Transcript Two-Sample T-Test for Difference Between Sample Means

SOLVING THE PROBLEM
The one sample t-test compares two values for the population mean of a single
variable. The two-sample test of a population means compares the population means
for two groups of subjects on a single variable. The null hypothesis for this test is:
there is no difference between the population mean of the variable for one group of
subjects and the population mean of the same variable for a second group of subjects.
In addition to our concern with the assumption of normality for each group and the
number of cases in each group if we are to apply the Central Limit Theorem, but this
test also requires us to examine the spread or dispersion of both groups so that the
measure of standard error used in the t-test fairly represents both group.
While there is a test of Equality of Variance and a formula to use when the test is
satisfied and a formula to use when the test is violated, the authors of our text suggest
we always use the formula that assumes the test is violated. If we use this version of
the statistic when the variances are in fact equal, the results of the test are
comparable to what we would obtain using the formula for equal variances.
We will the authors advice and restrict our attention to the “Equal variances not
assumed” row in the SPSS output table without examining the Levene test of equality
of variance.
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Slide 1
The introductory statement in the question indicates:
• The data set to use (GSS2000R)
• The variables to use in the analysis: socioeconomic
index [sei] for groups of survey respondents defined
by the variable sex [sex]
• The task to accomplish (two-sample t-test for the
difference between sample means)
• The level of significance (0.05, two-tailed)
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Slide 2
The first statement asks about
the level of measurement.
A two-sample t-test for the difference
between sample means requires a
quantitative dependent variable and a
dichotomous independent variable.
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Slide 3
"Socioeconomic index" [sei] is quantitative, satisfying the
level of measurement requirement for the dependent
variable. "Sex" [sex] is dichotomous, satisfying the level of
measurement requirement for the independent variable.
Mark the statement as correct.
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Slide 4
To justify the use of probabilities based on a normal
sampling distribution in testing hypotheses, either the
distribution of the variable must satisfy the nearly
normal condition or the size of the sample must be
sufficiently large to generate a normal sampling
distribution under the Central Limit Theorem.
A two-sample t-test for the difference between sample means
requires that the distribution of the variable satisfy the nearly
normal condition for both groups. We will operationally define
the nearly normal condition as having skewness and kurtosis
between -1.0 and +1.0 for both groups, and not having any
outliers with standard scores equal to or smaller than -3.0 or
equal to or larger than +3.0 in the distribution of scores for
either group.
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Slide 5
To evaluate the
variables conformity to
the nearly normal
condition, we will use
descriptive statistics
and standard scores.
We will first compute
the standard scores.
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To compute the standard
scores, select the Descriptive
Statistics > Descriptives
command from the Analyze
menu.
Slide 6
First, move the variable for the
analysis sei to the Variable(s) list box.
Third, click on the
OK button to
produce the output.
Second, mark the check box Save
standardized values as variables.
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Slide 7
Sort the column Zsei in ascending
order to show any negative
outliers at the top of the column.
There were no outliers
that had a standard score
less than or equal to -3.0.
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Slide 8
Sort the column Zsei in descending
order to show any positive outliers
at the top of the column.
There were no outliers that
had a standard score greater
than or equal to +3.0.
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Slide 9
Next, we will use the
Explore procedure to
generate descriptive
statistics for each
gender..
To compute the descriptive
statistics, select the
Descriptive Statistics > Explore
command from the Analyze
menu.
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Slide 10
First, move the
dependent variable sei
to the Dependent List.
Fourth, click on the
OK button to
produce the output.
Second, move the
group variable sex
to the Factor List.
Third, mark the option button
to display Statistics only.
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Slide 11
For survey respondents who were male,
"socioeconomic index" satisfied the criteria
for a normal distribution. The skewness of
the distribution (0.539) was between -1.0
and +1.0 and the kurtosis of the distribution
(-0.852) was between -1.0 and +1.0.
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Slide 12
For survey respondents who were female,
"socioeconomic index" satisfied the criteria
for a normal distribution. The skewness of
the distribution (0.610) was between -1.0
and +1.0 and the kurtosis of the distribution
(-0.921) was between -1.0 and +1.0.
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Slide 13
For survey respondents who were male, "socioeconomic index" satisfied the criteria
for a normal distribution. The skewness of the distribution (0.539) was between -1.0
and +1.0 and the kurtosis of the distribution (-0.852) was between -1.0 and +1.0. For
survey respondents who were female, "socioeconomic index" satisfied the criteria for
a normal distribution. The skewness of the distribution (0.610) was between -1.0 and
+1.0 and the kurtosis of the distribution (-0.921) was between -1.0 and +1.0.
There were no outliers that had a standard score less than or equal to -3.0 or greater
than or equal to +3.0.
Mark the statement
as correct.
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Slide 14
Though we have satisfied the nearly normal
condition and do not need to utilize the
Central Limit Theorem to justify the use of
probabilities based on the normal distribution,
we will still examine the sample size.
To apply the Central Limit Theorem for a
two-sample t-test for the difference
between sample means requires that
both groups defined by the independent
variable have 40 or more cases.
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Slide 15
There were 110 valid cases for
survey respondents who were male
and 145 valid cases for survey
respondents who were female.
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Slide 16
Both groups had 40 or more cases, so the Central Limit Theorem
would be applicable. However, since the distribution of
"socioeconomic index" satisfied the nearly normal condition, we do
not need to rely upon the Central Limit Theorem to satisfy the
sampling distribution requirements of a two-sample t-test for the
difference between sample means.
Mark the statement as correct.
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Slide 17
The next statement asks us to
identify the mean for each group in
the sample data and the standard
error of the sampling distribution.
To answer this question, we
need to produce the output
for the two-sample t-test.
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Slide 18
To produce the two-sample t-test
(which SPSS calls IndependentSamples T-Test), select the Compare
Means > Independent Samples T Test
command from the Analyze menu.
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Slide 19
First, move the variable sei to
the Test Variable(s) list box.
Second, move the grouping variable
sex to the text box.
SPSS adds ?’s after the variable name
to remind us that we need to specify
the numeric codes for the groups.
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Third, click on the
Define Groups button to
enter the group codes.
Slide 20
First, enter 1 for
males as Group 1.
Third, click on the Continue
button to close the dialog box.
First, enter 2 for
females as Group 2.
If I did not remember the code
numbers for male and female, I
would look them up in the Variable
View of the SPSS Data Editor.
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Slide 21
Third, click on the
OK button to
produce the output.
SPSS replaces the question
marks with the codes I entered.
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Slide 22
The mean "socioeconomic
index" for survey respondents
who were male was 50.29 and
the mean for survey respondents
who were female was 47.51
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The standard error of the differences
between group means was 2.446.
Slide 23
The mean "socioeconomic index" for survey respondents
who were male was 50.29 and the mean for survey
respondents who were female was 47.51. The standard
error of the differences between group means was 2.446.
Mark the question as correct.
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Slide 24
The next statement asks us about the
null hypothesis for the one-sample t-test.
We should check to make certain the
relationship is stated correctly.
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Slide 25
The null hypothesis for the test is: there is no difference
between the population mean of "socioeconomic index"
for survey respondents who were male and the
population mean of "socioeconomic index" for survey
respondents who were female.
Since the hypothesis is stated correctly, mark the
question as correct.
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Slide 26
The next statement asks
us to relate the t-test to
the data in our problem.
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Slide 27
Following the convention in
the text book, we will only
focus on the “Equal variances
not assumed” option. Within
this option, the difference
and standard error are
correctly identified.
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The t-test statistic is based on the difference
between the means of the two groups
(2.777) relative to the standard error of the
differences between sample means (2.446).
Slide 28
The statement is correct and contains the correct
values for both the difference in means and the
sampling error that we would typically expect to find
in the sampling distribution for differences in means.
Mark the statement as correct.
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Slide 29
The next statement asks about the probability for the
comparison made by the t-test. i.e. what is the probability
that the population means for each group are not different.
In the last question, the difference in means was only
slightly larger than the standard error of the differences, so
we should expect a ratio near one and a high value for the
probability.
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Slide 30
The probability that the population mean for
survey respondents who were male (50.3) was
not different from the population mean for
survey respondents who were female (47.5)
was p = .257.
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Slide 31
The probability that the population mean for
survey respondents who were male (50.3) was not
different from the population mean for survey
respondents who were female (47.5) was p = .257.
Since the probability was correctly stated, mark
the question as true.
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Slide 32
When the p-value for the statistical test is less
than or equal to alpha, we reject the null
hypothesis and interpret the results of the
test. If the p-value is greater than alpha, we
fail to reject the null hypothesis and do not
interpret the result.
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Slide 33
The p-value for this test (p = .257) is larger than the
alpha level of significance (p = .050) supporting the
conclusion to fail to reject the null hypothesis.
The check box is not marked.
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Slide 34
The final statement asks us to
interpret the result of our statistical
test as a finding in the context of
the problem we created.
We only interpret the results when
the null hypothesis is rejected.
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Slide 35
If we had a significant p-value, we
would have looked at the means of
the two groups to identify the
direction of the relationship.
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Slide 36
Since we did not have a significant p-value,
we cannot reject the null hypothesis and
interpret the relationship.
The check box is not marked.
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Slide 37
Dependent variable is
quantitative?
No
Yes
Independent variable
is dichotomous?
Do not mark
check box.
No
Mark only “None of
the above.”
Stop.
Yes
Mark statement
check box.
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Slide 38
Nearly normal:
• Skewness and kurtosis
between -1.0 and +1.0
for both groups
• Z-scores between -3.0
and +3.0
Nearly normal
distribution?
No
Do not mark
check box.
Yes
Mark statement
check box.
CLT stands for Central
Limit Theorem.
CLT applicable
(Sample size ≥ 40 in
each group)?
No
Yes
Mark statement
check box.
Do not mark
check box.
Stop.
If the variable is not normal
and the sample size is less than
40, the test is not appropriate.
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Slide 39
Nearly normal:
• Skewness and kurtosis
between -1.0 and +1.0
for both groups
• Z-scores between -3.0
and +3.0
Nearly normal
distribution?
No
Do not mark
check box.
Yes
Mark statement
check box.
CLT stands for Central
Limit Theorem.
We will check the
applicability of the
Central Limit Theorem
based on sample size,
even when our data
satisfies the nearly
normal condition.
CLT applicable
(Sample size ≥ 40 in
each group)?
No
Yes
Mark statement
check box.
Do not mark
check box.
Stop.
If the variable is not normal
and the sample size is less than
40, the test is not appropriate.
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Slide 40
Sample means and
standard error correct?
No
Do not mark
check box.
Yes
Mark statement
check box.
H0: no difference
between sample means
No
Do not mark
check box.
Yes
Mark statement
check box.
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Slide 41
T-test accurately
described?
No
Do not mark
check box.
No
Do not mark
check box.
Yes
Mark statement
check box.
P-value (sig.) stated
correctly?
Yes
Mark statement
check box.
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Slide 42
Reject H0 is correct
decision (p ≤ alpha)?
No
Do not mark
check box.
Stop.
Yes
We interpret results
only if we reject
null hypothesis.
Mark statement
check box.
Interpretation is
stated correctly?
No
Do not mark
check box.
Yes
Mark statement
check box.
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Slide 43