Su'aalo iyo Doodo Tusaale ah oo Dib-u-noqosho Toosan ah
Dib-u-noqoshada toosan waa hab tirakoob oo loo isticmaalo in lagu go'aamiyo xiriirka ka dhexeeya laba ama in ka badan oo doorsoomayaal ah. Habkan waxaa si weyn loogu isticmaalaa dhinacyo kala duwan, oo ay ku jiraan dhaqaalaha, ganacsiga, cilmiga bulshada, iyo cilmiga dabiiciga ah. Maqaalkan, waxaan kaga hadli doonnaa dib-u-noqoshada toosan, sida loo xisaabiyo, waxaanan bixin doonnaa dhowr tusaale oo dhibaatooyin ah oo leh sharraxaadyo si ay akhristayaasha uga caawiyaan inay si qoto dheer u fahmaan fikraddan.
Fahmidda Dib-u-dhaca Toosan
Dib-u-noqoshada toosan waa hab falanqayn ah oo loo isticmaalo in lagu qaabeeyo xiriirka ka dhexeeya hal ama in ka badan oo doorsoomayaal madax-bannaan (saadaaliyayaal) iyo doorsoome ku-tiirsan (jawaab). Dib-u-noqoshada toosan ee fudud waxay ku lug leedahay hal doorsoome madax-bannaan iyo hal doorsoome ku-tiirsan, halka dib-u-noqoshada toosan ee badan ay ku lug leedahay in ka badan hal doorsoome madax-bannaan.
Isle'egta xariiqda dib-u-celinta toosan ee fudud waa:
\[ Y = a + bX \]
Halkee:
– \( Y \) waa doorsoomaha ku tiirsan.
– \( X \) waa doorsoomaha madaxbannaan.
– \( a \) waa dhexda, kaas oo ah qiimaha Y marka X = 0.
– \( b \) waa isku-xidhka dib-u-celinta, taas oo ah, inta Y isbeddelo haddii X isbeddelo hal cutub.
Tallaabooyinka Dib-u-noqoshada Toosan
1. Ururi Xogta: Marka hore, ururi xogta la falanqeynayo.
2. Xogta Sheekadda: Samee sawir kala firdhisan si aad u aragto haddii uu jiro xiriir toosan oo u dhexeeya doorsoomayaasha.
3. Xisaabi Isku-xidhka Dib-u-Celinta: Adeegso habka ugu yar ee labajibbaaran si aad u go'aamiso khadka ugu fiican.
4. Tijaabinta Qaabka: Tijaabi muhiimadda isku-xidhka dib-u-celinta iyadoo la adeegsanayo tijaabada t-test oo go'aami qiimaha R-squared si aad u aragto sida qaabku ugu habboon yahay xogta.
Su'aalo iyo Doodo Tusaale ah
Su'aal Tusaale ah 1: Dib-u-dhac Fudud oo Toosan
Su'aal:
Cilmi-baare ayaa doonaya inuu ogaado xiriirka ka dhexeeya tirada saacadaha waxbarashada (X) iyo dhibcaha imtixaanka ardayda (Y). Xogta la helay waa sidan soo socota:
| Saacadaha Waxbarashada (X) | Dhibcaha Imtixaanka (Y) |
|——————–|———————–|
| 2 | 70 |
| 3 | 75 |
| 5 | 80 |
| 7 | 85 |
| 8 | 90 |
Ka samee isla'eg dib-u-celin toosan xogtan!
Dood:
1. Xisaabinta Celceliska:
\[
\bar{X} = \frac{2 + 3 + 5 + 7 + 8}{5} = 5
\]
\[
\bar{Y} = \frac{70 + 75 + 80 + 85 + 90}{5} = 80
\]
2. Xisaabinta Isugeynta Dib-u-Celinta \( b \):
\[
b = \frac{\sum (X_i – \bar{X})(Y_i – \bar{Y})}{\sum (X_i – \bar{X})^2}
\]
\[
\sum (X_i – \bar{X})(Y_i – \bar{Y}) = (2 – 5)(70 – 80) + (3 – 5)(75 – 80) + (5 – 5)(80 – 80) + (7 – 5)(85 – 80) + (8 – 5)(90 – 80)
\]
\[
= (-3)(-10) + (-2)(-5) + (0)(0) + (2)(5) + (3)(10) = 30 + 10 + 0 + 10 + 30 = 80
\]
\[
\sum (X_i – \bar{X})^2 = (2 – 5)^2 + (3 – 5)^2 + (5 – 5)^2 + (7 – 5)^2 + (8 – 5)^2
\]
\[
= 9 + 4 + 0 + 4 + 9 = 26
\]
\[
b = \frac{80}{26} \qiyaastii 3.08
\]
3. Xisaabinta Jarista \( a \):
\[
a = \bar{Y} – b\bar{X}
\]
\[
a = 80 – 3.08 \ jeer 5 = 80 – 15.4 = 64.6
\]
4. Isleegta Dib-u-noqoshada:
\[
Y = 64.6 + 3.08X
\]
Markaa, isle'egta dib-u-celinta toosan ee xogta waa \( Y = 64.6 + 3.08X \). Taas macnaheedu waa in saacad kasta oo dheeraad ah oo daraasad ah la filayo inay dhibcaha imtixaanka ku kordhiso 3.08 dhibcood.
Su'aal Tusaale 2: Imtixaanka Moodeelka iyo Fasiraadda
Su'aal:
Iyada oo la sii wadayo isla xogtaas, xisaabi qiimaha R-squared (R²) si aad u cabbirto sida qaabku ugu habboon yahay xogta. Sidoo kale, tijaabi muhiimadda isku-dhafka dib-u-celinta \( b \).
Dood:
1. Xisaabi Wadarta Wadarta Labajibbaarane (SST), Wadarta Dib-u-Celinta Labajibbaarane (SSR), iyo Wadarta Khaladka Labajibbaarane (SSE):
\[
SST = \sum (Y_i – \bar{Y})^2
\]
\[
SST = (70 – 80)^2 + (75 – 80)^2 + (80 – 80)^2 + (85 – 80)^2 + (90 – 80)^2 = 100 + 25 + 0 + 25 + 100 = 250
\]
\[
SSR = \sum (\koofiyad{Y}_i – \bar{Y})^2
\]
Halkee \( \hat{Y}_i \) uu yahay qiimaha la saadaaliyay ee isleegta dib-u-celinta:
\[
\hat{Y}_i = 64.6 + 3.08X_i
\]
\[
\ koofiyadda {Y} = [67.76, 70.84, 76.0, 82.16, 85.24]
\]
\[
\bar{Y} = 80
\]
\[
SSR = (67.76 – 80)^2 + (70.84 – 80)^2 + (76.0 – 80)^2 + (82.16 – 80)^2 + (85.24 – 80)^2
\]
\[
SSR = (-12.24)^2 + (-9.16)^2 + (-4.0)^2 + 2.16^2 + 5.24^2 = 149.8
\]
2. Xisaabinta SSE:
\[
SSE = SST – SSR = 250 – 149.8 = 100.2
\]
3. Xisaabinta R-squared:
\[
R^2 = \frac{SSR}{SST} = \frac{149.8}{250} \qiyaastii 0.6
\]
Qiimaha R-squared ee 0.6 wuxuu muujinayaa in qaabkani uu sharraxayo qiyaastii 60% kala duwanaanshaha xogta. Tani waxay muujinaysaa in xariiqda dib-u-celinta ay si fiican ugu habboon tahay xogta.
4. t-Tijaabada Muhiimadda Isku-xidhka \( b \):
\[
t = \frac{b}{SE(b)}
\]
\[
SE(b) = \sqrt{\frac{SSE}{n-2}} / \sqrt{\sum (X_i – \bar{X})^2}
\]
\[
SE(b) = \sqrt{\frac{100.2}{5-2}} / \sqrt{26}
\]
\[
SE(b) = \sqrt{33.4} / \sqrt{26} \qiyaastii 1.13
\]
\[
t = \frac{3.08}{1.13} \qiyaastii 2.73
\]
Marka la eego a \( t-statistik \qiyaastii 2.73 \), haddii aan isticmaalno heer caadi ah oo muhiimad ah (α = 0.05), waxaan barbar dhignaa t-table. Tusaale ahaan, for \( df = 3 \), critical \( t \) waa qiyaastii 2.353. Kadib \( t-observed > t-critical \), taasoo muujinaysa in isku-xidhku uu muhiim yahay.
Gabagabo
Maqaalkan, waxaan ku soo qaadanay aasaaska dib-u-celinta toosan, sida loo xisaabiyo isku-xidhka dib-u-celinta iyo dhex-galka, iyo sida loo fasiro natiijooyinka iyadoo la adeegsanayo masalooyinka tusaalaha ah. Ku celcelinta joogtada ah ee xogaha kala duwan ayaa lagama maarmaan u ah si aad u noqoto mid ku xeel dheer isticmaalka habkan. Dib-u-celinta toosan waa qalab qiimo leh oo lagu falanqeeyo xogta waxayna bixin kartaa aragtiyo qoto dheer oo ku saabsan xiriirka ka dhexeeya doorsoomayaasha.
