# The Central Limit Theorem

## The Central Limit Theorem for Sample Means (Averages)

Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution). Using a subscript that matches the random variable, suppose
1. 
 
  
 = the mean of X
2. 
 
  
 = the standard deviation of X
If you draw random samples of size n, then as n increases, the random variable 
, which consists of sample means, tends to be normally distributed and
The central limit theorem for sample means says that if you keep drawing larger and larger samples (such as rolling one, two, five, and finally, ten dice) and calculating their means, the sample means form their own normal distribution (the sampling distribution). The normal distribution has the same mean as the original distribution and a variance that equals the original variance divided by the sample size. The variable n is the number of values that are averaged together, not the number of times the experiment is done.
To put it more formally, if you draw random samples of size n, the distribution of the random variable 
, which consists of sample means, is called the sampling distribution of the mean. The sampling distribution of the mean approaches a normal distribution as n, the sample size, increases. 
The random variable 
 has a different z-score associated with it from that of the random variable X. The mean 
 is the value of 
 in one sample.
μ is the average of both X and 
.

 = standard deviation of 
 and is called the standard error of the mean.

### References

Baran, D. (2010). 20 percent of Americans have never used email. WebGuild. Retrieved from http://www.webguild.org/20080519/20-percent-of-americans-have-never-used-email
The Flurry Blog. (2013). Retrieved from http://blog.flurry.com
U.S. Department of Agriculture. (n.d.). Retrieved from https://www.usda.gov/

### Chapter Review

In a population whose distribution may be known or unknown, if the size (n) of the sample is sufficiently large, the distribution of the sample means will be approximately normal. The mean of the sample means will equal the population mean. The standard deviation of the distribution of the sample means, called the standard error of the mean, is equal to the population standard deviation divided by the square root of the sample size (n).

### Formula Review

Central limit theorem for sample means:

 ~ N

Mean 
 : μ
Central limit theorem for sample means z-score and standard error of the mean: 

Standard error of the mean (standard deviation (
 )): 
 
Use the following information to answer the next six exercises: Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews take her approximately four hours each to do with a population standard deviation of 1.2 hours. Let Χ be the random variable representing the time it takes her to complete one review. Assume Χ is normally distributed. Let 
 be the random variable representing the mean time to complete the 16 reviews. Assume that the 16 reviews represent a random set of reviews.

### Homework
