Download 5.4 Sampling Distributions and the Central Limit Theorem

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Central limit theorem wikipedia, lookup

History of statistics wikipedia, lookup

Student's t-test wikipedia, lookup

Taylor's law wikipedia, lookup

Bootstrapping (statistics) wikipedia, lookup

Gibbs sampling wikipedia, lookup

Sampling (statistics) wikipedia, lookup

Transcript
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)