1
There are different participants in each group and there was no blocking. Therefore
treatment is a between-subjects factor.

This is the SAS program for the descriptive statistics and the t test.
DATA;
INPUT GROUP 1 TEST 2-3;
CARDS;
2 62 72 8other data go here
115
115
125
proc sort; by group;proc means maxdec=3; by group;
PROC TTEST;CLASS GROUP;
run;
quit;

Coding is
1=restricted
2=unrestricted2--------------------------------- GROUP=1 ---------------------------------The MEANS Procedure
Analysis Variable : TESTN
Mean
Std Dev
Minimum
Maximumƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
21
10.0954.560
4.000
25.000ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ

--------------------------------- GROUP=2 ---------------------------------Analysis Variable : TESTN
Mean
Std Dev
Minimum
Maximumƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
21
16.3338.333
6.000
35.000ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ

3

The TTEST Procedure
Statistics

Variable
TEST
TEST
TEST

GROUP

N
12

2121

Diff (1-2)

Lower CL
Mean

Mean

Upper CL
Mean

Lower CL
Std Dev

Std Dev

8.0197
12.54-10.4310.09516.333-6.23812.171
20.126-2.0493.48846.375
5.51444.55978.33276.7165

Statistics

VariableTESTTESTTESTGROUP
12Diff (1-2)

Upper CL
Std Dev

Std Err

Minimum

Maximum

6.584512.0338.59380.995
1.81832.07284625
35

means, mean difference, confidence limits for mean difference, pooled standard deviation   S P  , standard error of mean
1 1 
difference  S P
 n1 n2 4

T-Tests
Variable

MethodVariances

DFt Value

Pr > |t|

TEST
TEST

PooledSatterthwaite

Equal
Unequal4031-3.01-3.01

0.00450.0052Equality of Variances
VariableMethod

TEST

Folded F

Num DF

Den DFF Value

Pr > F2020

3.34

0.0096

5This is the SAS program for the confidence interval for the effect size.

*This program computes a CI for the effect size in a
between-subject design with two groups.
m1 and m2 are the means for the two groups
s1 and s2 are the standard deviations for the two groups
n1 and n2 are the sample sizes for the two groups
prob is the confidence level;data;
m1=10.095;
These are the commands that must be changed
m2=16.333;in the program downloaded from
s1=4.560;
http://plaza.ufl.edu/algina/
s2=8.333 ;n1=21 ;
n2=21 ;
prob=.95;
v1=s1**2;
v2=s2**2;
pvar=((n1-1)*v1+(n2-1)*v2)/(n1+n2-2);
se=sqrt(pvar*(1/n1+1/n2));nchat=(m1-m2)/se;
es=(m1-m2)/(sqrt(pvar));
df=n1+n2-2;
ncu=TNONCT(nchat,df,(1-prob)/2);
ncl=TNONCT(nchat,df,1-(1-prob)/2);
ll=(sqrt(1/n1+1/n2))*ncl;
ul=(sqrt(1/n1+1/n2))*ncu;
output;proc print;
title1 'll is the lower limit and ul is the upper limit';
title2 'of a confidence interval for the effect size';var es ll ul;
run;
quit;

6ll is the lower limit and ul is the upper limitof a confidence interval for the effect size

Obsesllul

1-0.92871-1.56145-0.28551

71.

1=restricted
2=unrestricted
H 0 : 1  2  0
H1 : 1  2  0

1=unrestricted
2=restricted
H 0 : 1  2  0
H1 : 1  2  0

2.

t , N  2  t.05,42 2  1.68t , N  2  t.05,42 2  1.68

t distribution : df = 40
0.4

t distribution : df = 40
0.4

0.3

0.3

0.2

0.2

0.1

0.1

- 1.681.68

81=restricted
2=unrestricted
H 0 : 1  2  0
H1 : 1  2  0
3

1=unrestricted
2=restricted
H 0 : 1  2  0
H1 : 1  2  0t

10.095  16.333
1 16.717

21 21
 3.01t

Y1  Y2
1 1Sp
n1 n2
16.333  10.095
t
1 16.717
21 21
 3.01

9H 0 : 1  2  0
H1 : 1  2  0
4.

1=restricted
2=unrestricted

prob  t  .0045 . The hypothesized sign of 1  2 and the sign of Y1  Y2 match. So the directional

p value is p 1
.0045  .0023
2t distribution : df = 40
0.4.0023- 3.01
5.
6.

H 0 : 1  2  0
H1 : 1  2  0

1=unrestricted
2=restricted

t distribution : df = 40
0.4

0.3

0.3

0.2

0.2

0.1

0.1.0023
3.01

Reject H 0
We can conclude that learning is more efficient (requires fewer cues) when mistakes are permitted

107.

Y  Y    t1

10.095  16.333

2



/2, N  2

 S  n1  n1 p116.333  10.095
2

 1 1   t.05/2, 42 2  6.717   21 21 

 1 1   t.05/2, 42 2  6.717   21 21 

10.095  16.333

16.333  10.095

 1 1   2.021 6.717  2121 

 1 1   2.021 6.717  2121 

10.43,  2.049
2.049,10.43,
The CI implies that the in the population being allowed to make mistake results in, on average, requiring between 2 and 10.4 cues
fewer to solve a problem. Without more information about the nature of the cues this interval is difficult to interpret.

118.

Y Y
d 1 2SP

10.095  16.333
16.333  10.095
d
6.717
6.717
 0.93 0.93
1.56, .29.29,1.56,
The CI implies that the population value Cohen’s effect size could be between .29 and 1.56. Because .29 is a smallish effect size and1.6 is a very large effect size, the CI implies we have not estimated the effect size accurately.
d