Download Chapter 8 Sampling Distributions and Hypothesis Testing

Chapter 8
Sampling Distributions
and Hypothesis Testing
Fundamental Statistics for the
Behavioral Sciences, 5th edition
David C. Howell
©2003 Brooks/Cole Publishing Company/ITP
2
Chapter 8 Sampling Distributions
Major Points
• An example
• Sampling distribution
• Hypothesis testing
 The null hypothesis
 Test statistics and their distributions
 The normal distribution and testing
• Important concepts
3
Chapter 8 Sampling Distributions
Media Violence
• Does violent content in a video affect
later behavior?
 Bushman (1998)
• Two groups of 100 subjects saw a video
 Violent video versus nonviolent video
• Then free associated to 26 homonyms
with aggressive & nonaggressive forms.
 e.g. cuff, mug, plaster, pound, sock
Cont.
4
Chapter 8 Sampling Distributions
Media Violence
• Results
 Number of aggressive free associates to the
homonym as a function of video:
 saw violent video
mean = 7.10
 saw nonviolent video
mean = 5.65
• Is this difference large enough to
conclude that type of video affected
results?
Chapter 8 Sampling Distributions
A Simplified Version of Study
• One-group study is easier to start with
than two-group study.
• Convert to one-group study
 People normally give 5.65 aggressive
associates to homonyms. (a pop. parameter)
 A group who watched violent videos give 7.10
aggressive associates. (a sample statistic)
 Is this sufficiently more than expected to
conclude that violent video has effect?
5
Chapter 8 Sampling Distributions
What is the Question?
• Is the difference between 7.10 and 5.65
large enough to lead us to conclude that
it is a real difference?
 Would we expect a similar kind of difference
with a repeat of this experiment?
• Or...
 Is the difference due to “sampling error?”
6
7
Chapter 8 Sampling Distributions
Sampling Error
• The normal variability that we would
expect to find from one sample to
another, or one study to another
• Random variability among observations
or statistics that is just due to chance
Chapter 8 Sampling Distributions
How Could we Assess
Sampling Error?
• Take many groups of 100 subjects who
did not see a violent video.
• Record the number of aggressive
responses to 26 homonyms.
• Plot the distribution and record its mean
and standard deviation.
• This distribution is a “Sampling
Distribution.”
8
Chapter 8 Sampling Distributions
Sampling Distribution
• The distribution of a statistic over
repeated sampling from a specified
population.
• Possible result for this example.
 See next slide.
 Shows the kinds of means we expect to find
when people don’t view a violent video.
9
10
Chapter 8 Sampling Distributions
Sampling Distribution
Number of Aggressive Associates
1400
1200
Fr equency
1000
800
600
400
Std. Dev = .45
200
Mean = 5.65
0
N = 10000.00
7.
7.
6.
6.
6.
6.
5.
5.
5.
5.
4.
4.
4.
4.
3.
25
00
75
50
25
00
75
50
25
00
75
50
25
00
75
Mean Number A ggr essive Ass ociates
Chapter 8 Sampling Distributions
What Do We Conclude?
• When people don’t view violent video,
they average between about 4.5 and 6.5
aggressive interpretations of homonyms.
• Our violent video group averaged 7.10
aggressive interpretations.
 Our subjects’ responses were not like
normal.
• Conclude that the violent video increased
aggressive associations.
11
Chapter 8 Sampling Distributions
Hypothesis Testing
• A formal way of doing what we just did
• Start with hypothesis that subjects are
normal.
 The null hypothesis
• Find what normal subjects do.
• Compare our subjects to that standard.
12
Chapter 8 Sampling Distributions
The Null Hypothesis
• The hypothesis that our subjects came
from a population of normal responders.
• The hypothesis that watching a violent
video does not change mean number of
aggressive interpretations.
• The hypothesis we usually want to reject.
13
Chapter 8 Sampling Distributions
Steps in Hypothesis Testing
• Define the null hypothesis.
• Decide what you would expect to find if
the null hypothesis were true.
• Look at what you actually found.
• Reject the null if what you found is not
what you expected.
14
Chapter 8 Sampling Distributions
Important Concepts
• Concepts critical to hypothesis testing
 Decision
 Type I error
 Type II error
 Critical values
 One- and two-tailed tests
15
16
Chapter 8 Sampling Distributions
Decisions
• When we test a hypothesis we draw a
conclusion; either correct or incorrect.
 Type I error
• Reject the null hypothesis when it is actually
correct.
 Type II error
• Retain the null hypothesis when it is actually
false.
17
Chapter 8 Sampling Distributions
Type I Errors
• Assume violent videos really have no
effect on associations
• Assume we conclude that they do.
• This is a Type I error
 Probability set at alpha ()
•  usually at .05
 Therefore, probability of Type I error = .05
18
Chapter 8 Sampling Distributions
Type II Errors
• Assume violent videos make a difference
• Assume that we conclude they don’t
• This is also an error (Type II)
 Probability denoted beta ()
• We can’t set beta easily.
• We’ll talk about this issue later.
• Power = (1 - ) = probability of correctly
rejecting false null hypothesis.
19
Chapter 8 Sampling Distributions
Critical Values
• These represent the point at which we
decide to reject null hypothesis.
• e.g. We might decide to reject null when
(p|null) < .05.
 Our test statistic has some value with p =
.05
 We reject when we exceed that value.
 That value is the critical value.
Chapter 8 Sampling Distributions
20
One- and Two-Tailed Tests
• Two-tailed test rejects null when
obtained value too extreme in either
direction
 Decide on this before collecting data.
• One-tailed test rejects null if obtained
value is too low (or too high)
 We only set aside one direction for rejection.
Cont.
Chapter 8 Sampling Distributions
One- & Two-Tailed Example
• One-tailed test
 Reject null if violent video group had too
many aggressive associates
• Probably wouldn’t expect “too few,” and
therefore no point guarding against it.
• Two-tailed test
 Reject null if violent video group had an
extreme number of aggressive associates;
either too many or too few.
21
Chapter 8 Sampling Distributions
22
Review Questions
• Define a sampling distribution.
• How would you create a sampling distribution
of mean number of aggressive associates if the
null were true?
• What is sampling error?
• What does sampling error have to do with all of
this?
Cont.
Chapter 8 Sampling Distributions
Review Questions--cont.
• What are the steps in hypothesis testing?
• What is the probability we’d conclude
violent videos cause aggression if they
really don’t?
• Distinguish between Type I and Type II
errors.
• Distinguish between one-tailed and twotailed tests.
23