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```Samples & the Sampling
Distributions of the Means
Chapter 7
Homework: 1 (a-i), 2-8
sketch: use mean & standard deviation
or
mean & standard error
Sampling
Goal of sampling: describe population
 Sample: subset of population
 Could take many different samples
 error introduced

X 
XA  XB
Sampling
Want representative sample
 members reflect characteristics of
the population
 not extremes
 best chance for representative...
 choose members at random ~

Random Samples
Each member has equal chance of being
selected for sample
 independent of selection of other
members
 Helps avoid biases of experimenters
 Focus: simple & stratified random
sampling

also
other methods ~
Simple Random Sampling
All members of population treated
equally
 regardless of characteristics
 e.g., bag of M&Ms
 Set of random digits better
 computer-generated
 table of random digits

Table
A.10 ~
Simple Random Sampling: Procedure
1. Assign # to each population member
2. Go to random digit table
3. Quasi-randomly select “seed”
e.g., N=20, use 2 digits
 Read L --> R, ignore spaces
 If # already used or not in range
 discard & got to next ~
Simple Random Sampling
1
2
3
4
5
6
7
8
9
10
Population
Jack
Susan
Ann
Bill
Steve
Sara
Jane
Julia
Dave
Ellen
Draw random sample: n = 5
Table A.10
2000 Random digits
1
6
11
16
21
31
36
1
10480
15011
01536
02011
81647
91646
69179
2
22368
46573
25595
85393
30995
89198
27982
3
24130
48360
22527
97265
76393
64809
15179
4
42167
93093
06243
61680
07856
16376
39440
5
37570
39975
81837
16656
06121
91782
60468
Stratified Random Sampling
If population has subgroups of interest
 representative sample has same
proportion of subgroups
 Number subjects within each group

females:
1, 2, 3, 4, ....
males: 1, 2, 3, 4, ...

Use same procedures as simple random
sampling
 new seed for each group ~
Stratified Random Sampling
Draw random sample: n = 5
Population
Jack
Susan
Ann
Bill
Steve
Sara
Jane
Julia
Dave
Ellen
1
2
3
4
5
6
Females
Susan
Ann
Sara
Jane
Julia
Ellen
1
2
3
4
Males
Jack
Bill
Steve
Dave
Proportion females =
Proportion males
=
Sampling from a Population

Repeatedly draw random samples
 will differ from population
 different shape
 similar mean
 larger sample ---> closer to  ~
The Sampling Distribution of the Means

Distribution of sample means
from

a single population
Distribution of X ,
not X
has  and s
 Find exact values
 take all possible samples
 or apply Central Limit Theorem ~

Notation
Mean
 X
sample
 
population
 X
population of sample means
 Standard deviation
 s
sample
 s
population
 sX
population of sample means
standard error of the mean ~

Central Limit Theorem
Describes sampling distribution of mean
 Specifies shape, center, width
1. It is a normal distribution
 even if parent population not normal

if
n > 30
2. X = 
3. Can calculate standard error of mean
s
X

s
n
Distributions: Variable vs Means
 = 100
s = 15
n=9
f
s
70
85
IQ Score
100
115
X
130
mean IQ Score

Standard Error of the Mean: Magnitude

Desirable to have small
s
X
sample means close 
 Depends on n and s
 large sample size & small s
 little control s
 can increase sample size

divide
by larger number ~
Sampling Distribution of the Means: Use

Conducting an experiment
 randomly selecting...
one X
 for sample size n
 from population of X

with mean 
 & standard error

s
X

s
n
~
How close is X to ?
means are normally distributed
 Use area under curve
 between mean and 1 standard error
from the mean
 .34
 Same rules as any normal distribution
 compute z score ~

Z scores, X & proportions

Calculate just like values of X

except use X and s X
z
X 
sX
X  zs X  
Know/want Diagram:
Sampling Distribution of Means
X  zs X  
Table: column A or B
z score
X
z
X 
sX
area under
distribution
Table: z column
```
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