Lipotso le Lipuisano tsa Mohlala oa ho Fokotsa Moeli
Ho khutlela morao ka mola ke mokhoa oa lipalo-palo o sebelisoang ho fumana kamano pakeng tsa lintho tse peli kapa ho feta. Mokhoa ona o sebelisoa haholo mafapheng a fapaneng, ho kenyeletsoa moruo, khoebo, mahlale a sechaba le mahlale a tlhaho. Sehloohong sena, re tla buisana ka ho khutlela morao ka mola, mokhoa oa ho e bala, le ho fana ka mehlala e 'maloa ea mathata ka litlhaloso ho thusa babali ho utloisisa mohopolo ona ka botebo.
Ho Utloisisa Khatello ea Linear
Ho kgutlisa mola ka mola ke mokgwa wa tlhahlobo o sebediswang ho etsa mohlala wa kamano pakeng tsa diphetoho tse ikemetseng tse le nngwe kapa tse ngata (diponelopele) le phetoho e itshetlehileng (karabo). Ho kgutlisa mola ka mola ho kenyeletsa phetoho e le nngwe e ikemetseng le phetoho e le nngwe e itshetlehileng, ha ho kgutlisa mola ka mola ka mola ho kenyeletsa diphetoho tse ikemetseng tse fetang e le nngwe.
Tekanyo ea mola o bonolo oa ho khutlela morao o otlolohileng ke:
\[ Y = a + bX \]
Di mana:
– \( Y \) ke phetoho e itshetlehileng.
– \( X \) ke phetoho e ikemetseng.
– \( a \) ke intercept, e leng boleng ba Y ha X = 0.
– \( b \) ke coefficient ya regression, ke hore, Y e fetoha hakae haeba X e fetoha ka yuniti e le nngwe.
Mehato ea ho Fokotsa Methapo
1. Bokella Lintlha: Taba ea pele, bokella lintlha tse lokelang ho hlahlojoa.
2. Lintlha tsa ho beha lintlha: Etsa moralo oa ho hasanya ho bona hore na ho na le kamano e otlolohileng pakeng tsa li-variable.
3. Bala Coefficient ea Regression: Sebelisa mokhoa oa lisekoere tse nyane ho fumana mola o motle ka ho fetisisa.
4. Ho Lekola Mohlala: Leka bohlokoa ba di-coefficient tsa regression ka teko ya t mme o fumane boleng ba R-squared ho bona hore na mohlala o dumellana hantle hakae le data.
Lipotso tsa Mehlala le Puisano
Mohlala oa Potso ea 1: Ho Fokotsa ho Bonolo ha Linear
Potso:
Mofuputsi o batla ho tseba kamano pakeng tsa palo ea lihora tsa ho ithuta (X) le lintlha tsa tlhahlobo ea baithuti (Y). Lintlha tse fumanoeng ke tse latelang:
| Lihora tsa Thuto (X) | Lintlha tsa Tlhahlobo (Y) |
|———————–|——————–|
| 2 | 70 |
| 3 | 75 |
| 5 | 80 |
| 7 | 85 |
| 8 | 90 |
Etsa equation ea regression e otlolohileng ho tsoa ho data ena!
Puisano:
1. Ho Bala Karolelano:
\[
\bar{X} = \frac{2 + 3 + 5 + 7 + 8}{5} = 5
\]
\[
\bar{Y} = \frac{70 + 75 + 80 + 85 + 90}{5} = 80
\]
2. Ho bala Coefficient ea Regression \( b \):
\[
b = \frac{\sum (X_i – \bar{X})(Y_i – \bar{Y})}{\sum (X_i – \bar{X})^2}
\]
\[
\sum (X_i – \bar{X})(Y_i – \bar{Y}) = (2 – 5)(70 – 80) + (3 – 5)(75 – 80) + (5 – 5)(80 – 80) + (7 – 5)(85 – 80) + (8 – 5)(90 – 80)
\]
\[
= (-3)(-10) + (-2)(-5) + (0)(0) + (2)(5) + (3)(10) = 30 + 10 + 0 + 10 + 30 = 80
\]
\[
\sum (X_i – \bar{X})^2 = (2 – 5)^2 + (3 – 5)^2 + (5 – 5)^2 + (7 – 5)^2 + (8 – 5)^2
\]
\[
= 9 + 4 + 0 + 4 + 9 = 26
\]
\[
b = \frac{80}{26} \hoo e ka bang 3.08
\]
3. Ho bala Intercept \( a \):
\[
a = \bar{Y} – b\bar{X}
\]
\[
a = 80 – 3.08 \makhetlo a 5 = 80 – 15.4 = 64.6
\]
4. Tekanyo ea ho Fokotsa Khatello:
\[
Y = 64.6 + 3.08X
\]
Kahoo, equation ea regression e otlolohileng bakeng sa data ke \( Y = 64.6 + 3.08X \). Sena se bolela hore hora e 'ngoe le e 'ngoe e eketsehileng ea thuto e lebelletsoe ho eketsa lintlha tsa teko ka lintlha tse 3.08.
Mohlala oa Potso ea 2: Teko ea Mohlala le Tlhaloso
Potso:
Ha o tswela pele ka data e tshwanang, bala boleng ba R-squared (R²) ho lekanya hore na mohlala o dumellana hantle hakae le data. Hape, leka bohlokwa ba coefficient ya regression \( b \).
Puisano:
1. Bala Kakaretso ea Lisekwere (SST), Kakaretso ea Lisekwere (SSR), le Kakaretso ea Lisekwere tsa Liphoso (SSE):
\[
SST = \sum (Y_i – \bar{Y})^2
\]
\[
SST = (70 – 80)^2 + (75 – 80)^2 + (80 – 80)^2 + (85 – 80)^2 + (90 – 80)^2 = 100 + 25 + 0 + 25 + 100 = 250
\]
\[
SSR = \sum (\hat{Y}_i – \bar{Y})^2
\]
Moo \( \hat{Y}_i \) e leng boleng bo boletsoeng esale pele ba equation ea regression:
\[
\hat{Y}_i = 64.6 + 3.08X_i
\]
\[
\hat{Y} = [67.76, 70.84, 76.0, 82.16, 85.24]
\]
\[
\bar{Y} = 80
\]
\[
SSR = (67.76 – 80)^2 + (70.84 – 80)^2 + (76.0 – 80)^2 + (82.16 – 80)^2 + (85.24 – 80)^2
\]
\[
SSR = (-12.24)^2 + (-9.16)^2 + (-4.0)^2 + 2.16^2 + 5.24^2 = 149.8
\]
2. Ho bala SSE:
\[
SSE = SST – SSR = 250 – 149.8 = 100.2
\]
3. Ho bala R-squared:
\[
R^2 = \frac{SSR}{SST} = \frac{149.8}{250} \hoo e ka bang 0.6
\]
Boleng ba R-squared ba 0.6 bo bontša hore mohlala ona o hlalosa hoo e ka bang 60% ea phapang datang. Sena se bontša hore mola oa regression o lumellana hantle le data.
4. Teko ea t bakeng sa Bohlokoa ba Coefficient \( b \):
\[
t = \frac{b}{SE(b)}
\]
\[
SE(b) = \sqrt{\frac{SSE}{n-2}} / \sqrt{\sum (X_i – \bar{X})^2}
\]
\[
SE(b) = \sqrt{\frac{100.2}{5-2}} / \sqrt{26}
\]
\[
SE(b) = \sqrt{33.4} / \sqrt{26} \hoo e ka bang 1.13
\]
\[
t = \frac{3.08}{1.13} \hoo e ka bang 2.73
\]
Ka \( t-statistic \approx 2.73 \), haeba re sebelisa moeli o tloaelehileng bakeng sa bohlokoa (α = 0.05), re o bapisa le tafole ea t. Mohlala, bakeng sa \( df = 3 \), critical \( t \) e ka ba 2.353. Ebe \( t-observed > t-critical \), e bontšang hore coefficient e bohlokoa.
Qetello
Sehloohong sena, re akarelitse metheo ea ho khutlela morao ka tatellano, mokhoa oa ho bala coefficient ea ho khutlela morao le ho kenella, le mokhoa oa ho toloka liphetho u sebelisa mathata a mohlala. Ho ikoetlisa khafetsa ka lihlopha tse fapaneng tsa data ho bohlokoa ho ba le boiphihlelo ba ho sebelisa mokhoa ona. Ho khutlela morao ka tatellano ke sesebelisoa sa bohlokoa tlhahlobong ea data 'me ho ka fana ka temohisiso e tebileng mabapi le likamano lipakeng tsa li-variable.
