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Satterthwaite’s Approximation for Degrees of Freedom
Art Instruction Effect on Reading Development
J.C. Mills (1973), “The Effect of Art Instruction Upon a Reading Development Test: An Experimental Study with Rural Appalachian Children,” Studies in Art Education, Vol. 14, #3, pp.4-8
Setting • Comparing Means from 2 Normal Distributions• Small Samples (Computer Packages Solve for any sizes)• Distributions have Possibly Different Variances
Case 1 – Variances are Equal
111
~11
11
2
2
~2
1,0~11
~1
,~
2
112,1
1
,~,...,,~,...,
2,2/
21
2
2121
2,2/
2
21
2
2121
2
2
21
2
2121
2
2
21
2
221
222
221
21
2
2121
212
22
21
222
21121
2
21
2122
221
222221
211111
2121
21
21
21
nn
p
nn
nn
pp
W
p
p
W
nnp
nii
iii
pi
n
j
iij
ii
n
jij
i
nn
t
nnS
YYtP
t
nnS
YY
S
nn
YY
df
W
ZS
nn
Snn
df
W
WZSnn
WN
nn
YYZ
Sn
nNY
nn
SnSnSi
n
YY
Sn
Y
Y
YYNYYNYY
i
ii
Case 2 – Variances are Unequal - I
*
2
1
21
1
211*1
*
2,12,~ :Aside
freedom. of degrees itsby divided square-chi a NOT is which *
:consider, variancesestimatedr with Denominato Replacing :Problem
1,0~
1
2,12
1,1
1~
1
,~,...,,~,...,
2
42
221
41
21
2
2
22
1
21*
2
22
1
21
2
22
1
21
2
22
1
21
2
22
1
21
*
2
2
22
1
21
2
22
1
21
*
2
22
1
21
2121
4222
2
2
2
22
12
2
2122
21
222221
211111 21
nnnn
nn
df
WV
nn
nn
nn
nS
nS
Edf
WE
WV
WEWVWEW
nn
nS
nS
df
W
N
nn
YYZ
nSVSEn
SnVn
SnE
Sn
YYNYYNYY
WW
W
i
iiiii
i
iii
i
iin
i
ii
nn
i
Case 2 – Variances are Unequal - II
4 41 2
2 2 22 2* 1 1 2 21 2
1 2
24 4 2 21 2 1 2
2 21 1 2 2 1 2
2 2 22 2 2 21 2 1 1 2 2
1 2 1 2
2 2* 1 1 1 2
1 1 *
21 12
**
1 1
Replacing the unknown variance
W
WVdf n n n n
n n
n n n n n n
n nn n n n
2 22 2 21 2
^11 2 2
22 2 22 21 1 2 2
11 2
s with their estimates:
1* where: 1
1 1
So, we have the approximate degrees of freedom if our denominator
i ii
i i i i iii i
i i
S Sg MS
n ng MS S n
ng MSS n S n
n n
were the square root
of the ratio of a chi-square to its degrees of freedom
Example – Art Instruction Effect on Reading
• Experiment to Determine Effect of Art Instruction on a Reading Development Test 52 Children Given Baseline Reading Test 26 Received Art Instruction (Trt), 26 Did not (Control) Y=Post-Test – Pre-Test Score
020.2:23.41
33.0
77.13
252600.26
252649.70
2626
2649.70
*
85.4
2600.26
2649.70
)58.1(77.7:..::
23.21,max:71.2:..::
2600.2658.1
2649.7077.7
23.41,025.22
2
^
220
25,25,025.2
22222
0
2
2
ttRR
n
S
nS
YYtSTHH
FFFRRS
SFSTHH
nSY
nSY
obs
C
C
T
T
CT
obsCTACT
obsobsC
TobsCTACT
CCC
TTT