Central Limit Theorem
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
variances
The differences between planned amounts and actual amounts
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
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.
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
population parameters,, mean: Sd:
μ σ
standard error equation
(σ / √n)
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 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)
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 Given #1
The random variable x has a distribution (which may or may not be normal) with mean µ and standard deviation ø
CLT Conclusion #3
The standard deviation of the sample means will approach (O/√n)
sampling distribution of the mean
X(bar) ~ N(μ, (σ / √n))
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
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
standard error
is usually smaller than the standard deviation of our population,, but is usually equal to or close to our sample
dependent samples t-test
means of two conditions, when the same sample is used for both
z test
test of the data compared to a general population with population parameters
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
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
sample parameters,, mean: Sd:
x bar s
