section 6-5
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
Section 6-5
The Central Limit Theorem
THE CENTRAL LIMIT THEOREMGiven:
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.
THE CENTRAL LIMIT THEOREM
1. The distribution of sample means will, as the sample size increases, approach a normal distribution.
2. The mean of the sample means will be the population mean µ.
3. The standard deviation of the sample means will approach
Conclusions:
COMMENTS ON THE CENTRAL LIMIT THEOREM
1. The population distribution. (This is what we studied in Sections 6-1 through 6-3.)
2. The distribution of sample means. (This is what we studied in the last section, Section 6-4.)
The Central Limit Theorem involves two distributions.
PRACTICAL RULESCOMMONLY USED
1. 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.
2. If the original population is itself normally distributed, then the sample means will be normally distributed for any sample size n (not just the values of n larger than 30).
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.
NOTATION FOR THE SAMPLING DISTRIBUTION OF x
A NORMAL DISTRIBUTIONAs we proceed from n = 1 to n = 50, we see that the distribution of sample means is approaching the shape of a normal distribution.
A UNIFORM DISTRIBUTIONAs we proceed from n = 1 to n = 50, we see that the distribution of sample means is approaching the shape of a normal distribution.
A U-SHAPED DISTRIBUTIONAs we proceed from n = 1 to n = 50, we see that the distribution of sample means is approaching the shape of a normal distribution.
As the sample size increases, the
sampling distribution of sample
means approaches a
normal distribution.
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
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.