Download §7.3--The Sampling Distribution of the Sample Mean

§7.3–The Sampling Distribution of the Sample Mean
Tom Lewis
Fall Term 2009
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
1/9
Outline
1
Normal data
2
The central limit theorem
3
A summary of sampling
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
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Normal data
Theorem
Suppose that the variable x of a population is normally distributed with a
mean µ and a standard deviation σ. Then, for samples of size n, the
variable x is normally distributed and has mean µ and standard deviation
√
σ/ n.
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
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Normal data
Problem
An IQ test has a mean of 100 and a standard deviation of 16. The scores
of 25 randomly selected people are collected and their sample mean, x, is
found.
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
4/9
Normal data
Problem
An IQ test has a mean of 100 and a standard deviation of 16. The scores
of 25 randomly selected people are collected and their sample mean, x, is
found.
What is the distribution of x.
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
4/9
Normal data
Problem
An IQ test has a mean of 100 and a standard deviation of 16. The scores
of 25 randomly selected people are collected and their sample mean, x, is
found.
What is the distribution of x.
What is the chance that x exceeds 106?
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
4/9
Normal data
Problem
An IQ test has a mean of 100 and a standard deviation of 16. The scores
of 25 randomly selected people are collected and their sample mean, x, is
found.
What is the distribution of x.
What is the chance that x exceeds 106?
If x were 110, would you be suspicious?
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
4/9
The central limit theorem
Problem
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
5/9
The central limit theorem
Problem
Toss a fair coin three times and let x count the number of heads
tossed. What is the distribution of x.
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
5/9
The central limit theorem
Problem
Toss a fair coin three times and let x count the number of heads
tossed. What is the distribution of x.
Toss a fair coin four times and let x count the number of heads
tossed. What is the distribution of x.
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
5/9
The central limit theorem
Problem
Toss a fair coin three times and let x count the number of heads
tossed. What is the distribution of x.
Toss a fair coin four times and let x count the number of heads
tossed. What is the distribution of x.
Repeat the above for 100 tosses!
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
5/9
The central limit theorem
Tossing coins
Here is the distribution for the number of heads in the tossing of 100 coins.
0.08
Binomial Distribution: Trials = 100, Probability of success = 0.5
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0.04
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0.02
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0.00
Probability Mass
0.06
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35
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40
45
50
55
60
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65
Number of Successes
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
6/9
The central limit theorem
The central limit theorem
Roughly speaking, the central limit theorem states that a large sum of
independent and identically distributed data will be approximately normally
distributed.
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
7/9
The central limit theorem
The central limit theorem
Roughly speaking, the central limit theorem states that a large sum of
independent and identically distributed data will be approximately normally
distributed.
Independent means that the value of one piece of data has no
statistical influence on another piece of data. For example, a students
SAT math score and SAT verbal score would not be independent data.
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
7/9
The central limit theorem
The central limit theorem
Roughly speaking, the central limit theorem states that a large sum of
independent and identically distributed data will be approximately normally
distributed.
Independent means that the value of one piece of data has no
statistical influence on another piece of data. For example, a students
SAT math score and SAT verbal score would not be independent data.
Identically distributed means that the data is being drawn from the
same distribution; thus, for example, you would not want to mix data
on heights with data on weights in studying the population of adult
males.
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
7/9
The central limit theorem
The central limit theorem
Roughly speaking, the central limit theorem states that a large sum of
independent and identically distributed data will be approximately normally
distributed.
Independent means that the value of one piece of data has no
statistical influence on another piece of data. For example, a students
SAT math score and SAT verbal score would not be independent data.
Identically distributed means that the data is being drawn from the
same distribution; thus, for example, you would not want to mix data
on heights with data on weights in studying the population of adult
males.
Approximately normally distributed means that we do not expect the
sum of the data to be exactly normally distributed. However, if the
sum is sufficiently large, we would expect a very close approximation
to the distribution by a normal distribution.
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
7/9
A summary of sampling
A summary
Suppose that a variable x of a population has a mean µ and a standard
deviation σ. Then, for samples of size n,
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
8/9
A summary of sampling
A summary
Suppose that a variable x of a population has a mean µ and a standard
deviation σ. Then, for samples of size n,
the mean of x is µ.
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
8/9
A summary of sampling
A summary
Suppose that a variable x of a population has a mean µ and a standard
deviation σ. Then, for samples of size n,
the mean of x is µ.
√
the standard deviation of x is σ/ n
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
8/9
A summary of sampling
A summary
Suppose that a variable x of a population has a mean µ and a standard
deviation σ. Then, for samples of size n,
the mean of x is µ.
√
the standard deviation of x is σ/ n
if x is normally distributed, then so is x, regardless of the sample size
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
8/9
A summary of sampling
A summary
Suppose that a variable x of a population has a mean µ and a standard
deviation σ. Then, for samples of size n,
the mean of x is µ.
√
the standard deviation of x is σ/ n
if x is normally distributed, then so is x, regardless of the sample size
if the sample size is large and the conditions of the CLT are in place,
then x is approximately normally distributed, regardless of the
distribution of x
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
8/9
A summary of sampling
Textbook problem
Work problem #7.62 from page 351 of the textbook.
Tom Lewis ()
§7.3–The Sampling Distribution of the Sample Mean
Fall Term 2009
9/9