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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
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Sampling
Goal of sampling: describe population
Sample: subset of population Could take many different samples
error introduced
X BA XX
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Sampling
Want representative sample members reflect characteristics of
the population not extremes
best chance for representative... choose members at random ~
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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 ~
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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 ~
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Simple Random Sampling: Procedure
1. Assign # to each population member
2. Go to random digit table
3. Quasi-randomly select “seed”
4. Start with seed, read # digits required
e.g., N=20, use 2 digits Read L --> R, ignore spaces If # already used or not in range
discard & got to next ~
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JackSusanAnnBillSteveSaraJaneJuliaDaveEllen
1 2 3 4 5 6 7 8 910
Simple Random Sampling
Draw random sample: n = 5
Population
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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
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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 ~
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JackSusanAnnBillSteveSaraJaneJuliaDaveEllen
Stratified Random SamplingDraw random sample: n = 5
JackBillSteveDave
1 2 3 4
Males
SusanAnnSaraJaneJuliaEllen
1 2 3 4 5 6
FemalesPopulation
Proportion females =
Proportion males =
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Sampling from a Population Repeatedly draw random samples
will differ from population different shape similar mean larger sample ---> closer to ~
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The Sampling Distribution of the Means
Distribution of sample means from a single population
Distribution of , not X
X
has and Find exact values
take all possible samples or apply Central Limit Theorem ~
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Notation Mean
X sample population X population of sample means
Standard deviation s sample population X population of sample means
standard error of the mean ~
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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
nX
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Distributions: Variable vs Means
115 1301008570
f
IQ Score
= 100 = 15n = 9
X
mean IQ Score
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Standard Error of the Mean: Magnitude
sample means close Depends on n and
large sample size & small little control can increase sample size
divide by larger number ~
Desirable to have small X
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with mean & standard error ~
Sampling Distribution of the Means: Use
Conducting an experiment randomly selecting...
nX
one X for sample size n
from population of X
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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 ~
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Z scores, X & proportions
Calculate just like values of X
X
Xz
except use X and X
X
zX
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Know/want Diagram: Sampling Distribution of Means
z score area under distribution
Table: column A or B
Table: z column
X
Xz
X
X
zX