Download Basic Statistics for the Behavioral Sciences

Chapter 7
OVERVIEW OF STATISTICAL HYPOTHESIS
TESTING: THE z-TEST
Going Forward
Your goals in this chapter are to learn:
• Why the possibility of sampling error causes
researchers to perform inferential statistics
• When experimental hypotheses lead to
either one-tailed or a two-tailed tests
• How to create the null and alternative
hypotheses
• When and how to perform the z-test
Going Forward
• How to interpret significant and nonsignificant
results
• What Type I errors, Type II errors, and power
are
The Role of Inferential Statistics
in Research
Inferential Statistics
Inferential statistics are used to decide whether
sample data represent a particular relationship
in the population.
Parametric Statistics
• Parametric statistics are inferential
procedures requiring certain assumptions
about the raw score population being
represented by the sample
• Two assumptions are common to all
parametric procedures:
– The population of dependent scores should be at
least approximately normally distributed
– The scores should be interval or ratio
Nonparametric Procedures
Nonparametric statistics are inferential
procedures not requiring stringent assumptions
about the populations being represented.
Setting up Inferential Procedures
Experimental Hypotheses
Experimental hypotheses describe the possible
outcomes of a study.
Predicting a Relationship
• A two-tailed test is used when we do not
predict the direction in which dependent
scores will change
• A one-tailed test is used when we do predict
the direction in which dependent scores will
change
Designing a
One-Sample Experiment
To perform a one-sample experiment, we must
already know the population mean for
participants tested under another condition of
the independent variable.
Alternative Hypothesis
The alternative hypothesis (Ha) describes the
population parameters the sample data
represent if the predicted relationship exists in
nature.
Null Hypothesis
The null hypothesis (H0) describes the
population parameters the sample data
represent if the predicted relationship does not
exist in nature.
A Graph Showing the
Existence of a Relationship
The Logic
When a relationship is indicated by the sample
data, it may be because
• The relationship operates in nature and it
produced our data
OR
• We are misled by sampling error
A Graph Showing a Relationship
Does Not Exist
Performing the z-Test
The z-Test
The z-test is the procedure for computing a zscore for a sample mean on the sampling
distribution of means.
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 another condition of the independent variable
4. We know the true standard deviation of the
population ( X ) described by the null hypothesis
Setting up for a Two-Tailed Test
 Create the sampling distribution of means from the
underlying raw score population that H0 says our
sample represents
 Choose the criterion, symbolized by a (alpha)
 Locate the region of rejection which, for a two-tailed
test, involves defining an area in both tails
 Determine the critical value by using the chosen a
to find the zcrit value resulting in the appropriate
region of rejection
Two-Tailed Hypotheses
• In a two-tailed test, the null hypothesis states the
population mean equals a given value. For
example, H0: m = 100.
• In a two-tailed test, the alternative hypothesis
states the population mean does not equal
the same given value as in the null
hypothesis. For example, Ha: m  100.
A Sampling Distribution for H0 Showing the Region
of Rejection for a = 0.05 in a Two-tailed Test
Computing z
• The z-score is computed using the same
formula as before
zobt 
X m
X
where
X 
X
N
Comparing Obtained z
• In a two-tailed test, reject H0 and accept Ha if
the z-score you computed is
– Less than the negative of the critical z-value
OR
– Greater than the positive of the critical z-value
• Otherwise, fail to reject H0
Interpreting Significant and
Nonsignificant Results
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 the
results are unlikely to occur if the predicted
relationship does not exist in the population.
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
the results are likely to occur if the predicted
relationship does not exist in the population.
Nonsignificant Results
• When we fail to reject H0, we do not prove H0
is true
• Nonsignificant results provide no convincing
evidence the independent variable does not
work
Summary of the z-Test
1. Determine the experimental hypotheses and
create the statistical hypothesis
2. Select a, locate the region of rejection, and
determine the critical value
3. Compute X and zobt
4. Compare zobt to zcrit
The One-Tailed Test
One-Tailed Hypotheses
In a one-tailed test, if it is hypothesized the independent
variable causes an increase in scores, then the null
hypothesis states the population mean is less than or equal
to a given value and the alternative hypothesis states the
population mean is greater than the same value. For
example:
H 0 : m  50
H a : m  50
One-Tailed Hypotheses
In a one-tailed test, if it is hypothesized the independent
variable causes a decrease in scores, then the null hypothesis
states the population mean is greater than or equal to a
given value and the alternative hypothesis states the
population mean is less than the same value. For example:
H 0 : m  50
H a : m  50
A Sampling Distribution Showing the Region of Rejection
for a One-tailed Test of Whether Scores Increase
A Sampling Distribution Showing the Region of Rejection
for a One-tailed Test of Whether Scores Decrease
Choosing One-Tailed Versus
Two-Tailed Tests
Use a one-tailed test only when it is the
appropriate test for the independent variable.
That is, when the independent variable can
“work” only if scores go in one direction.
Errors in Statistical Decision Making
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 we
conclude the predicted relationship exists when it
really does not
• The theoretical probability of a Type I error equals a
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 we conclude the predicted
relationship does not exist when it really does
Power
Power is
• The probability of rejecting H0 when it is false
• The probability of not making a Type II error
• The probability that we will detect a
relationship and correctly reject a false null
hypothesis (H0)
Example
Use the following data set and conduct a twotailed z-test to determine if m = 11 and 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
Example
1. H 0 : m  11 ; H a : 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
13.67  11

 2.764
0.966
Example
Since zobt lies within the rejection region, we
reject H0 and accept Ha. Therefore, we conclude
m does not equal 11.