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Qdidactic » didactica & scoala » matematica » statistica Corelatia tetracorica, corelatia partiala, ecuatii de regresie |
Corelatia tetracorica, corelatia partiala, ecuatii de regresie
Corelatia tetracorica
intre 2 variabile artificial dihotomice (impartirea in 2 clase este artificiala – ex: inteligenta medie – inteligenta ridicata )
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; 
Corelatia partiala
cand vrem sa stim corelatia intre 2 variabile cu excluderea influentei celei de a treia:
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r12=0,63; r13=0,56; r23=0,70;
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;
Ecuatii de regresie
- predictia liniara simpla: Y= a + bXY*X, unde
Y este scorul prezis (criteriul)
X este scorul la testul predictor,
a este o constanta calculata dupa formula:
,
bXY este coeficientul de regresie: 
(rXY –
coeficientul de corelatie intre predictor si criteriu)
Aceasta ecuatie este folosita pentru note brute. Pentru note standard ecuatia devine: zY=rXY*zX
Exemplu:
test de
inteligenta (predictor): 
; sX=7,8
performanta la
matematica: 
; sY=1,3
corelatia: rXY=0,71;
Nota unui subiect la testul de inteligenta: X=43
Note brute
b= 0,71*1,3/7,8=0,12;
a=7,25 – 48,2*0,12=1,55;
Y=1,55 + 0,12*43=6,63
Deci, putem prognoza nota la matematica 6,63 pentru un subiect care are nota bruta de 43 la testul de inteligenta
Eroarea standard de
estimare: 
=0,92
Putem spune ca 68% din subiectii care au nota bruta la predictor 43 vor avea nota la matematica in intervalul: 6,63±0,92 adica (5,71; 7,55)
95% din subiecti vor avea nota in intervalul: 6,63±1,96*0,92; 99% din subiecti – in intervalul 6,63±2,58*0,92
Note standard
zX=(43 – 48,2)/7,8= - 0,67;
zY=0,71*(-0,67)=-0,47 (Y=(-0,47)*1,3+7,25=6,63)
- regresia multipla – se realizeaza predictia unei variabile dependente (criteriu) in dunctie de mai multe variabile independente (predictori):
Se urmareste predictia notei pe semestrul I a unor studenti la Informatica pe baza unei Baterii de teste, care cuprinde urmatoarele probe:
Comprehensiune Verbala (CV);
Rationament (RA);
Operatori Logici (OL);
Aptitudine Numerica (AN);
Diagrame (DG)
Matricea de corelatii intre probe (predictori) + corelatiile probelor cu media pe sem I (criteriul):
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|
CV |
RA |
OL |
AN |
DG |
|
CV |
1 |
0.775 |
0.67 |
0.625 |
0.733 |
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|
0 |
0 |
0 |
0 |
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47 |
47 |
47 |
47 |
47 |
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RA |
0.775 |
1 |
0.6 |
0.571 |
0.593 |
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0 |
|
0 |
0 |
0 |
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47 |
47 |
47 |
47 |
47 |
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OL |
0.67 |
0.6 |
1 |
0.489 |
0.608 |
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|
0 |
0 |
|
0 |
0 |
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|
47 |
47 |
47 |
47 |
47 |
|
AN |
0.625 |
0.571 |
0.489 |
1 |
0.295 |
|
|
0 |
0 |
0 |
|
0.044 |
|
|
47 |
47 |
47 |
47 |
47 |
|
DG |
0.733 |
0.593 |
0.608 |
0.295 |
1 |
|
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0 |
0 |
0 |
0.044 |
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|
47 |
47 |
47 |
47 |
47 |
|
medie semestrul 1 |
0.667 |
0.626 |
0.369 |
0.447 |
0.655 |
Algoritmul lui Aitken pentru aflarea coeficientilor beta redusi:
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1 (CV) |
2 (RA) |
3 (OL) |
4 (AN) |
5 (DG) |
6 |
7 |
8 |
9 |
10 |
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1 |
1.00 |
0.82 |
0.86 |
0.75 |
0.83 |
1.00 |
0.00 |
0.00 |
0.00 |
0.00 |
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2 |
0.82 |
1.00 |
0.63 |
0.60 |
0.69 |
0.00 |
1.00 |
0.00 |
0.00 |
0.00 |
|
|
3 |
0.86 |
0.63 |
1.00 |
0.84 |
0.78 |
0.00 |
0.00 |
1.00 |
0.00 |
0.00 |
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|
4 |
0.75 |
0.60 |
0.84 |
1.00 |
0.64 |
0.00 |
0.00 |
0.00 |
1.00 |
0.00 |
|
|
5 |
0.83 |
0.69 |
0.78 |
0.64 |
1.00 |
0.00 |
0.00 |
0.00 |
0.00 |
1.00 |
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6 |
0.67 |
0.63 |
0.37 |
0.45 |
0.66 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
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7 |
(3.03) |
0.33 |
-0.07 |
-0.02 |
0.01 |
-0.82 |
1.00 |
0.00 |
0.00 |
0.00 |
1-2 |
|
8 |
|
1.00 |
-0.22 |
-0.05 |
0.03 |
-2.48 |
3.03 |
0.00 |
0.00 |
0.00 |
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9 |
|
-0.07 |
0.27 |
0.20 |
0.07 |
-0.86 |
0.00 |
1.00 |
0.00 |
0.00 |
1-3 |
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10 |
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-0.02 |
0.20 |
0.44 |
0.02 |
-0.75 |
0.00 |
0.00 |
1.00 |
0.00 |
1-4 |
|
11 |
|
0.01 |
0.07 |
0.02 |
0.31 |
-0.83 |
0.00 |
0.00 |
0.00 |
1.00 |
1-5 |
|
12 |
|
0.08 |
-0.20 |
-0.05 |
0.10 |
-0.67 |
0.00 |
0.00 |
0.00 |
0.00 |
1-6 |
|
13 |
|
(4.01) |
0.25 |
0.20 |
0.07 |
-1.04 |
0.22 |
1.00 |
0.00 |
0.00 |
8-9 |
|
14 |
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|
1.00 |
0.79 |
0.29 |
-4.17 |
0.90 |
4.01 |
0.00 |
0.00 |
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15 |
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|
0.20 |
0.44 |
0.02 |
-0.79 |
0.05 |
0.02 |
1.00 |
0.00 |
8-10 |
|
16 |
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|
0.07 |
-1.04 |
-0.79 |
-0.81 |
-0.03 |
0.00 |
0.00 |
1.00 |
8-11 |
|
17 |
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-0.18 |
-0.05 |
0.10 |
-0.47 |
-0.24 |
0.00 |
0.00 |
0.00 |
8-12 |
|
18 |
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(3.52) |
0.28 |
-0.04 |
0.03 |
-0.13 |
-0.77 |
1.00 |
0.00 |
14-15 |
|
19 |
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|
1.00 |
-0.13 |
0.10 |
-0.44 |
-2.71 |
3.52 |
0.00 |
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20 |
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|
-0.04 |
-0.81 |
-0.51 |
-0.09 |
-0.29 |
0.00 |
1.00 |
14-16 |
|
21 |
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|
0.10 |
0.15 |
-1.24 |
-0.08 |
0.74 |
0.00 |
0.00 |
14-17 |
|
22 |
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|
(-0,16) |
-0.82 |
-0.51 |
-0.11 |
-0.38 |
0.13 |
1.00 |
19-20 |
|
23 |
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1.00 |
0.62 |
0.13 |
0.47 |
-0.16 |
-1.22 |
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24 |
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0.16 |
-1.25 |
-0.03 |
1.00 |
-0.34 |
0.00 |
19-21 |
|
25 |
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-1.35 |
-0.06 |
0.92 |
-0.31 |
0.20 |
23-24 |
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β1 |
β2 |
β3 |
β4 |
β5 |
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R = 0,984
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