Sampling Distributions & Standard Error
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Sampling Distributions & Standard Error
Lesson 7
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Populations & Samples
Research goals Learn about population Characteristics that widely apply Impossible/impractical to directly study
Research methods Study representative sample Introduce sampling error ~
X
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Sampling Error
Difference between sample
statistic and population parameter result of choosing random sample
Many potential samples With different ~
X
sandX
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Sampling Distributions
Samples from a single population Repeatedly draw random samples Every possible combination
Calculate a test statistic (e.g., t test) One-sample: or Independent samples:
Results sampling distribution and ~
X
21 XX
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The Distribution of Sample Means Distribution of means for many samples from
a single population Repeatedly draw random samples Calculate
Sampling variation (or sampling error) will differ from population different shape similar mean larger sample closer to~
sandX
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Samples: n=10
#1 #2
#3 #4
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Law of Large Numbers
Large sample size (n) give better estimates of parameters i.e., better fit
of estimatebetter a becomesX:
of estimatebetter a becomess:
estimate good ,30 if* n:
estimate good moderately ,6 if* n:
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Parameters: Distribution of X
Results in narrower distribution Has and
Find exact values take all possible samples or apply Central Limit Theorem ~
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Central Limit Theorem
1.
2.
nX
APA style: SE Also SEM ~
X ofon distributi of
X ofon distributi of
)(mean theoferror standard calledX
orn
sX
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Central Limit Theorem 3. As sample size (n) increases
the sampling distribution of means approaches a normal distribution
even if parent population not normaldistribution of variable (or X)
Very Important! In n ≥ 6, then… probabilities from standard normal
distribution useful Because we study samples ~
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Distributions: Xi vs X
1301008570
f
IQ Score
= 100 = 15n = 9
9
15
nM
5
11595 105
mean IQ Score90 110
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Standard Error of the Mean: Magnitude
Small standard error better fit sample means close m More representative sample
Depends on n and large sample size & small little control can increase sample size
increase value of denominator ~
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Using the distribution of X
Use samples to describe populations is it representative of population? how close is ?
Sample means normally distributed Use z table
find area under curve only slight difference in z formula ~
toX
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Conducting an experiment
Same as randomly selecting...
nX
For a sample size n
with mean =
& standard error
XX ofon distributi from one
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Calculating z scores
X
Xz
X means samplefor
X
z
Xscores raw for
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How close is X to ?
means are normally distributed Use area under curve
between mean and 1 standard error above the mean
34% Same rules as any normal distribution
compute z score ~
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Distribution of Sample Means is Normal
1 20-1-2
.34
.14
f
standard error of mean
.02
.34
.14
.02
)(X
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z scores & Distribution of X
What are z scores that define boundaries of middle 95% of ? p in left & right tails = .025 + .025 Look up z scores Left tail = - 1.96; right tail = + 1.96
Boundaries for middle 99% of ? ~
X
X
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Distribution of Sample Means is Normal
1 20-1-2
f
z scores )(X
Boundaries for middle 95% (or .95) of sample means?
-1.96 +1.96
for middle 99% (or .95) of sample means?
-2.58 +2.58
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Using z scores
Sample Mean z score
area under curveor
proportionOr
probabilityor
percentage
Table: large/smaller portion column
Table: z column
X
Xz
X
zX
X