Adolescent Development Psychology 242 Professor Jean Rhodes.
1 Dr. David McKirnan, [email protected] Psychology 242 Introduction to Research Dr. McKirnan,...
-
Upload
damon-bryer -
Category
Documents
-
view
216 -
download
2
Transcript of 1 Dr. David McKirnan, [email protected] Psychology 242 Introduction to Research Dr. McKirnan,...
1
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research Dr. McKirnan, Psychology 242
Introduction to statistics:Calculate t
Revised 4/10/10
grp2
grp2
grp22
grp1
grp1
grp12
n1-n
M-X
n1-n
M-X
(Mgroup1 - Mgroup2) - 0
t =
Click anywhere to proceed.If this does not open as a running show, please go to “Slide Show” and click “run show”
2
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research
Statistical Hypothesis Testing
We use the t test (or any statistic) to test our hypothesis. Part of the operational definition of our variables is the numbers
we use to represent them.
What is our (statistical) hypothesis?
a. That the mean score (M) for the experimental group is greater than (or less than…) the M for the control group…
b. …by more than we might expect by chance alone.
What is the “null” hypothesis? Any difference between the M for the experimental group and the M for
the control group is by chance alone.
Mexperimental – Mcontrol = 0, except for chance (error variance)
The research question (in statistical terms): In our study, is the difference between the group Means (Mexp – Mcontrol)
greater than (or less than…) 0 by more than we would expect by chance alone?
3
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research
Statistical Hypothesis Testing
For a t-test:The experimental effect is the difference between the Ms of the experimental & control groupsThe error variance is the square root of the summed variances of the groups, similar to a two-group standard deviation.
= = t(Mexp - Mcontrol) - 0
The concept underlying the t test is the critical ratio:
How strongly did the independent variable affect the outcome?
How much error variance [“uncertainty”, “noise”] is there in the data
4
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research t-test
Difference between groups standard error of M
t = =
grp2
grp2
grp1
grp1
n
Variance
n
Variance
(Mgroup1 - Mgroup2) - 0
grp2
grp2
grp2
grp1
grp1
grp1
ndf
SS
ndf
SS
(Mgroup1 - Mgroup2) - 0
t =
grp2
grp2
grp22
grp1
grp1
grp12
n1-n
M-X
n1-n
M-X
(Mgroup1 - Mgroup2) - 0
t =
➔ How strong is the experimental effect?
➔ How much error variance is there
5
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research t-test
Difference between groups standard error of M
t = =
grp2
grp2
grp1
grp1
n
Variance
n
Variance
(Mgroup1 - Mgroup2) - 0
grp2
grp2
grp2
grp1
grp1
grp1
ndf
SS
ndf
SS
(Mgroup1 - Mgroup2) - 0
t =
Standard error:
➔ Calculate the variance for for group 1
➔ Sum of squares
➔ Divided by degrees of freedom (n-1)
➔Divide by n for group 1
➔ Repeat for group 2
➔ Add them together
➔ Take the square root
6
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research t-test
Difference between groups standard error of M
t = =
grp2
grp2
grp1
grp1
n
Variance
n
Variance
(Mgroup1 - Mgroup2) - 0
grp2
grp2
grp2
grp1
grp1
grp1
ndf
SS
ndf
SS
(Mgroup1 - Mgroup2) - 0
t =
grp2
grp2
grp22
grp1
grp1
grp12
n1-n
M-X
n1-n
M-X
(Mgroup1 - Mgroup2) - 0
t =
The expanded version…
7
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research Compute a t score
grp2
grp2
grp22
grp1
grp1
grp12
n1-n
M-X
n1-n
M-X
(Mgroup1 - Mgroup2) - 0
t =
Compute the Experimental Effect:
Calculate the Mean for each group, subtract Mgroup2 from Mgroup1.
Compute the Standard Error Calculate the variance for each group
8
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research Calculate the Variance using the box method:
2. Calculate the Mean.
3. Calculate Deviation scores:
Simple deviations: Σ (X – M) = 0
Square the deviations to create + values:
Σ Squares = Σ(X - M)2 = 52
4. Degrees of freedom:
df = [n – 1] = [10 – 1] = 9
X
7
6
2
1
4
1
7
4
2
6
M4
4
4
4
4
4
4
4
4
4
X - M
3
2
-2
-3
0
-3
3
0
-2
2
Σ = 0
(X - M)2
9
4
4
9
0
9
9
0
4
4
Σ = 52n = 10Σ= 40
M = 40/10 = 4
1. Enter the Scores.
5. Apply the Variance formula:
9
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research Compute a t score
grp2
grp2
grp22
grp1
grp1
grp12
n1-n
M-X
n1-n
M-X
(Mgroup1 - Mgroup2) - 0
t =
Compute the Experimental Effect:
Calculate the Mean for each group, subtract group2 M from group1 M.
Compute the Standard Error Calculate the variance for each group
Divide each variance by n for the group
Add those computations
Take the square root of that total
Compute t Divide the Experimental Effect
effect
error
by the Standard Error
10
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research
Examples of deriving t values
M1 – M2 = 4 – 2.5 = 1.5
Standard error = .75
1.5
.75= = 2t =
M1 – M2 = 4 – 2.5 = 1.5
Standard error = 1.75
1.5
1.75= =.86t =
M = 4M = 2.5
M = 4M = 2.5
11
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research Clicker!
Why does this have a t value = 2?
a. The variance within each group is large relative to the difference between the group means.
b. The M of the larger group = 4 and there are 2 groups
c. The difference between the group means is large relative to the variance within each group
d. t is a random number
M = 4M = 2.5
12
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research Clicker!
Why does this have a t value = 2?
a. The variance within each group is large relative to the difference between the group means.
b. The M of the larger group = 4 and there are 2 groups
c. The difference between the group means is large relative to the variance within each group
d. t is a random number
M = 4M = 2.5
13
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research Clicker, 2
Why does this have a t value = .86?
M = 4M = 2.5
a. The variance within each group is large relative to the difference between the group means.
b. The M of the larger group = 4 and there are 2 groups
c. The difference between the group means is large relative to the variance within each group
d. t is a random number
14
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research Clicker, 2
Why does this have a t value = .86?
M = 4M = 2.5
a. The variance within each group is large relative to the difference between the group means.
b. The M of the larger group = 4 and there are 2 groups
c. The difference between the group means is large relative to the variance within each group
d. t is a random number
15
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research
Sampling distribution of t scores
Sampling distribution & statistical significance
Any 2 group Ms differ at least slightly by chance.
Any t score is therefore > 0 or < 0 by chance alone.
We assume that a t score with less than 5% probability of occurring [p < .05] is not by chance alone
We calculate the probability of a t score by comparing it to a sampling distribution
16
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research The Sampling Distribution
-3 -2 -1 0 +1 +2 +3
Z or t Scores
(standard deviation units)
34.13% of scores from Z = 0 to Z = +1
andfrom Z = 0 to Z = -1
13.59% of scores+
13.59% of scores
2.25% of scores+
2.25% of scores
We can segment the population into standard deviation units from the mean.
These are denoted as Z or tM = 0,
Each segment takes up a fixed % of cases (or “area under the curve”).
each standard deviation represents Z = 1
17
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research
Sampling distribution of t scores
t scores and statistical significance, 1
M1 – M2 = 4 – 2.5
Standard error
1.5
.75= = 2t =
t = 2.0
Comparing t to a sampling distribution:
About 98% of t values are lower than 2.0
About 98% of t scores
18
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research
Sampling distribution of t scores
t scores and statistical significance, 1
t = .88About 81% of the distribution of t
scores are below .88.
(area under the curve = .81)
About 81% of scores
M1 – M2 = 4 – 2.5
Standard error
1.5
1.75= .86t = =
19
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research
Sampling distribution of t scores
t = .86 t = 2.0
Between v. within group variance: t-test logic
About 98% of t scores; p < .05
About 81% of scores
The difference between Ms is the same in the two
data sets.
Since the variances differ…
We get different t values
We make differ judgments about whether these t scores occurred by chance.
20
Dr. David McKirnan, [email protected]
Psychology 242Introductionto Research Continue…
Continue this series by clicking on the module for The Central Limit Theorem.