Central Limit Theorem

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The Central Limit Theorem says that if you have a random sample and the sample size is large enough (usually bigger than 30), then the sample mean  follows a normal distribution with mean = µ and standard deviation =  . This comes in really handy when you haven't a clue what the distribution is or it is a distribution you're not used to working with like, for instance, the Gamma distribution. Since  follows a normal distribution, we can convert it to Z:

where Z is the standard Normal distribution with m = 0 and s = 1.

Example

The amount of time it takes people in a certain city to drive to work is normally distributed with a mean of 45 minutes and standard deviation of 8.25 minutes. For a random sample of 100 people in this city, what is the probability their average time to drive to work is more than 46 minutes?

Solution

First, we convert to Z:

Then

There is also the central limit theorem for proportions. The symbol for the sample proportion is  which is called p-hat. It states that if X follows a binomial distribution with number of trials n and probability of success p, then  follows a normal distribution with mean = p and standard deviation =  provided n is greater than 30, and the mean plus/minus 3 standard deviations is between 0 and 1. As above, we can convert to Z:

Example

Suppose 25% of people own a certain electronic gadget. For a random sample of 200 people, what is the probability the percentage of people in this sample who own this gadget is more than 28%?

Solution

First, we convert to Z:

Then = 0.5 – 0.3365 = 0.1635 = 16.35%

 

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