Sampling and sampling distibutions

Sampling from a finite and an infinite population

• Simple random sample (finite population)– Population size N, sample size n– Each possible sample of size n has the same

probability of being selected• Random sample (infinite population)– Each element comes from the same population

• Making sure all the cereal boxes have the same weight at the plant

– Each element is selected independently• Sampling customers of a McDonalds restaurant

The number of different possible samples

• N=2500, n = 30• N!/(n!*(N-n)!)=2.57*10^69

Point estimation

• Finding the sample mean • Population mean = $51,800• Population standard deviation = $4000

Sampling distributions

• Distribution function for the sample means – from 500 simple random samples of size 30 each

• is the sample mean• The relationship between the standard deviation of

the population and the standard deviation of the sample mean – Infinite population (a.k.a. the standard error of the mean) – Finite population

• Rule: If for a finite population, n/N <= 0.05, then we can use the formula from infinite population

An example, sample size 30, population standard deviation 4000

• = 4000/sqrt(30)=730.3

Distribution function of sample mean

• When the population has a normal distribution the distribution function of is normal for any sample size

• Central limit theorem– In selecting random samples of size n from a

population, the distribution function of can be approximated by a normal distribution as the sample size becomes large

Central limit theorem

TETC-110B

Sampling distributions

• is the sample proportion• The relationship between the standard deviation

of the population proportion and the standard deviation of the sample proportion– Infinite population – Finite population

• Rule: If for a finite population, n/N <= 0.05, then we can use the formula from infinite population

The distribution function for sample proportion

• Can be approximated by a normal distribution– If np>=5– and n(1-p)>=5