Sampling Distribution of the Mean

1
Sampling Distribution of the Mean
Central Limit Theorem
Given population with and the
sampling distribution will have
A mean
A variance
Standard Error (mean)
As N increases, the shape of the distribution
becomes normal (whatever the shape of the
population)
2
Testing Hypothesis Known and
Remember
We could test a hypothesis concerning a
population and a single score by
Obtain
and use z table
We will continue the same logic
Given Behavior Problem Score of 10 years olds
Sample of 10 year olds under stress
Because we know and , we can use the
Central Limit Theorem to obtain the Sampling
Distribution when H0 is true.
3
Sampling Distribution will have
We can find areas under the distribution by
referring to Z table
We need to know
Minor change from z score
NOW
or
With our data
Changes in formula because we are dealing with
distribution of means NOT individual scores.
4
From Z table we find
is 0.0901
Because we want a two-tailed test we double
0.0901 (2)0.0901 0.1802
NOT REJECT H0
or
is
5
One-Sample t test
Popn known unknown we must
estimate with
Because we use S, we can no longer declare the
answer to be a Z, now it is a t
Why?
Sampling Distribution of t
-
S2 is unbiased estimator of
-
The problem is the shape of the S2 distribution
positively skewed
6
thus
S2 is more likely to UNDERESTIMATE (especially
with small N)
thus
t is likely to be larger than Z (S2 is in
denominator)
t - statistic
and substitute S2 for
To treat t as a Z would give us too many
significant results
7
Guinness Brewing Company (student)
Students t distribution we switch to the t Table
when we use S2
Go to Table
Unlike Z, distribution is a function of
with
Degrees of Freedom
For one-sample cases,
lost because we used (sample mean) to
calculate S2
all x can vary save for 1
8
Example One-Sample Unknown
Effect of statistic tutorials
(no tutorials)
Last 100 years
(tutorials)
this years
N 20, S 6.4
9
Go to t-Table
t-Table
-
not area (p) above or below value of t
-
gives t values that cut off critical areas, e.g.,
0.05
-
t also defined for each df
N20 df (N-1) 20-1 19
Go to Table
t.05(19) is
2.093
critical value
reject
10
Factors Affecting Magnitude of t Decision
Difference between and
1.
the larger the numerator, the larger the t value
2.
Size of S2
as S2 decreases, t increases
3.
Size of N
as N increases, denominator decreases, t increases
4.
level
5.
One-, or two-tailed test
11
Confidence Limits on Mean
Point estimate
Specific value taken as estimator of a parameter
Interval estimates
A range of values estimated to include parameter
Confidence limits
Range of values that has a specific (p) of
bracketing the parameter. End Points
confidence limits.
How large or small could be without
rejecting if we ran a t-test on the
obtained sample mean.
12
Confidence Limits (C.I.)
We already know , S and
We know critical value for t at
We solve for
Rearranging
Using 2.993 and -2.993
13
Two Related Samples t
Related Samples
Design in which the same subject is observed
under more than one condition (repeated measures,
matched samples)
Each subject will have 2 measures and
that will be correlated. This must be taken into
account.
Promoting social skills in adolescents
Before and after intervention
before
after
Difference Scores
Set of scores representing the difference between
the subjects performance or two occasions
14
our data can be the D column
from
we are testing a hypothesis using ONE sample
15
Related Samples t
remember
now
N of D scores
Degrees of Freedom
same as for one-sample case
(N - 1) (15 - 1) 14
our data
Go to table
16
Advantages of Related Samples
Avoids problems that come with subject to subject
variability.
1.
The difference between(x1) 26 and (x2) 24 is
the same as between (x1) 6 and (x2) 4
(increases power)
(less variance, lower denominator, greater t)
2.
Control of extraneous variables
3.
Requires fewer subjects
Disadvantages
1.
Order effects
2.
Carry-over effects
17
Two Independent Samples t
Sampling distribution of differences between means
Suppose
2 popns
and
and
and
draw pairs of samples sizes N1, and N2
record means and and the differences
between , and for
each pair of samples
repeat times
18
Mean Difference
Mean
Variance
Standard
Error
Variance Sum Law
Variance of a sum or difference of two
INDEPENDENT variables sum of their variances
The distribution of the differences is also normal
19
t Difference Between Means
We must estimate with
Because
or
20
is O.K. only when the Ns are the same size
When we need a better estimate of
We must assume homogeneity of variance
Rather than using or to estimate
, we use their average.
Because
we need a Weighted Average
weighted by their degrees of freedom
Pooled Variance
21
Now
come from formula for Standard Error
Degrees of Freedom
two means have been used to calculate
22
Example
23
Example
We have numerator
18.00 15.25
We need denominator
???????
Pooled Variance because
Denominator becomes

24
Go to Table
25
Summary
If and are known, then treat as
in Z score formula replaces
If is known and is unknown, then
replaces
in
If two related samples, then replaces
and replaces
26
If two independent samples, and Ns are of equal
size, then is replaced by
If two independent samples, and Ns are NOT equal,
then and are replaced by