I.     Student's t Test for Correlated Groups

A. Correlated Groups

1.               

2.               

B. Difference

1.               

2.               

C.Ho: No difference between groups =

1.              Note,

D.Formula: tobt =

E. Caclulate Difference Scores, Squared Difference Scores, and Mean of Difference Scores

F. Calculate tobt

G.           Evaluate the Statistic, df = N-1 use Appendix D

H.Compare to Sign Test

1.               

2.               

I.      Assumptions

1.              Sampling distribution of D - normal

a)              N >= 30

b)             Or, Population Normal

II.  Z and t tests for Independent Groups

A. Independent Groups -

1.               

2.               

3.               

4.               

5.               

B. Hypotheses

1.               

2.               

3.              Ho:

4.              H₁:

C.Z test for independent groups

1.              Sampling Distribution of the Difference Between Sample Means

a)              All possible samples of a given size drawn for population 1 and population 2

b)             Difference scores for each possible pair of means

c)              Calculate probability of each mean difference

d)             Has the following characteristics

(1)          If populations are normal, then distribution of difference between sample means is also normal

(2)          The mean of the distribution of the difference between sample means = mean of population one minus population two

(3)          The standard deviation of the sampling distribution of the difference between sample means, or the standard error of the difference between sample means = the square root of the variance of sampling distribution one - the variance of the sampling distribution two

(a)          Therefore, because we assume that the variances of the two populations are equal, we can simplify the equation for standard deviation of mean difference to:

e)              Calculate zobt for mean differences:

f)                Remember can rarely know sd for population, but can estimate it from the sample – leads to t Test for Independent Groups

III.          Student's t Test for Independent Groups

A. Same equation except use weighted estimate of sd

B. Sw2 =

C.Tobt =

D.Tobt reduces to =

because assume mean1 = mean 2 (Ho)

E. Need to know the t distribution for independent samples

F. Example

G.           When n1 = n2, equation can be simplified to:

H.Example

I.      Assumptions

1.              Sampling Distribution of Mean Differences is normally distributed.

2.              Homogeneity of Variance - iv affects means, not sds

3.              Violations: Robust if

a)              n1 = n2 and

b)             n1 >= 30

J.  Effect Size

1.              Recall r2

2.              Use omega squared =

3.              Where

K.Power

1.              Larger tobt

2.              Greater effect of iv on dv

3.              Increasing sample size

4.              High sample variability

IV.         Comparing Correlated Groups and Independent Groups Designs

A.  

B.  

C.