Transcript 5.4 Sampling Distributions and the Central Limit Theorem

5.4 Sampling Distributions and the Central
Limit Theorem
• Key Concepts:
– How to find sampling distributions and verify
their properties
– The Central Limit Theorem
– Using the Central Limit Theorem to find
probabilities
5.4 Sampling Distributions and the Central
Limit Theorem
• What is a sampling distribution?
– The probability distribution of a sample
statistic that is formed when samples of size n
are repeatedly taken from a population.
• Examples of sample statistics:
2
X , s, s , and p
5.4 Sampling Distributions and the Central
Limit Theorem
• Properties of sampling distributions of sample
means:
1. The mean of the sample means,  X , is equal to the
population mean  :
X  
2. The standard deviation of the sample means,  X ,
is proportional to σ :
X 
• Note:
X

n
(valid when n ≤ 0.05N)
is known as the standard error of the mean.
5.4 Sampling Distributions and the Central
Limit Theorem
• According to Forbes Magazine, the top five
wealthiest women in the world in 2010 were:
Christy Walton:
Alice Walton:
Liliane Bettencourt:
Birgit Rausing:
Savitri Jindal:
$22.5 billion
$20.6 billion
$20.0 billion
$13.0 billion
$12.2 billion
1. Let X = wealth (in billions of dollars). Find the
mean and standard deviation of X.
5.4 Sampling Distributions and the Central
Limit Theorem
2. List all possible
samples of size 2 from
this population of 5 and
list the sample mean for
each sample.
3. Find the mean and
standard deviation of
the sample means in
column three.
Sample
Wealth
Sample
Mean
{C, A }
22.5, 20.6
21.55
{ C,L }
22.5, 20.0
21.25
{ C,B }
22.5, 13.0
17.75
{ C,S }
22.5,12.2
17.35
{ A,L }
20.6, 20.0
20.3
{ A,B}
20.6, 13.0
16.8
{ A,S }
20.6, 12.2
16.4
{ L,B }
20.0, 13.0
16.5
{ L,S }
20.0, 12.2
16.1
{ B,S }
13.0, 12.2
12.6
5.4 Sampling Distributions and the Central
Limit Theorem
• Where does the Central Limit Theorem fit in?
– We use the CLT to make a statement about
the shape of the distribution of the sample
means (see p. 263)
• If samples of size 30 or more are drawn from any
population with mean µ and standard deviation σ,
then the sampling distribution of the sample means
will be approximately normal.
• If the population itself is normally distributed, then
the sampling distribution of the sample means is
normally distributed for any sample size n.
5.4 Sampling Distributions and the Central
Limit Theorem
• Practice using the Central Limit Theorem
#10 p. 270 (Annual Snowfall)
#24 p. 271 (Canned Vegetables)
#30 p. 272 (Gas Prices: California)
#36 p. 272 (Make a Decision)