Sampling

Distributions

Chapter 9

Central Limit Theorem

Central Limit Theorem

Take a random sample of size n from any

population with mean and standard

deviation . When n is large, the sampling

distribution of the sample mean is close to

the normal distribution.

How large a sample size is needed depends

on the shape of the population distribution.

Uniform distribution

Sample size 1

Uniform distribution

Sample size 2

Uniform distribution

Sample size 3

Uniform distribution

Sample size 4

Uniform distribution

Sample size 8

Uniform distribution

Sample size 16

Uniform distribution

Sample size 32

Triangle distribution

Sample size 1

Triangle distribution

Sample size 2

Triangle distribution

Sample size 3

Triangle distribution

Sample size 4

Triangle distribution

Sample size 8

Triangle distribution

Sample size 16

Triangle distribution

Sample size 32

Inverse distribution

Sample size 1

Inverse distribution

Sample size 2

Inverse distribution

Sample size 3

Inverse distribution

Sample size 4

Inverse distribution

Sample size 8

Inverse distribution

Sample size 16

Inverse distribution

Sample size 32

Parabolic distribution

Sample size 1

Parabolic distribution

Sample size 2

Parabolic distribution

Sample size 3

Parabolic distribution

Sample size 4

Parabolic distribution

Sample size 8

Parabolic distribution

Sample size 16

Parabolic distribution

Sample size 32

Loose ends

An unbiased statistic falls sometimes above

and sometimes below the actual mean, it

shows no tendency to over or underestimate.

As long as the population is much larger than

the sample (rule of thumb, 10 times larger),

the spread of the sampling distribution is

approximately the same for any size

population.

Loose ends

As the sampling standard deviation continually decreases, what conclusion can we make regarding each individual sample mean with respect to the population mean ?

As the sample size increases, the mean of the observed sample gets closer and closer to . (law of large numbers)