Sampling and Estimation

Sampling error of the mean = sample mean – population mean  –  μ

The distribution of these estimates of the mean is the sampling distribution of the mean

Central Limit Theorem

Mean of Sample Distribution = μ

Variance of distribution of sample mean = σ²/N    where N > 30

Standard Error of the sample mean is the standard deviation of the distribution of the sample mean.

or    

s is the sample standard deviation ( if σ is not known )

n is the size (number of observations) of the sample.

σ is the standard deviation of the population.

“As the sample size increases , the sample mean gets closer , on average , to the true mean of the population”

“The distribution of the sample means about the population mean gets smaller and smaller, so the standard error of the sample mean decreases”

Point Estimate : Examples include mean , variance , etc

Confidence interval Estimates  = Point Estimates ± (reliability factor * Standard Error)

where reliability factor = the probability that the point estimate falls in the confidence interval  ( 1 – α)

Standard Error = Std Error of the point estimate

T – Distribution

Symmetrical Distribution centred about zero

Flatter and has thicker tails than the Standard Normal Distribution

hence hypothesis testing using t-distribution makes it more difficult to reject the null relative to hypothesis testing using the z-distribution.

Normal Distribution with a known variance =  ± Z α/2  σ/ root(n)

Normal Distribution with unknown variance n > 30  = ± Z α/2  σ/ root(n)

Normal Distribution with unknown variance n < 30  =  ± T α/2  σ/ root(n)

Read Full Post »