1 dr. david mckirnan, [email protected] psychology 242 introduction to research dr. mckirnan,...

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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 =

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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?


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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


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Psychology 242Introductionto Research t-test

Difference between groups standard error of M

t = =

grp2

grp2

grp1

grp1

n

Variance

n

Variance


grp2

grp2

grp2

grp1

grp1

grp1

ndf

SS

ndf

SS


t =

grp2

grp2

grp22

grp1

grp1

grp12

n1-n

M-X

n1-n

M-X


t =

➔ How strong is the experimental effect?

➔ How much error variance is there


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t = =

grp2

grp2

grp1

grp1

n

Variance

n

Variance


grp2

grp2

grp2

grp1

grp1

grp1

ndf

SS

ndf

SS


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


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t = =

grp2

grp2

grp1

grp1

n

Variance

n

Variance


grp2

grp2

grp2

grp1

grp1

grp1

ndf

SS

ndf

SS


t =

grp2

grp2

grp22

grp1

grp1

grp12

n1-n

M-X

n1-n

M-X


t =

The expanded version…


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Psychology 242Introductionto Research Compute a t score

grp2

grp2

grp22

grp1

grp1

grp12

n1-n

M-X

n1-n

M-X


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


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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:


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Psychology 242Introductionto Research Compute a t score

grp2

grp2

grp22

grp1

grp1

grp12

n1-n

M-X

n1-n

M-X


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


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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


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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


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Psychology 242Introductionto Research Clicker!

Why does this have a t value = 2?





M = 4M = 2.5


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Psychology 242Introductionto Research Clicker, 2

Why does this have a t value = .86?

M = 4M = 2.5






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Psychology 242Introductionto Research Clicker, 2

Why does this have a t value = .86?

M = 4M = 2.5






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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


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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


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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


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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 = =


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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.


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Psychology 242Introductionto Research Continue…

Continue this series by clicking on the module for The Central Limit Theorem.


1 dr. david mckirnan, [email protected] psychology 242 introduction to research dr. mckirnan,...

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