Download Chapter 12 Hypothesis Tests: One Sample Mean

Chapter 12
Hypothesis Tests: One
Sample Mean
Fundamental Statistics for the
Behavioral Sciences, 4th edition
David C. Howell
©1999 Brooks/Cole Publishing Company/ITP
Chapter 12 Hypothesis Tests: One
Sample Mean
2
Major Points
• An example
• Sampling distribution of the mean
• Testing hypotheses: sigma known
 An example
• Testing hypotheses: sigma unknown
 An example
Cont.
Chapter 12 Hypothesis Tests: One
Sample Mean
Major Points--cont.
• Factors affecting the test
• Confidence limits
3
Chapter 12 Hypothesis Tests: One
Sample Mean
An Example
• Returning to the effect of violent videos
on subsequent behavior
• The first of several modifications of
Bushman study from Chapter 8
4
Chapter 12 Hypothesis Tests: One
Sample Mean
5
Media Violence
• Does violent content in a video affect
subsequent responding?
• 100 subjects saw a video containing
considerable violence.
• Then free associated to 26 homonyms
that had an aggressive & nonaggressive
form.
 e.g. cuff, mug, plaster, pound, sock
Cont.
Chapter 12 Hypothesis Tests: One
Sample Mean
Media Violence--cont.
• Results
 Mean number of aggressive free associates
= 7.10
• Assume we know that without aggressive
video the mean would be 5.65, and the
standard deviation = 4.5
 These are parameters (m and s)
• Is 7.10 enough larger than 5.65 to
conclude that video affected results?
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Chapter 12 Hypothesis Tests: One
Sample Mean
7
Sampling Distribution of the
Mean
• Formal solution to example given in
Chapter 8.
• We need to know what kinds of sample
means to expect if video has no effect.
 i. e. What kinds of means if m = 5.65 and
s = 4.5?
 This is the sampling distribution of the mean.
Cont.
Chapter 12 Hypothesis Tests: One
Sample Mean
8
Sampling Distribution of the
Mean--cont.
• In Chapter 8 we saw exactly what this
distribution would look like.
• It is called Sampling Distribution of the
Mean.
 Why?
 See next slide.
Cont.
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Chapter 12 Hypothesis Tests: One
Sample Mean
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
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Mean Number A ggr essive Ass ociates
Cont.
Chapter 12 Hypothesis Tests: One
Sample Mean
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Sampling Distribution of the
Mean--cont.
• The sampling distribution of the mean
depends on
 Mean of sampled population
• Why?
 St. dev. of sampled population
• Why?
 Size of sample
• Why?
Cont.
Chapter 12 Hypothesis Tests: One
Sample Mean
Sampling Distribution of the
mean--cont.
• Shape of the sampled population
 Approaches normal
• Why?
 Rate of approach depends on sample size
• Why?
• Basic theorem
 Central limit theorem
11
Chapter 12 Hypothesis Tests: One
Sample Mean
Central Limit Theorem
• Given a population with mean = m and
standard deviation = s, the sampling
distribution of the mean (the distribution
of sample means) has a mean = m, and a
standard deviation = s /n. The
distribution approaches normal as n, the
sample size, increases.
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Chapter 12 Hypothesis Tests: One
Sample Mean
13
Demonstration
• Let population be very skewed
• Draw samples of 3 and calculate means
• Draw samples of 10 and calculate means
• Plot means
• Note changes in means, standard
deviations, and shapes
Cont.
14
Chapter 12 Hypothesis Tests: One
Sample Mean
Parent Population
Skewed Population
3000
Frequency
2000
1000
Std. Dev = 2.43
Mean = 3.0
N = 10000.00
0
.0
20
.0
18
.0
16
.0
14
.0
12
.0
10
0
8.
0
6.
0
4.
0
2.
0
0.
X
Cont.
15
Chapter 12 Hypothesis Tests: One
Sample Mean
Sampling Distribution n = 3
Sampling Distribution
Sample size = n = 3
Frequency
2000
1000
Std. Dev = 1.40
Mean = 2.99
N = 10000.00
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Sample Mean
Cont.
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Chapter 12 Hypothesis Tests: One
Sample Mean
Sampling Distribution n = 10
Sampling Distribution
Sample size = n = 10
1600
1400
Frequency
1200
1000
800
600
400
Std. Dev = .77
200
Mean = 2.99
N = 10000.00
0
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Sample Mean
Cont.
Chapter 12 Hypothesis Tests: One
Sample Mean
Demonstration--cont.
• Means have stayed at 3.00 throughout-except for minor sampling error
• Standard deviations have decreased
appropriately
• Shapes have become more normal--see
superimposed normal distribution for
reference
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Chapter 12 Hypothesis Tests: One
Sample Mean
Testing Hypotheses: s known
• H0: m = 5.65
• H1: m  5.65 (Two-tailed)
• Calculate p(sample mean) = 7.10 if
m = 5.65
• Use z from normal distribution
• Sampling distribution would be normal
18
Chapter 12 Hypothesis Tests: One
Sample Mean
19
Using z To Test H0
• Calculate z
X m
7.1  5.65 1.45
z


 3.22
s
4.5
.45
n
100
• If z > + 1.96, reject H0
• 3.22 > 1.96
 The difference is significant.
Cont.
20
Chapter 12 Hypothesis Tests: One
Sample Mean
z--cont.
• Compare computed z to histogram of
sampling distribution
• The results should look consistent.
• Logic of test
 Calculate probability of getting this mean if
null true.
 Reject if that probability is too small.
Chapter 12 Hypothesis Tests: One
Sample Mean
Testing When s Not Known
• Assume same example, but s not known
• Can’t substitute s for s because s more
likely to be too small
 See next slide.
• Do it anyway, but call answer t
• Compare t to tabled values.
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Chapter 12 Hypothesis Tests: One
Sample Mean
Sampling Distribution of the
Variance
1400
1200
138.89
Population
variance = 138.89
Frequency
1000
n=5
800
10,000 samples
600
58.94% < 138.89
400
200
0
0
0.
80 0
0.
75 0
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70 0
0.
65 0
0.
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55 0
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50 0
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45 0
0.
40 0
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35 0
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30 0
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25
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20 0
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15 0
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.0
50
0
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Sample variance
Chapter 12 Hypothesis Tests: One
Sample Mean
t Test for One Mean
• Same as z except for s in place of s.
• For Bushman, s = 4.40
X  m 7.1  5.65 1.45
t


 3.30
s
4.40
.44
n
100
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Chapter 12 Hypothesis Tests: One
Sample Mean
Degrees of Freedom
• Skewness of sampling distribution of
variance decreases as n increases
• t will differ from z less as sample size
increases
• Therefore need to adjust t accordingly
• df = n - 1
• t based on df
24
25
Chapter 12 Hypothesis Tests: One
Sample Mean
t Distribution
Two-Tailed Significance Level
df
10
15
20
25
30
100
.10
1.812
1.753
1.725
1.708
1.697
1.660
.05
2.228
2.131
2.086
2.060
2.042
1.984
.02
2.764
2.602
2.528
2.485
2.457
2.364
.01
3.169
2.947
2.845
2.787
2.750
2.626
Chapter 12 Hypothesis Tests: One
Sample Mean
Conclusions
• With n = 100, t.0599 = 1.98
• Because t = 3.30 > 1.98, reject H0
• Conclude that viewing violent video leads
to more aggressive free associates than
normal.
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Chapter 12 Hypothesis Tests: One
Sample Mean
Factors Affecting t
• Difference between sample and
population means
• Magnitude of sample variance
• Sample size
27
Chapter 12 Hypothesis Tests: One
Sample Mean
Factors Affecting Decision
• Significance level a
• One-tailed versus two-tailed test
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Chapter 12 Hypothesis Tests: One
Sample Mean
Confidence Limits on Mean
• Sample mean is a point estimate
• We want interval estimate
 Probability that interval computed this way
includes m = 0.95
CI .95  X  t.025 s X
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Chapter 12 Hypothesis Tests: One
Sample Mean
For Our Data
CI
 X t
sX
.95
.025
 7.1  1.98  0.44
 7.1  0.87
 6.23  m  7.97
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Chapter 12 Hypothesis Tests: One
Sample Mean
Confidence Interval
• The interval does not include 5.65--the
population mean without a violent video
• Consistent with result of t test
• What can we conclude from confidence
interval?
31
Chapter 12 Hypothesis Tests: One
Sample Mean
32
Review Questions
• What is the sampling distribution of the
mean?
• Why do we need it?
• Why did I assume that the population
mean was 5.65 and its st. dev. was 0.45?
• What is the problem when the population
variance (or st. dev.) is not known?
Cont.
Chapter 12 Hypothesis Tests: One
Sample Mean
33
Review Questions--cont.
• What does the shape of the sampling
distribution of the variance have to do
with anything?
• How does t differ from z
• What are degrees of freedom (df )?
• What factors affect the value of t ?
Cont.
Chapter 12 Hypothesis Tests: One
Sample Mean
Review Questions--cont.
• What factors affect our conclusions?
• What is the difference between
confidence limits and a confidence
interval?
• How do we interpret a computed
confidence interval?
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