# The Central Limit Theorem

## Using the Central Limit Theorem

It is important for you to understand when to use the central limit theorem. If you are being asked to find the probability of the mean, use the clt for the means. If you are being asked to find the probability of a sum or total, use the clt for sums. This also applies to percentiles for means and sums.

### Examples of the Central Limit Theorem


### Law of Large Numbers

The law of large numbers says that if you take samples of larger and larger sizes from any population, then the mean 
  of the samples tends to get closer and closer to μ. From the central limit theorem, we know that as n gets larger and larger, the sample means follow a normal distribution. The larger n gets, the smaller the standard deviation gets. (Remember that the standard deviation for 
  is 
 .) This means that the sample mean 
  must be close to the population mean μ. We can say that μ is the value that the sample means approach as n gets larger. The central limit theorem illustrates the law of large numbers.

### Central Limit Theorem for the Mean and Sum Examples


### References

The Wall Street Journal. (n.d.). Retrieved from https://www.wsj.com/
Centers for Disease Control and Prevention. (2017, April 16). National health and nutrition examination survey. National Center for Health Statistics.  Retrieved from http://www.cdc.gov/nchs/nhanes.htm

### Chapter Review

The central limit theorem can be used to illustrate the law of large numbers. The law of large numbers states that the larger the sample size you take from a population, the closer the sample mean, 
 , gets to μ.
Use the following information to answer the next 10 exercises: A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.
 The length of time a particular smartphone's battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken.
 A uniform distribution has a minimum of six and a maximum of ten. A sample of 50 is taken.

### Homework

 Richard’s Furniture Company delivers furniture from 10 a.m. to 2 p.m. continuously and uniformly. We are interested in how long (in hours) past the 10 a.m. start time that individuals wait for their delivery.
Use the following information to answer the next two exercises: The time to wait for a particular rural bus is distributed uniformly from zero to 75 minutes. One hundred riders are randomly sampled to learn how long they waited.
 The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of $4.59 and a standard deviation of $0.10. Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations.
For each problem, wherever possible, provide graphs and use a calculator.