Transcript Chapter 1

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?
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.
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 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
Sample statistics are least squares,
unbiased, consistent estimates of their
population parameters.
We’ll get to this in a minute, in detail.
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 the correlation
coefficient.
What is to come – CH. 8
In Chapter 8, you will learn to predict scores on
one variable 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.
In this chapter, we will
focus on estimating
population parameters
from sample statistics.
Estimation 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.
Estimation underlies both correlational and
Experimental research.
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.
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
Estimated variance
The estimate of 2 is called the mean squared
error and is represented by MSW.
It is also a least squares, unbiased, consistent
estimate.
SSW = (X - X)2
MSW = (X - X)2 / (n-k)
Estimated standard
deviation
The estimate of  is called s.
s = MSW
In English
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
(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
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
 You use the deviation scores as the basis of estimating
sigma2 with MSW.
 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.
 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 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.
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.
A philosophical point
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.
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, 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 tells us how much
variation among the means we should
have 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.