1 / 12

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.

isaura
Télécharger la présentation

Section 6-5

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

More Related