Transcript MS W - Rutgers

Chapter 5
Introduction
to Inferential
Statistics
Definition
infer - vt., arrive at a decision by or
opinion by reasoning from known facts or
evidence.
Sample
A sample comprises a part of the population
selected for a study.
Random Samples
If every score in the population has an equal
chance of being selected each time you chose
a score, then it is called a random sample.
Random samples, and only random samples,
are representative of the population from
which they are drawn.
Q: ON WHAT MEASURES IS A
RANDOM SAMPLE
REPRESENTATIVE OF THE
POPULATION?
Q: ON WHAT MEASURES IS A
RANDOM SAMPLE
REPRESENTATIVE OF THE
POPULATION?
A: ON EVERY MEASURE.
REPRESENTATIVE ON
EVERY MEASURE
The mean of the random sample’s height
will be similar to the mean of the
population.
The same holds for weight, IQ, ability to
remember faces or numbers, the size of
their livers, self-confidence, how many
children their aunts had, etc., etc., etc. ON
EVERY MEASURE THAT EVER WAS OR
CAN BE.
All sample statistics are
representative of their
population parameters
The sample mean is a least squares,
unbiased, consistent estimate of the
population mean.
MSW is a least squares, unbiased,
consistent estimate of the population
variance, sigma2.
In Chapter 7, you will learn
about the population
correlation coefficient,
rho, and its estimate based
on a random sample called
r.
Based on what you know about sample
statistics so far, what would you say about
the relationship of r and rho?
r should be (and is) a least
squares, unbiased,
consistent estimate of rho.
Assume you start with a
random sample.
On what measures will
sample statistics be least
squares, unbiased, and
consistent estimates of
their population
parameters?
REPRESENTATIVE ON
measures of central
tendency (the mean), on
measures of variability
(e.g., sigma2), and on all
derivative measures
For example, the way scores fall around the
mean of a random sample (as indexed by MSW)
will be similar to the way scores fall around the
mean of the population (as indexed by sigma2).
THERE ARE OCCASIONAL
RANDOM SAMPLES THAT
ARE POOR
REPRESENTATIVES OF
THEIR POPULATION
But 1.) we will take that into account
And 2.) most samples are fairly to very
good representatives of their
populations
Population Parameters and
Sample Statistics: Nomenclature
The characteristics of a population are called
population parameters. They are usually
represented by Greek letters (, ).
The characteristics of a sample are called
SAMPLE STATISTICS. They are usually
represented by the English alphabet ( X , s).
Three things we can do
with random samples
Estimate population parameters. This is called
estimation research.
Estimate the relationship between variables in
the population from their relationship in a
random sample. This is called correlational
research.
Compare the responses of random samples
drawn from the same population to different
conditions. This is called experimental research.
Estimating population
parameters from a random
sample.
As noted before, when you study a
random sample, sample statistics are least
squares, unbiased, consistent estimates of
their population parameters.
Correlational Research
We observe the relationship among variables
in a random sample. We are unlikely to find strong
relationships purely by chance. When you study a
sample and the relationship between two variables
is strong enough, you can infer that a similar
relationship between the variables will be found
in the population as a whole.
This is called correlational research.
For example, height and weight are co-related.
Another way to describe
correlational research
A key datum in psychology is that individuals
differ from each other in fairly stable ways.
For example, some people learn foreign
languages easily, while others find it more
difficult.
Correlational research allows us to determine
whether individual differences on one variable
(called the X variable) are related to individual
differences on a second variable (called Y).
What is to come- CH. 6 & 7
In Chapter 6, you will learn to turn scores on
different measures from a sample into t scores,
scores that can be directly compared to each
other. (You will also learn to use the t
distribution to create confidence intervals and
test hypotheses.)
In Chapter 7, you will learn to compute a single
number that describes the direction and
consistency of the relationship between two
variables. That number is called Pearson’s r, the
correlation coefficient.
What is to come – CH. 8
In Chapter 8, you will learn to predict scores on
one variable from scores on another variable
when you know (or can estimate) the
correlation coefficient.
In Chapter 8, you will also learn when not to do
that and to go back to predicting that everyone
will score at the mean of their distribution.
What is to come - CH 9 – 11
Experimental Research
In Chapters 9 – 11 you will learn about experiments.
In an experiment, we start with samples that can be
assumed to be similar and then treat them differently.
Then we measure response differences among the samples
and make inferences about whether or not similar
differences would occur in response to similar treatment
in the whole population.
For example, we might expose randomly selected groups
of depressed patients to different doses of a new drug to
see which dose produces the best result.
If we got clear differences, we might suggest that all
patients be treated with that dose.
The logic of experimentation
We start off with groups that are all random
samples from a single population.
As we add scores to each group, the group
means become more and more similar to the
population mean and to each other.
The variation of each group’s scores around its
own mean becomes more and more similar to
sigma2. Thus, the mean squares in each group
become more and more similar to each other.
If nothing changes things,
the groups become more
alike in every way, on
every possible measure.
At the beginning of an experiment, the different
groups are alike.
HOWEVER, they then get treated differently.
We then determine whether they differ more
after being treated differently than they would if
they all were treated the same.
Note that individual
differences underlie our
ability to do experimental
research as well as
correlational research
Many people believe that a science of
psychology is impossible because people
differ too much to be the subject of a
science.
It is obvious that individual
differences underlie our ability
to do correlational research, but
why does that work for
experimental research as well?
Why do the groups become more alike? Because
each person’s score is different and tends to
correct each sample statistic back towards its
population parameter.
This happens on every measure.
Thus, taken together, individual differences
make random samples from the same
population similar in every way.
TO REPEAT, in order to do experiments, we
must start with groups that are similar to each
other in every way.
Why are the groups alike at the beginning of an
experiment? Because each person’s score is
different and each different, randomly selected
score tends to correct each group’s sample
statistic back towards its population parameter
and toward those of the other groups.
This happens on every conceivable measure.
Thus, as you add people to your samples,
individual differences make random samples
from the same population similar in every way.
In summary, individual
differences underlie our
ability to compose groups
that are the same, not
different at the beginning
of the experiment.
This is important enough for us to go over
it yet again at the end of class.
In this chapter, we will
focus on estimating
population parameters
from sample statistics.
Least squares, unbiased,
and consistent estimates in
more detail.
Definition
A least square estimate is a number that is
the minimum average squared distance from
the number it estimates.
We will study sample statistics that are least
squares estimates of their population parameters.
Definition
An unbiased estimate is one around which
deviations sum to zero.
We will study sample statistics that are unbiased
estimates of their population parameters.
Definition
A consistent estimator is one where the larger
the number of randomly selected scores underlying
the sample statistic, the closer the statistic will tend
to come to the population parameter.
We will study sample statistics that are consistent
estimates of their population parameters.
Estimation research underlies
both correlational and
experimental research.
We measure the characteristics of a random
sample and then we infer that they are similar
to the characteristics of the population.
Characteristics are things like the mean and
standard deviation. We use sample statistics that
are ____ _____, ________, and _______ predictors
to estimate their population parameters.
Estimation research underlies
both correlational and
experimental research.
We measure the characteristics of a random
sample and then we infer that they are similar
to the characteristics of the population.
Characteristics are things like the mean and
standard deviation. We use sample statistics that
are least squares, consistent and unbiased
predictors to estimate their population parameters.
The sample mean
The sample mean is called X-bar and is
represented by X.
X is the best estimate of , because it is a least
squares, unbiased, consistent estimate.
X = X / n
Most people think Xbar is
important mostly because
it is the mean of the
sample. That’s wrong.
The most important thing about the mean of a
random sample is that it is a least squares,
unbiased, consistent estimate of mu.
So, when we find deviations from a sample
mean we are actually finding out how much
scores differ from our best estimate of mu!
Estimated variance
The sum of squares within group is found by
finding the deviation of each score from the mean
of its own The estimate of 2 is called the mean
squared error or the mean square within groups,
represented by the symbol “MSW.”
To compute MSW, you find the sum of the squared
deviations around the sample mean (which is your
best estimate of mu), then divide by n-k.
SSW = (X - X)2
MSW = (X - X)2 / (n-k)
Estimated variance
k is the number of groups you are studying
and n is the total number of scores in all the
groups together.
MSW is a least squares, unbiased, consistent
estimate of sigma2.
SSW = (X - X)2
MSW = (X - X)2 / (n-k)
Estimated standard
deviation
The estimate of  is called s. Just as you
find  by taking the square root of 2, you
find s by taking the square root of MSW.
s = MSW
Fill in the blanks
We estimate the population mean by finding the
____________________ (phrase).
We estimate the population variance (sigma2)
with ___ by first finding the ________ __
______ (phrase) between our best estimate of
mu (which is the _____ ____ ) and each score.
Then, we divide the sum of squares by ___
where ___ is the number of scores and ___ is
the number of groups in our sample.
We estimate sigma by taking a square root of
____, our best estimate of sigma2.
In English – fill in the blanks
We estimate the population mean by finding the
mean of the sample.
We estimate the population variance (sigma2)
with MSW by first finding the sum of the
squared differences between our best
estimate of mu (which is the sample mean)
and each score. Then, we divide the sum of
squares by n-k where n is the number of scores
and k is the number of groups in our sample.
We estimate sigma by taking a square root of
MSW, our best estimate of sigma2.
Estimating mu and sigma
– single sample
S#
A
B
C
X
6
8
4
X=18
N= 3
X=6.00
X
6.00
6.00
6.00
(X - X)2
0.00
4.00
4.00
(X - X)
0.00
2.00
-2.00
(X-X)=0.00
(X-X)2=8.00 = SSW
MSW = SSW/(n-k) = 8.00/2 = 4.00
s=
MSW = 2.00
Group1
1.1
1.2
1.3
1.4
X
50
77
69
88
X
71.00
71.00
71.00
71.00
Group2
2.1
2.2
2.3
2.4
(X - X)
-21.00
+6.00
-2.00
+17.00
(X-X1)=0.00
(X - X)2
441.00
36.00
4.00
289.00
(X-X1)2= 770.00
78
57
82
63
70.00
70.00
70.00
70.00
Group3
3.1
3.2
3.3
3.4
8.00
-13.00
12.00
-7.00
(X-X2)=0.00
64.00
169.00
144.00
49.00
(X-X2)2= 426.00
74
70
63
81
72.00
72.00
72.00
72.00
2.00
-2.00
-9.00
9.00
(X-X3)=0.00
4.00
4.00
81.00
81.00
(X-X3)2= 170.00
X1 = 71.00
X2 = 70.00
X3 = 72.00
MSW = SSW/(n-k) = 1366.00/9 = 151.78
s = MSW = 151.78 = 12.32
You do one.
There were two small (n=4) samples
taken from a population. In the first
sample, Al scored 7, Barbara scored 9,
Charlie scored 11 and Donna scored 17.
In the second sample Ed scored 8,
Francine scores 11, George scored 14,
and Hannah scored 19. Estimate sigma2
with these data.
Group1
1.1
1.2
1.3
1.4
X
7
9
11
17
X
11.00
11.00
11.00
11.00
Group2
2.1
2.2
2.3
2.4
(X - X)
-4.00
-2.00
0.00
+6.00
(X-X1)=0.00
(X - X)2
16.00
4.00
0.00
36.00
(X-X1)2= 56.00
8
11
14
19
13.00
13.00
13.00
13.00
-5.00
-2.00
+1.00
+6.00
(X-X2)=0.00
25.00
4.00
1.00
36.00
(X-X2)2= 66.00
X1 = 11.00
X2 = 13.00
MSW = SSW/(n-k) =
122.00/6 = 20.33
s = MSW = 20.33 = 4.51
Why n-k?
This has to do with “degrees of freedom.”
As you saw last chapter, each time you
add a score to a sample, you pull the
sample statistic toward the population
parameter.
Any score that isn’t free to
vary does not tend to pull the
sample statistic toward the
population parameter.
One deviation in each group is
constrained by the rule that deviations
around the mean must sum to zero. So
one deviation in each group is not free to
vary.
Deviation scores underlie our computation
of SSW, which in turn underlies our
computation of MSW.
n-k is the number of degrees
of freedom for MSW and s.
You use the deviation scores as the basis of
estimating sigma2 with MSW and sigma with s.
Scores that are free to vary are called degrees
of freedom.
 Since one deviation score in each group is
not free to vary, you lose one degree of
freedom for each group - with k groups
you lose k*1=k degrees of freedom.
n-k is the number of degrees
of freedom for MSW
 There are n deviation scores in total. k are not free to
vary. That leaves n-k that are free to vary, n-k degrees
of freedom for MSW, for your estimate of sigma2.
 The precision or “goodness” of an estimate is based on
degrees of freedom. The more df, the closer the
estimate tends to get to its population parameter.
 Look again at the three group example. See how one
score in each group becomes constrained, not free to
vary, after we find the mean of each group. This is
because deviations around each group mean must sum
to zero.
Group1
1.1
1.2
1.3
1.4
X
50
77
69
88
X
71.00
71.00
71.00
71.00
Group2
2.1
2.2
2.3
2.4
(X - X)
-21.00
+6.00
-2.00
+17.00
(X-X1)=0.00
(X - X)2
441.00
36.00
4.00
289.00
(X-X1)2= 770.00
78
57
82
63
70.00
70.00
70.00
70.00
Group3
3.1
3.2
3.3
3.4
8.00
-13.00
12.00
-7.00
(X-X2)=0.00
64.00
169.00
144.00
49.00
(X-X2)2= 426.00
74
70
63
81
72.00
72.00
72.00
72.00
2.00
-2.00
-9.00
9.00
(X-X3)=0.00
4.00
4.00
81.00
81.00
(X-X3)2= 170.00
X1 = 71.00
X2 = 70.00
X3 = 72.00
MSW = SSW/(n-k) = 1366.00/9 = 151.78
s = MSW = 151.78 = 12.32
More scores that are free
to vary = better estimates:
the mean as an example.
Each time you add a randomly selected score to your sample,
it is most likely to pull the sample mean closer to mu, the
population mean.
Any particular score may pull it further from mu.
But, on the average, as you add more and more scores, the odds
are that you will be getting closer to mu..
Book example
Population is 1320 students taking a test.
 is 72.00,  = 12
Unlike estimating the variance (where df=n-k) when estimating the
mean, all the scores are free to vary.
So each score in the sample will tend to make the sample mean a better
estimate of mu.
Let’s randomly sample one student at a time and see what happens.
Test Scores
F
r
e
q
u
e
n
c
y
Standard
deviations
Scores
Mean
score
3
2
1
0
1
2
3
36
48
60
72
84
96
108
Sample scores:
102
Means:
72
87
66
80
76
79
66
76.4
78
69
76.7 75.6
63
74.0
Consistent estimators
This tendency to pull the sample mean back to the population
mean is called “regression to the mean”.
We call estimates that improve when you add scores
to the sample consistent estimators.
Recall that the statistics that we will learn are:
consistent,
least squares, and
unbiased.
Individual differences,
again.
Once again, in regard to
individual differences:
Many intro psych students wonder about
psychologists efforts to understand and
predict how the average person will
respond to specific situations or
conditions.
They feel that people are too different for
us to really determine such laws.
While psychology as a
“science” can be criticized
on a number of grounds,
this is not one of them.
The random differences among individuals
form one of the bases of our science. Let’s
go over the importance of individual
differences again.
The effect of individual
differences
As you know, each time you add a score that is
free to vary to a sample, the sample statistics
become better estimates of their population
parameters.
This happens, in part, because individuals differ
randomly . So, each person’s score corrects the
sample statistics back towards their population
parameters.
These include the mean, standard deviation and
other statistics that you have yet to learn, such
as r, the correlation coefficient that describes
the strength and direction of a relationship
between two variables.
Individual differences are central
in correlational research
In correlational research, we compare
individuals who differ on two variables.
We see whether differences on one
variable are related to differences on the
other.
If individuals did not have (reasonably)
stable differences from each other, there
could be no such correlational research.
Individual differences underlie
experimental research
In experimental work, we need to do two
things.
First, we need to compose groups that are
similar to each other.
The key to this is randomly selecting members
for each experimental group.
That way, as you add individuals who
randomly differ to each group, the groups
increasingly resemble the population
(and, therefore, each other) in all possible
regards.
Second, in experimental work, we
examine the differences among
the means of experimental
groups.
Before extrapolating to the
population from differences
observed among group means,
we must be sure that we are not
simply seeing the results of
sampling fluctuation.
We must have an index of how much
variation among the means simply
reflects sampling fluctuation.
In most of the designs we will study in
Chapters 9-11, MSW is the basis that
lets us determine how much the
means should vary from each other
simply from sampling fluctuation.
Individual differences play a large part
in determining MSW.
Thus, rather than making a
science of human behavior
impossible, the fact that
individuals differ plays a
critical role in the
research designs and
statistical tools that have
been developed.