Download Ch 6 The Normal Distribution and Sampling Distributions

Ch 9 Sampling Theory and Some Important Sampling Distributions
9.1 Introduction to Sampling Distributions
population – set of measurements.
sample – a subset of the population
parameter – a numerical descriptive measure of a population.
statistic – a numerical descriptive measure of a sample
Since the population is often not available, we use statistics to estimate population
parameters.
In statistical application, we take a random sample from the population. We compute a
statistic, say x .
The value of the statistic x depends on which items are selected for the sample.
Therefore, x is a random variable. Different samples yield different values of x .
The distance between a statistic and the parameter it is estimating is called the sampling
error.
In order to evaluate the reliability of x , we need to know the probability distribution of
x.
The probability of a statistic over all possible samples is known as its sampling
distribution.
statistics estimate parameters
x estimates the population mean 
We hope x is very close to 
We want the sampling distribution to be centered at the value of the parameter and to
have little variation.
The statistic is an unbiased estimator if it is centered about the parameter of interest.
Otherwise, the estimator has bias.
An statistic is a minimum variance unbiased estimator if it is and unbiased estimator and
has less variance than all other unbiased estimators.
Draw picture of three distributions (which is best)
9.2 Sampling Distribution of the Sample Mean
x is a random variable whose value depends on which members of the population are
selected in the sample
Facts about the sampling distribution of x
x  
 
2
x
The average value of x across all possible samples is  , the population mean
2
n
The variance of the sampling distribution is the population variance divided by n.
 x   x2 
2


n
n
The standard deviation of the sampling distribution is the square root of the
variance
The standard error of an estimate is the standard deviation of its sampling distribution

n
is the standard error of x
If we sample without replacement from a finite population containing N elements, then
we must use the finite population correction factor when calculating the variance of the
sampling distribution.
 x2 
 2  N n


n  N 1 
Notice that as n increases the sample to sample variability in x decreases.
Notice that as the  2 decreases so does  x2 .
9.3 The Central Limit Theorem
If our sample comes from a normal distribution with mean  and standard deviation 
x
then: Z 
has a standard normal distribution

n
Central Limit Theorem
If we sample from a population with mean  and standard deviation  then Z 
x

n
is approximately standard normal for large n .
If n  30 or larger, the central limit theorem will apply in almost all cases
Example
A population of soft drink cans has amounts of liquid following a normal distribution
with   12 and   0.2 oz.
What is the probability that a single can is between 11.9 and 12.1 oz.
P(11.9  X  12.1)
 P(0.5  Z  0.5)
 P( Z  0.5)  P( Z  0.5)
 .6915  .3085
 .3830
What is the probability that x is between 11.9 and 12.1 for n = 16 cans
P(11.9  x  12.1)
 P(2  Z  2)
 P( Z  2)  P( Z  2)
 .9772  .0228
 .9544
The statement for x is approximately correct even if the amounts of liquid do not follow
a normal distribution
The statement for x is not correct if the amount of liquid does not follow a normal
distribution.
Example
A population of trees have heights that have a mean of 110 feet and a standard deviation
of 20 feet.
A sample of 100 trees is selected
Find P( x  108 feet)
Draw Picture
P( x  108)
 P( Z  1)
 1  P( Z  1)
 1  .1587
.8413
What about P ( X  108) ? Cannot say without normality!!!
9.5 Sampling Distribution of the Sample Proportion
Population Proportion
# in population with characteri stic
p
# in population
Sample Proportion
# in sample with characteri stic
p̂ 
n
p̂ is a point estimate of p
 pˆ  p
pq
 p2ˆ 
n
pq
 pˆ 
n
If we sample from a population with a proportion of p, then Z 
standard normal for large n.
pˆ  p
pq
n
is approximately