Central Limit Theorem

Central Limit Theorem, CLT

for any given population with a mean μ and a standard deviation σ with samples of size n, the distribution of sample means for samples of size n will have a mean of μ and a standard deviation of σ and will approach a normal distribution as n approaches infinity

CLT

statistical theory that states that given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population.

Central Limit Theorem (CLT) tells us that for any population distribution, if we draw many samples of a large size, nn, then the distribution of sample means, called the sampling distribution, will:

Be normally distributed.

Have a mean equal to the population mean, μ.

Have a standard deviation equal to the standard error of the mean,

σ / n‾ √σ/n

CLT equation

Regardless of the distribution of a population, as n increases, the distribution of the means of random samples from the population will approach a normal distribution, specifically:

N(μ, (σ / √n))

- sampling distribution of the means

CLT Given #1

The random variable x has a distribution (which may or may not be normal) with mean µ and standard deviation ø

CLT Given #2

Samples all of the same size n are randomly selected from the population of x values.

CLT Conclusion #1

The distribution of sample x̄ will, as the sample size increases, approach a normal distribution

CLT Conclusion #2

The mean of the sample means will be the population mean µ

CLT Conclusion #3

The standard deviation of the sample means will approach (O/√n)

CLT Practical Rules Commonly 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 get better as the sample size n becomes larger.

CLT Practical Rules Commonly Used #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 value of n larger than 30)

If we calculate the 95% confidence interval, we would expect

the true population mean to be found in its range in 95% of our samples

t distribution

when we have a sample and don't know

the population variance

What happens when the degrees of freedom increases

the peak starts to get closer to the peak of a standard normal distribution

The tail starts to get thicker

the average value of n independent instances of random variables from ANY probability distribution will have approximately a t-distribution when

after subtracting its mean and dividing by its standard deviation

and the n is sufficiently large

z test

test of the data compared to a general population with population parameters

single sample t-test

a statistic to evaluate whether a sample mean statistically differs from a specific value

independent samples t-test

a test to determine if there is a difference between two separate, independent groups; conducted when researchers wish to compare mean values of two groups

dependent samples t-test

means of two conditions, when the same sample is used for both

variances

The differences between planned amounts and actual amounts

standard error

the standard deviation of a sampling distribution, simply put the standard deviation from a point

sum of squares, SS

the sum of the squared deviation scores

population parameters,,

mean:

Sd:

μ

σ

sample parameters,,

mean:

Sd:

x bar

s

standard error

is usually smaller than the standard deviation of our population,, but is usually equal to or close to our sample

standard error equation

(σ / √n)

sampling distribution of the mean

X(bar) ~ N(μ, (σ / √n))