Download Sampling and sampling distibutions

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
• 𝑥 is the sample mean
• The relationship between the standard deviation of the
population and the standard deviation of the sample mean
– Infinite population 𝜎𝑥 =
mean)
𝜎
𝑛
– Finite population 𝜎𝑥 =
𝑁−𝑛
𝑁−1
(a.k.a. the standard error of the
𝜎
𝑛
• 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
• 𝜎𝑥 =
𝜎
𝑛
= 4000/sqrt(30)=730.3
Distribution function of sample mean
𝑥
• When the population has a normal
distribution the distribution function of 𝑥 is
normal for any sample size
• Central limit theorem
– In selecting random samples of size n from a
population, the distribution function of 𝑥 can be
approximated by a normal distribution as the
sample size becomes large
Central limit theorem
TETC-110B
Sampling distributions
• 𝑝 is the sample proportion
• The relationship between the standard deviation
of the population proportion and the standard
deviation of the sample proportion
– Infinite population 𝜎𝑝 =
– Finite population 𝜎𝑝 =
𝑝(1−𝑝)
𝑛
𝑁−𝑛
𝑁−1
𝑝(1−𝑝)
𝑛
• 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