Download Chapter Ten - Bakersfield College

Chapter Ten
Introduction to
Hypothesis
Testing
New Statistical Notation
• The symbol for greater than is >.
• The symbol for less than is <.
• The symbol for greater than or equal to
is ≥.
• The symbol for less than or equal to
is ≤.
• The symbol for not equal to is ≠.
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Chapter 10 - 2
The Role of Inferential
Statistics in Research
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Chapter 10 - 3
Sampling Error
Remember:
Sampling error results when random
chance produces a sample statistic that
does not equal the population parameter
it represents.
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Chapter 10 - 4
Parametric Statistics
• Parametric statistics are procedures that
require certain assumptions about the
characteristics of the populations being
represented. Two assumptions are common
to all parametric procedures:
– The population of dependent scores forms a
normal distribution
and
– The scores are interval or ratio.
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Chapter 10 - 5
Nonparametric Procedures
• Nonparametric statistics are
inferential procedures that do not
require stringent assumptions about the
populations being represented.
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Chapter 10 - 6
Robust Procedures
Parametric procedures are robust. If the
data don’t meet the assumptions of the
procedure perfectly, we will have only a
negligible amount of error in the
inferences we draw.
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Chapter 10 - 7
Setting up Inferential Procedures
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Chapter 10 - 8
Experimental Hypotheses
Experimental hypotheses describe the
predicted outcome we may or may not
find in an experiment.
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Chapter 10 - 9
Predicting a Relationship
• A two-tailed test is used when we
predict that there is a relationship, but
do not predict the direction in which
scores will change.
• A one-tailed test is used when we
predict the direction in which scores will
change.
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Chapter 10 - 10
Designing a One-Sample Experiment
To perform a one-sample experiment, we
must already know the population mean
under some other condition of the
independent variable.
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Chapter 10 - 11
Alternative Hypothesis
The alternative hypothesis describes
the population parameters that the
sample data represent if the predicted
relationship exists.
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Chapter 10 - 12
Null Hypothesis
The null hypothesis describes the
population parameters that the sample
data represent if the predicted relationship
does not exist.
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Chapter 10 - 13
A Graph Showing the Existence of a
Relationship
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Chapter 10 - 14
A Graph Showing That a Relationship
Does Not Exist
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Chapter 10 - 15
Performing the z-Test
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Chapter 10 - 16
The z-Test
The z-test is the procedure for computing
a z-score for a sample mean on the
sampling distribution of means.
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Chapter 10 - 17
Assumptions of the z-Test
1. We have randomly selected one sample
2. The dependent variable is at least approximately
normally distributed in the population and involves an
interval or ratio scale
3. We know the mean of the population of raw scores
under some other condition of the independent
variable
4. We know the true standard deviation of the
population ( X ) described by the null hypothesis
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Chapter 10 - 18
Setting up for a Two-Tailed Test
1.Choose alpha. Common values are
0.05 and 0.01.
2.Locate the region of rejection. For a
two-tailed test, this will involve defining
an area in both tails of the sampling
distribution.
3.Determine the critical value. Using the
chosen alpha, find the zcrit value that
gives the appropriate region of rejection.
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Chapter 10 - 19
A Sampling Distribution for H0 Showing
the Region of Rejection for a = 0.05 in
a Two-tailed Test
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Chapter 10 - 20
Two-Tailed Hypotheses
• In a two-tailed test, the null hypothesis
states that the population mean equals
a given value. For example, H0: m =
100.
• In a two-tailed test, the alternative
hypothesis states that the population
mean does not equal the same given
value as in the null hypothesis. For
example, Ha: m  100.
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Chapter 10 - 21
Computing z
• The z-score is computed using the
same formula as before
zobt 
X m
where
X 
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X
X
N
Chapter 10 - 22
Rejecting H0
• When the zobt falls beyond the critical
value, the statistic lies in the region of
rejection, so we reject H0 and accept Ha
• When we reject H0 and accept Ha we
say the results are significant.
Significant indicates that the results are
too unlikely to occur if the predicted
relationship does not exist in the
population.
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Chapter 10 - 23
Interpreting Significant Results
• When we reject H0 and accept Ha, we
do not prove that H0 is false
• While it is unlikely for a mean that lies
within the rejection region to occur, the
sampling distribution shows that such
means do occur once in a while
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Chapter 10 - 24
Failing to Reject H0
• When the zobt does not fall beyond the
critical value, the statistic does not lie
within the region of rejection, so we do
not reject H0
• When we fail to reject H0 we say the
results are nonsignificant.
Nonsignificant indicates that the results
are likely to occur if the predicted
relationship does not exist in the
population.
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Chapter 10 - 25
Interpreting Nonsignificant Results
• When we fail to reject H0, we do not
prove that H0 is true
• Nonsignificant results provide no
convincing evidence—one way or the
other—as to whether a relationship
exists in nature
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Chapter 10 - 26
Summary of the z-Test
1.Determine the experimental hypotheses
and create the statistical hypothesis
2.Compute X and compute zobt
3.Set up the sampling distribution
4.Compare zobt to zcrit
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Chapter 10 - 27
The One-Tailed Test
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Chapter 10 - 28
One-Tailed Hypotheses
• In a one-tailed test, if it is hypothesized that
the independent variable causes an increase
in scores, then the null hypothesis is that the
population mean is less than or equal to a
given value and the alternative hypothesis is
that the population mean is greater than the
same value. For example:
– H0: m ≤ 50
– Ha: m > 50
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Chapter 10 - 29
A Sampling Distribution Showing the Region
of Rejection for a One-tailed Test of Whether
Scores Increase
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Chapter 10 - 30
One-Tailed Hypotheses
• In a one-tailed test, if it is hypothesized that
the independent variable causes a decrease
in scores, then the null hypothesis is that the
population mean is greater than or equal to a
given value and the alternative hypothesis is
that the population mean is less than the
same value. For example:
– H0: m ≥ 50
– Ha: m < 50
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Chapter 10 - 31
A Sampling Distribution Showing the Region of
Rejection for a One-tailed Test of Whether Scores
Decrease
[Insert Figure 10.8 here]
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Chapter 10 - 32
Choosing One-Tailed Versus
Two-Tailed Tests
Use a one-tailed test only when confident
of the direction in which the dependent
variable scores will change. When in
doubt, use a two-tailed test.
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Chapter 10 - 33
Errors in Statistical
Decision Making
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Chapter 10 - 34
Type I Errors
• A Type I error is defined as rejecting H0 when
H0 is true
• In a Type I error, there is so much sampling
error that we conclude that the predicted
relationship exists when it really does not
• The theoretical probability of a Type I error
equals a
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Chapter 10 - 35
Type II Errors
• A Type II error is defined as retaining H0
when H0 is false (and Ha is true)
• In a Type II error, the sample mean is so
close to the m described by H0 that we
conclude that the predicted relationship does
not exist when it really does
• The probability of a Type II error is b
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Chapter 10 - 36
Power
• The goal of research is to reject H0
when H0 is false
• The probability of rejecting H0 when it is
false is called power
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Chapter 10 - 37
Possible Results of Rejecting or Retaining
H0
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Chapter 10 - 38
Example
• Use the following data set and conduct
a two-tailed z-test to determine if m = 11
if the population standard deviation is
known to be 4.1
14
14
13
15
11
15
13
10
12
13
14
13
14
15
17
14
14
15
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Chapter 10 - 39
Example
1. H0: m = 11; Ha: m ≠ 11
2. Choose a = 0.05
3. Reject H0 if zobt > +1.965 or if zobt < -1.965.
X 
zobt 
X
4.1

 0.966
N
18
X m
X
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13.67  11

 2.764
0.966
Chapter 10 - 40
Example
Since zobt lies within the rejection region,
we reject H0 and accept Ha. Therefore, we
conclude that m ≠ 11.
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Chapter 10 - 41