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TRANSCRIPT
Recall: We just learned how Gossett
successfully used the t distribution, with a one-sample t-test, to compare
a sample of ale to the particular batch (i.e., the population) of ale it came from. However, Gossett soon
realized, however, that the problem arose if you have small samples from
two different populations and need to determine if they are significantly
different from one another.
Example 1:
Suppose we are interested in looking at the difference
in the number of disciplinary referrals
between children from single-parent homes and
children who live with both parents.
Answer:
Here, we can say our independent variable
would be "Home Environment," and there
would be two levels, "Single Parent" and
"Both Parents."
Example 2:
Suppose we might be interested in comparing the time it takes pigeons
to learn via operant conditioning or classical
conditioning.
Answer:
We would have one independent variable, type of reinforcement,
with two levels--operant conditioning and classical
conditioning.
Answer:
Our dependent variable would
represent the amount of time it took pigeons
to learn to do a particular task.
Did you know?
Both of these scenarios meet all of the criteria for an
independent- or dependent-sample t-test. Both have one independent variable with
two levels, one independent variable, and the data being
collected is quantitative.
Question:
How do we decide whether to use the
independent-sample t-test or the
dependent-sample t-test?
Independent-Sample t-Test
It compares the means between two unrelated groups on the same continuous, dependent variable. Being in one group excludes the possibility of being in the other.
The researcher would collect data from two groups, with each group
representing a unique population.
Question:
If we have samples from two independent populations, how do we know if they are
significantly different from one another?
Answer:
The general idea of hypothesis testing is the same but here we need to use a slightly different
sampling distribution. Instead of using the sampling distribution of
the means, which we talked about earlier, we will use the
sampling distribution of mean differences.
Sampling Distribution of Mean
Difference
Here, we are not interested in determining if a single sample is
different from the population from which it was drawn. Instead, we are
interested in determining if the mean difference between samples taken
from two like, but independent populations are significantly different from the mean difference of any other
two samples drawn from the same populations.
Going Back to Our Example
In order to demonstrate the SDMD, let's use the hypothesis about the
number of parents and disciplinary referrals.
Quality 1:
The shape of the distribution will be
different for different sample sizes. It will
generally be bell-shaped but will be "flatter" with
smaller sample sizes.
Quality 2:
As the number of samples plotted
grows, the distribution will be symmetrical around the mean.
Quality 4:
The measure of dispersion, the standard
error of the mean differences (SEMD), is
conceptually the same as the standard error of the
mean.
Quality 5:
The percentage of values under the curve depends on the alpha value and the degrees
of freedom (df).
Research Hypothesis:
There will be a significant
difference in the number of
referrals between students
from single-parent homes and
students who live with both
parents.
Null Hypothesis:
There will be no significant
difference in the number of
referrals between students
from single-parent homes and
students who live with both
parents.
Testing Our Hypothesis
As we saw, students from one-parent homes average three referrals, while their classmates average four. Again, they are different, but are they significantly
different?
Step 2:
Decide whether we are going to use the entire alpha value or not to determine our critical value.
Note:
Since we have a two-tailed hypothesis, we are going to divide
our alpha value by 2, giving us an alpha
value of .025.
Note:
We have to plot 2.101 as both a positive
and a negative value since we are testing a
two-tailed hypothesis.
Decision:In this case, we would fail to reject the null since our computed value of t (i.e., -.968) is within the range
created by our critical value of t.
Conclusion:
In this case, although the students with both parents at home have
a higher number of referrals, the difference
isn't significant.
Decision:We can see that p (i.e., Sig.
2-tailed) of .346 is much greater than our alpha
value, and we support our decision to not reject the
null hypothesis. Or, we fail to reject our null
hypothesis.
Note: Although there are columns for both
"Equal Variances Assumed" and "Equal Variances Not Assumed,"
we're only going to use the "Equal Variances Assumed" at this point. We will go into detail regarding the use
of these values after we have worked through a couple of examples and have a thorough understanding of the basic ideas of the independent-
sample t-test.
Effect Size .433--According to Cohen, this
is a medium effect size and, as might be expected given the
relatively large p value, indicates that the number of parents in a home does not have a large effect on the
number of referrals a given student receives.
Example 2:Let's say we are interested in
determining if there is a significant difference in the
frequency of nightmares between men and women. Let's
suppose we felt, from the outset, that men would have
more nightmares than women.
Null Hypothesis:
There will be no
significant difference
in the number of
nightmares between
males and females.
Collection of Data:
In order to test this hypothesis, we might ask
a representative sample of males and females to report how often they
have dreams of this type during one month.
Question:
What is our independent variable? How many levels? What are the levels? What is our
dependent variable?
Answer:
We have one independent variable, gender, with two levels, male and female. We have one dependent variable, frequency of
nightmares, and the data collected is quantitative.
Answer:
Since a given participant can fall into only one category, male and
female, it appears we will use an independent-
sample t-test.
Answer:
We are testing a directional hypothesis. Because we are dealing with a one-tailed hypothesis, however, we
need to refresh our memories on using our alpha value to
determine the correct critical value of t we need to use.
Recall: When you state a directional
hypothesis, you're saying one of the mean values you're comparing is
going to be significantly higher OR significantly lower than the other.
Since you're not looking at the probability of it being either, this means you're going to have to use your entire alpha value on one end
of the curve on the other.
Determine the Critical
Value of t
Using the t table, we have to use our entire value of
alpha, .05, and the degrees of freedom, which
is 18, hence our critical value of t is 1.734.
Answer:
In this case, our computed value of t (i.e., 4.16) is
greater than our critical value of t (i.e., 1.734). There seems to be a
significant difference in the values.
Note:
We have to be careful here. Remember, we are looking at a one-tailed hypothesis, and we
used our entire alpha value of t. Having changed those things,
our p value is also affected. We need to divide our p value by 2. Hence, it has dropped to .0005.
Answer:
Our p value (i.e., .0005) is less than our alpha value (i.e., .05), we do support the decision we made when we compared the
computed and critical values of t. Hence, we support our
research hypothesis and we reject our null hypothesis.
Please Remember My Dear
Students
We came to the decision to reject our hypothesis that is because we can see that our
computed value of t is greater than our critical value of t, and
our p value is less than our alpha value but we always
gave to be careful.
Please Remember My Dear
Students
Keep in mind, if you will use a statistical software, it doesn't know
your hypothesis; it simply works with the data you will supply. In
cases where the p value is less than alpha, you must make your mean
values match the direction you hypothesized before you decide to support your research hypothesis.
Deepening Our Knowledge
On the previous example, we hypothesized that males would have a significantly higher
number of nightmares than females. Our mean scores of 12 for males and 7 for females enable us to reject the null hypothesis. That
means we can support our research hypothesis. Suppose, however, the mean
values had been reversed. We have collected our data and found a mean score of 7 for
males and 12 for females. What do you think will happen? Will our computations change?
Are we going to support our research hypothesis?
Answer: If that's the case, the statistical software would still compute the
same p value. This means you would reject the null hypothesis, but you would not support the
research hypothesis because the mean difference is not in the order
you hypothesized. Be very, very careful!
The Case of the Cavernous Lab
A university lab manager has computers set up in two locations on
campus. One of these locations, the Old Library, is a dark, cavernous old
building; the other is the new brightly lighted, well decorated Arts and
Sciences building. The manager has noted that the equipment layout id
about the same in each building and both labs receive the same amount of
use.
The Case of the Cavernous Lab
Each term approximately 50 students use these labs for an "Introduction to Computer" course, with about half in each lab. The faculty member who teaches the
classes had noticed that the two groups of students are about the same in that they both have the same
demographic and achievement characteristics. Despite this, the teacher has noticed that the
students using the lab in the Old Library do not seem to perform as well as students working in the new
laboratory. The only thing the professor can figure is that "students always perform better in better surroundings." The teacher has asked the lab
manager to investigate and report the findings.
The Case of the Cavernous Lab
The manager decided the best thing to do is to collect final examination scores from all 50
students and compare the results of the students in the Library to those in the Arts and Sciences building. After collecting the
data, the manager realizes he will probably need to use some type of statistical tool to analyze the data, but isn't sure which one.
Because of that, the manager asks us to get involved; let's see what we can do to help
him.
The Problem:
The university lab manager has installed new computer labs, but their use seems to have affected student achievement. Not wanting his labs to play a part in students' grades, he
decides to investigate. He is certainly interested in a very manageable problem.
Since he installed the lab and is working with the teachers, he has the knowledge, time,
and resources necessary. He can easily collect numeric data to help ethically address a
problem with important practical significance.
Statement of the Problem
This study will
investigate whether
the location of a
computer lab affects
student achievement.
Step 2: State a Hypothesis
What kind of hypothesis should we
make base on the case and the
statement of the problem?
Research Hypothesis:
Students taking Introduction to
Computers in the new Arts and
Sciences laboratory will score
significantly higher on their final
examination than students
taking the same course in the
Old Library.
Null Hypothesis:
There will be no significant
difference in levels of achievement
between students taking
Introduction to Computers in the
new Arts and Sciences lab and
students taking the same course
in the Old Library.
Step 3: Identify the
Independent Variable
Based on our null hypothesis, what is our independent variable? What are the levels of
our independent variable?
The Independent Variable
The teacher is suggesting that the location of the laboratory (cause)
affects how students perform (effect) in the Introduction to
Computers class. This means our independent variable is location
and there are two levels: students in the Old Library and students in
the new Arts and Sciences buildings.
The Dependent Variable
The manager has decided to collect final examination scores from all of the students to help make a decision; this will be our dependent variable. Although it isn't explicitly stated, we can
assume the data will be quantitative, with scores ranging from 0 to 100.
Step 5: Choose the Right
Statistical Tool
Should we use independent-sample t-
test or dependent-sample t-test? Justify
your answer.
Statistical Tool:
If we consider all of the information, it appears that the independent-sample t-test is exactly what we need to use to
test our hypothesis.
Step 6: Computation and Data Analysis
to Test the Hypothesis
What is the computed t value? the critical t value?
What is our decision?
Testing the Hypothesis Using
the p Value
Up to this point, we've used only the "Equal Variances
Assumed" column to test our hypothesis. Now that we have a
good feeling for how the independent-sample t-test
works, it is important to understand the meaning of
these two values.
Testing Equal Variance in the
Groups
Although it rarely affects beginning statisticians, the amount of variance
within each group affects the manner in which the t
value is computed.
Levene's Test for Equality of
Variance
It is a test to determine whether researcher will
use the "Equal Variances Assumed" or "Equal
Variances Not Assumed" in hypothesis testing.
Hence:
In this case, the computed value of p is .913 and our alpha value is .05, we fail
to reject the null hypothesis.
Note:
Be careful here, we are still talking about the values in the rows for the Levene's Test, not the p values farther
down the table.
Interpretation:
This means there is no significant difference in
the variance between the two groups. Thus, we will use the column reading
"Equal Variances Assumed."
Equal Variances Assumed
The data enclosed here will be used in testing the
hypothesis for independent-sample t-test, if and only if,
there is no significant difference in the variance between the two groups.
Equal Variances Not Assumed
The data enclosed here will be used in testing the
hypothesis for independent-sample t-test, if and only if,
the computed p value for the Levene test was less than
.05. Hence, we would reject the null hypothesis.
Testing Hypothesis Using the p
Value
Remember, we have a one-tailed hypothesis, this means
we have to divide our Sig. (2-tailed) value of .029 by 2; this leaves us with one-tailed
p value of .0145. What will be our decision?
Computing for the Effect Size
Using the formula, we have .635 as the value of our Cohen's delta.
What assumption can we make?
Question:
What will happen if we have a degrees of freedom
of 48? How about 60? Anyway, when can we only have 48 and 60 as our degrees of freedom?
Answer:
If you look at the complete t distribution table, you'll see the values skip from 30 to 40, 40 to 60, 60 to 120, and then 120 to infinity. Once we get to
a certain number of degrees of freedom, the table can be abbreviated because the area under the curve for t changes so little between the differing degrees of freedom, there is no need to
have a value that is 100% correct.
Example:
Suppose the problem with 40 degrees of freedom and another problem has 60
degrees of freedom, what will be the critical values of t if we
will use alpha value of .05? What have you noticed to these
two critical values of t?
Note:
Since the difference between these two values is so small (i.e., .013) the table does not break the critical values down any further. Instead, when we're
manually computing statistics, we use the next highest df value in the table that is greater than
our actual df.
A Nonparametric Alternative
The use of the t-test mandates we have quantitative data that are normally (or
nearly normally) distributed (platykurtosis). Although the t-test is
robust enough to handle minor variances in the normality of the distribution, if the problem is bad enough we are forced to use the
nonparametric Mann-Whitney U test.
Mann-Whitney U test
The U test can be used for both quantitative data that is not normally distributed
and in cases where the researcher collects ordinal
(rank)-level data.