Transcript PPT Lecture Notes (Independent Samples T Test)

Independent
Samples
T-Test
Please refrain from typing, surfing or printing during our conversation! 
Outline of Today’s Discussion
1.
About T-Tests
2.
The One-Sample T-Test
3.
Independent Samples T-Tests
4.
Two Tails or One?
5.
Independent Samples T-Test: Excel
6.
Independent Samples T-Test: SPSS
Part 1
About T-Tests
Computing the ‘t’ Statistic
Recall One of Our Themes:
Correlational Research vs Differential Research
Computing the ‘t’ Statistic
When we evaluate the
difference between ANY two means
We can’t just look at the mean-difference alone.
We need to consider the mean-difference
in Context!!!
For t-tests…the context is the variability.
Computing the ‘t’ Statistic
The Effe ctive ne s s of Drug x
12
12
10
10
Mean Effectiveness
Mean Effectiveness
The Effe ctive ne s s of Drug x
8
6
4
8
6
4
2
2
0
0
Drug x
Placebo
Treatm e nt
Drug x
Placebo
Treatm e nt
Which graph makes a more convincing
case for Drug X, and why?
For t-tests…the context is the variability.
Computing the ‘t’ Statistic
Mantra:
T-Tests compare means.
Computing the ‘t’ Statistic
Mantra:
T-Tests compare means
(in the context of variability).
Part 2
The One Sample T-Test
The One Sample T-Test
1.
In ancient times –before the number 2 was inventedcaveman used a one sample t-test! ;)
2.
A one sample t-test is an evaluation of a single mean,
rather than two means.
3.
The sample mean is compared to a ‘test value’ that is of
interest to the research.
4.
Example 1: Consider the proportion of m&m’s colors in
the population. There would be a “test value” for each
color…
Test
Values
In the
Population:
24% blue
14% brown
16% green
20% orange
13% red
14% yellow
The One Sample T-Test
1.
Example 2: Assume that you are a therapist who has
received a ‘sample’ of 20 patients diagnosed with clinical
depression.
2.
After 5 weeks of your treatment, you might ask whether
the mean of your sample is statistically indistinguishable
from non-depressed patients, who have a mean of, say,
500 on a standard mood assessment.
The One Sample T-Test
1.
The null hypothesis would be as follows:
Ho: In the population, the mean mood score
of patients who have completed 5 weeks of
(my) therapy is equal to that of nondepressed people.
2.
In this example, the test value would be equal to
whatever the mean of the non-depressed population
is…let’s say ‘500’. So, the test value = 500 here.
The One Sample T-Test
1.
Example 3:
Ho: In the population, the mean age when
first married is equal to 20.
2.
So, the test value = 20 here.
3.
The steps in SPSS are simple, and are virtually identical
to the two-mean case…
The One Sample T-Test
SPSS Steps:
Analyze  Compare Means  One Sample T-Test 
Test Value Box  Enter the value that is to be compared to the sample mean
The sample mean of 22.79,
is compared to our test value,
Say, …20.
The ‘sig’ value indicates that we should reject Ho,
i.e., we should REJECT that the sample mean is equal to the test value.
The One Sample T-Test
SPSS Steps:
Analyze  Compare Means  One Sample T-Test 
Test Value Box  Enter the value that is to be compared to the sample mean
The sample mean of 22.79,
is compared to our test value,
Say, …22.7.
The ‘sig’ value indicates that we should retain Ho,
i.e., we should accept that the sample mean is equal to the test value.
The One Sample T-Test
SPSS Steps:
Analyze  Compare Means  One Sample T-Test 
Test Value Box  Enter the value that is to be compared to the sample mean
The sample mean of 22.79,
is compared to our test value,
Say, …22.7.
The mean difference of 0.092 is NOT significantly different from zero (Ho).
Note: When Ho is true, than the 95% confidence interval contains a zero.
The One Sample T-Test
1.
Typically, we prefer to run two samples…one
experimental group and one control group.
2.
However, that may not always be possible.
3.
The one sample t-test allows for a statistical
comparison to some abstract standard (the ‘test
value’), rather than to a control group.
Part 3
Independent Samples T-Tests
Computing the ‘t’ Statistic
Formula for the
independent samples “t” statistic
What does “x bar” represent, again?
Computing the ‘t’ Statistic
Denominator term in the
independent samples “t” statistic
What does this “s” represent, again?
Computing the ‘t’ Statistic
Denominator term in the
independent samples “t” statistic
SS1  SS 2  1 1 

*   
n1  n2  2  n1 n 2 
Simplified!
Computing the ‘t’ Statistic
Degrees of Freedom for the
independent samples “t” statistic
df = N - 2
Where N equals the sum of
the two sample sizes (n1 + n2).
Computing the ‘t’ Statistic
The Effe ctive ne s s of Drug x
12
12
10
10
Mean Effectiveness
Mean Effectiveness
The Effe ctive ne s s of Drug x
8
6
4
8
6
4
2
2
0
0
Drug x
Placebo
Treatm e nt
Drug x
Placebo
Treatm e nt
Which Graph would be associated with
the larger ‘t’ statistic, and why?
Computing the ‘t’ Statistic
1.
The independent samples “t” statistic is based on 3
assumptions.
2.
The first assumption is the distribution of scores should be
bell-shaped.
3.
The second assumption is that the two populations from
which the samples are selected must have (at least
approximately) equal variances.
4.
The third assumption is independence (the value of any
datum does not depend on the any other datum).
Computing the ‘t’ Statistic
1.
How might we quantitatively assess the first assumption,
i.e., the normalcy assumption? Hint: We’ve done it before
(kind of).
2.
There are ways to test the equal variance assumption
quantitatively. SPSS will help us with that later.
3.
The independence assumption requires that we investigate
how the data were obtained. (No stats here.)
The independence assumption does NOT pertain to the within-subject t-test.
Part 4
Two Tails or One?
Two-Tailed Case
Two Tails or One?
1.
Here’s are some hypotheses for t-tests
H0: In the population, the means for the control and
experimental groups will be equal.
H1: In the population, the means for the control and experimental
groups will NOT be equal.
2.
The alternate hypothesis is said to be “two tailed” because
it is non-directional…it does NOT state which of the two
means will be larger.
Two-Tailed Case
Two Tails or One?
Non-directional hypotheses are evaluated with a two-tailed test!
.025 is lower
than this
.025 is higher
than this
We could reject the null hypothesis whether the observed t-statistic
is in either critical region…but the observed “t” value must
be at least 47.5 percentiles away from the mean!
mean = 50th percentile:
50 - 47.5 = 2.5 percentile (left) : 50 + 47.5 = 97.5 percentile (right)
One-Tailed Case
Two Tails or One?
1. Here’s are some hypotheses for t-tests
H0: In the population, the means for the control and
experimental groups will be equal.
H1: In the population, the mean for the control group will be greater
than the mean for the experimental group.
2.
The alternate hypothesis is said to be “one tailed” because it
is directional…it states which mean will be larger.
One-Tailed Case
Two Tails or One?
Directional hypotheses are evaluated with a one-tailed test!
We can only reject the null hypothesis when the observed t-statistic
is in the predicted critical region… but the observed “t” value
only needs to be at least 45 percentiles away from the mean!
mean = 50th percentile:
50 + 45= 95 percentile (right)
Two Tails or One?
1.
When do we use two tails versus one tail?
2.
It depends!
3.
If you want to make a directional prediction, use one-tail,
otherwise use two-tails.
4.
Two-tails are often preferred because they are more
conservative (47.5 percentiles from mean for two-tailed
significance, versus a mere 45 percentiles from mean for
one tailed significance).
Part 5
Independent Sample
T-Tests in Excel
Part 6
Independent Sample
T-Tests in SPSS
Independent Samples T-Tests in SPSS
1.
SPSS Question – When conducting t-tests in SPSS, what
is a synonym for “Test Variable”? What is a synonym for
“Grouping Variable”?
2.
SPSS Question– Consider the four types of measurement
scales that we discussed earlier this semester. For which
scale or scales would a t-test be appropriate? Explain
your reasoning.
Independent Samples T-Tests in SPSS
1. SPSS Question – In your own words, explain why
it is necessary to consider Levene’s test?
2. SPSS Question – In your own words, explain how
to interpret Levene’s test in the SPSS output.