µ. • The standard deviation of the sample means will approach

  4. COMMENTS ON THE CENTRAL LIMIT THEOREM The Central Limit Theorem involves two distributions. • The population distribution. (This is what we studied in Sections 6-1 through 6-3.) • The distribution of sample means. (This is what we studied in the last section, Section 6-4.)

  5. PRACTICAL RULESCOMMONLY USED • For samples of size n larger than 30, the distribution of the sample means can be approximated reasonably well by a normal distribution. The approximation gets better as the sample size n becomes larger. • If the original population is itself normally distributed, then the sample means will be normally distributed for any sample sizen (not just the values of n larger than 30).

  6. NOTATION FOR THE SAMPLING DISTRIBUTION OF If all possible random samples of size n are selected from a population with mean μ and standard deviation σ, the mean of the sample means is denoted by , so Also, the standard deviation of the sample means is denoted by , so is often called the standard error of the mean.

  7. A NORMAL DISTRIBUTION As we proceed from n = 1 to n = 50, we see that the distribution of sample means is approaching the shape of a normal distribution.

  8. A UNIFORM DISTRIBUTION As we proceed from n = 1 to n = 50, we see that the distribution of sample means is approaching the shape of a normal distribution.

  9. A U-SHAPED DISTRIBUTION As we proceed from n = 1 to n = 50, we see that the distribution of sample means is approaching the shape of a normal distribution.

  10. As the sample size increases, the sampling distribution of sample means approaches a normaldistribution.

  11. CAUTIONS ABOUT THE CENTRAL LIMIT THEOREM • When working with an individual value from a normally distributed population, use the methods of Section 6-3. Use • When working with a mean for some sample (or group) be sure to use the value of for the standard deviation of sample means. Use

  12. RARE EVENT RULE If, under a given assumption, the probability of a particular observed event is exceptionally small, we conclude that the assumption is probably not correct.

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