Sampling Distributions & Standard Error Lesson 7 Populations & Samples Research goals Learn about population Characteristics that widely apply Impossible/impractical to directly study Research methods
Download ReportTranscript Sampling Distributions & Standard Error Lesson 7 Populations & Samples Research goals Learn about population Characteristics that widely apply Impossible/impractical to directly study Research methods
Sampling Distributions & Standard Error
Lesson 7
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
~
Sampling Error
X
Difference between sample statistic and population parameter
result of choosing random sample
Many potential samples
Sampling Distributions
Samples from a single population
Repeatedly draw random samples
Every possible combination
Calculate a test statistic (e.g., t test)
One-sample:
X
or
Independent samples: Results
and
s
~
X
1
X
sampling distribution
2
The Distribution of Sample Means
Distribution of means for many samples from a single population
Repeatedly draw random samples
Calculate
X and s
Sampling variation (or sampling error)
will differ from population
different shape
similar mean larger sample
closer to
~
#1 Samples: n=10 #2 #3 #4
Law of Large Numbers
Large sample size (n)
give better estimates of parameters
i.e., better fit
:
X
becomes a better estimate of :
s
becomes a better estimate of s : * if
n
30 , good estimate : * if
n
6 , moderately good estimate
Parameters: Distribution of
X
Results in narrower distribution
Has
and
s
Find exact values
take all possible samples
or apply Central Limit Theorem ~
Central Limit Theorem
1.
of distributi on of
X
2.
s of distributi on of
X
called s
X
standard s
n
error of the mean ( s X
or
s
X
s n
)
APA style: SE
Also SEM ~
Central Limit Theorem
3. As sample size (n) increases
the sampling distribution of means approaches a normal distribution
even if parent population not normal distribution of variable (or X)
Very Important! In n ≥ 6, then…
probabilities from standard normal distribution useful
Because we study samples ~
f
Distributions:
X i
vs
X
s
= 100 = 15 n = 9
s
M
s
n
15 9 5
70 85 90 95 IQ Score 100 105 110 115 130 mean IQ Score
Standard Error of the Mean: Magnitude
Small standard error
better fit
sample means close m
More representative sample Depends on n and
s
large sample size & small
s
little control
s
can increase sample size increase value of denominator ~
Using the distribution of X
Use samples to describe populations
is it representative of population?
Sample means normally distributed
Use z table
find area under curve
only slight difference in z formula ~
Conducting an experiment
Same as randomly selecting...
one
X
from distributi on of
X
For a sample size n
with mean =
& standard error
s
X
s
n
Calculating z scores for sample means
z
X
s
X
for raw scores z
X
s
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 ~
Distribution of Sample Means is Normal
f
.34
.34
.02
-2 .14
.14
-1 0 1 2 standard error of mean
( s
X
)
.02
z scores & Distribution of X
What are z scores that define
p in left & right tails = .025 + .025
Look up z scores
Left tail = - 1.96; right tail = + 1.96
f
Distribution of Sample Means is Normal
Boundaries for middle 95% (or .95) of sample means?
for middle 99% (or .95) of sample means?
-2 -2.58
-1.96
-1 0 z scores
( s
X
)
1 2 +1.96
+2.58
Sample Mean
X
Using z scores
X
z
s
X
Table: large/smaller portion column
z score area under curve or proportion Or probability or percentage
z
X
s
X
Table: z column