Transcript http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html

http://www.ruf.rice.edu/~lane/stat
_sim/sampling_dist/index.html
Problem
• What is the probability of drawing a card that is
higher than a jack or lower than a 3?
• If I gave only gave A’s to students scoring two
standard devastations above the mean on the next
test – what proportion of students would get an A?
• If the next test had a mean of 56 and a SD of 4.2 –
how many points would you need to get in order
to receive an A?
Example: IQ
• Mean IQ = 100
• Standard deviation = 15
• What is the probability that a person you
randomly bump into on the street has an IQ
of 110 or higher?
Step 1: Sketch out question
-3
-2
-1

1
2
3
Step 1: Sketch out question
110
-3
-2
-1

1
2
3
Step 2: Calculate Z score
(110 - 100) / 15 = .66
110
-3
-2
-1

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Step 3: Look up Z score in Table
Z = .66; Column C = .2546
110
.2546
-3
-2
-1

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Example: IQ
• You have a .2546 probability (or a 25.56%
chance) of randomly bumping into a person
with an IQ over 110.
Now. . . .
• What is the probability that the next 5
people you bump into on the street will
have a mean IQ score of 110?
• Notice how this is different!
Population
• You are interested in the average selfesteem in a population of 40 people
• Self-esteem test scores range from 1 to 10.
Population Scores
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•
•
•
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1,1,1,1
2,2,2,2
3,3,3,3
4,4,4,4
5,5,5,5
•
•
•
•
•
6,6,6,6
7,7,7,7
8,8,8,8
9,9,9,9
10,10,10,10
Histogram
10
9
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
What is the average self-esteem
score of this population?
• Population mean = 5.5
• What if you wanted to estimate this
population mean from a sample?
Group Activity
• Randomly select 5 people and find the
average score
Group Activity
• Why isn’t the average score the same as the
population score?
• When you use a sample there is always
some degree of uncertainty!
• We can measure this uncertainty with a
sampling distribution of the mean
EXCEL
Characteristics of a Sampling
Distribution of the means
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Every sample is drawn randomly from a population
The sample size (n) is the same for all samples
The mean is calculated for each sample
The sample means are arranged into a frequency
distribution (or histogram)
• The number of samples is very large
INTERNET EXAMPLE
Sampling Distribution of the
Mean
• Notice: The sampling distribution is
centered around the population mean!
• Notice: The sampling distribution of the
mean looks like a normal curve!
– This is true even though the distribution of
scores was NOT a normal distribution
Central Limit Theorem
For any population of scores, regardless of
form, the sampling distribution of the means
will approach a normal distribution as the
number of samples get larger. Furthermore,
the sampling distribution of the mean will
have a mean equal to  and a standard
deviation equal to / N
Mean
• The expected value of the mean for a
sampling distribution
• E (X) = 
Standard Error
• The standard error (i.e., standard deviation)
of the sampling distribution
x = /
N
Standard Error
• The  of an IQ test is 15. If you sampled 10
people and found an X = 105 what is the
standard error of that mean?
x = /
N
Standard Error
• The  of an IQ test is 15. If you sampled 10
people and found an X = 105 what is the
standard error of that mean?
x = 15/ 10
Standard Error
• The  of an IQ test is 15. If you sampled 10
people and found an X = 105 what is the
standard error of that mean?
4.74 = 15/ 3.16
Standard Error
• The  of an IQ test is 15. If you sampled 10
people and found an X = 105 what is the
standard error of that mean? What happens
to the standard error if the sample size
increased to 50?
4.74 = 15/ 3.16
Standard Error
• The  of an IQ test is 15. If you sampled 10
people and found an X = 105 what is the
standard error of that mean? What happens
to the standard error if the sample size
increased to 50?
4.74 = 15/ 3.16
2.12 = 15/7.07
Standard Error
• The bigger the sample size the smaller the
standard error
• Makes sense!
Question
• For an IQ test
•  = 100
•  = 15
• What is the probability that in a class the average
IQ of 54 students will be below 95?
• Note: This is different then the other “z”
questions!
Z score for a sample mean
Z = (X - ) / x
Step 1: Sketch out question
-3
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0
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Step 2: Calculate the Standard Error
15 / 54 = 2.04
-3
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-1
0
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Step 3: Calculate the Z score
(95 - 100) / 2.04 = -2.45
-3
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0
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Step 4: Look up Z score in Table
Z = -2.45; Column C =.0071
.0071
-3
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-1
0
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Question
• From a sample of 54 students the
probability that their average IQ score is 95
or lower is .0071
Practice
• Page 154
– 7.12
– 7.13
N
1
2
4
8
1
1
2
4
8
4
.5
1
2
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16
.25
.5
1
2
64
.125
.25
.5
1
As N (sample size) increases the standard error decreases!