unit 2 - chapter 7 - the central limit theorem Flashcards
central limit theorem
The more the samples you do the average will become more like a bell curve. Even if you start with a skew, you will end up with a normal distribution the more samples you have
doesn’t mean it has to be bell shaped but as n increases…
central limit theorem defined
from a population take all possible samples of a given size the result is as n increases the sampling distribution of the means approaches the normal distribution
If it’s skewed right, left, shaped weird, it works!! Always ends up as a normal distribution. Doesn’t matter what your population looks like if you have more samples it will work
more about the central limit theorem
- N should be 30% or more of the population
- We only need to take 1 sample
- We can talk about normal distribution and confidence interval.. can’t talk about the population
the central limit theorem gets us from…
The Central Limit Theorem gets us from descriptive statistics to inferential statistics
normal distribution formula/standardizing formula
z = (xbar - mew)/ (s/square root of n)
x bar = random variable
mew = avg
s/sq of n = standard error of the mean
central limit theorem from text
The astounding result is that it does not matter what the distribution of the original population is, or whether you even need to know it. The important fact is that the distribution of sample means tend to follow the normal distribution.
The size of the sample, n, that is required in order to be “large enough” depends on the original population from which the samples are drawn (the sample size should be at least 30 or the data should come from a normal distribution).
If the original population is far from normal, then more observations are needed for the sample means.
Sampling is done randomly and with replacement in the theoretical model.
sampling distribution
The sampling distribution is a theoretical distribution. It is created by taking many samples of size n from a population.
Each sample mean is then treated like a single observation of this new distribution, the sampling distribution.
The genius of thinking this way is that it recognizes that when we sample we are creating an observation and that observation must come from some particular distribution.
the central limit theorem answers this question:
The Central Limit Theorem answers the question: from what distribution did a sample mean come? If this is discovered, then we can treat a sample mean just like any other observation and calculate probabilities about what values it might take on.
the law of large numbers
The law of large numbers says that if you take samples of larger and larger size from any population, then the mean of the sampling distribution, 𝜇𝑥– tends to get closer and closer to the true population mean, μ. From the Central Limit Theorem, we know that as n gets larger and larger, the sample means follow a normal distribution.
This means that the sample mean 𝑥–must be closer to the population mean μ as n increases. 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.
two critical issues that flow from the Central Limit Theorem and the application of the Law of Large numbers to it.. these are:
- The probability density function of the sampling distribution of means is normally distributed regardless of the underlying distribution of the population observations and
- standard deviation of the sampling distribution decreases as the size of the samples that were used to calculate the means for the sampling distribution increases.
