Mafunso ndi Kukambirana kwa Chitsanzo cha Kubwerera M'mbuyo kwa Linear
Kubwerezabwereza kwa mzere ndi njira yowerengera yomwe imagwiritsidwa ntchito kudziwa ubale pakati pa zinthu ziwiri kapena zingapo. Njirayi imagwiritsidwa ntchito kwambiri m'magawo osiyanasiyana, kuphatikizapo zachuma, bizinesi, sayansi ya chikhalidwe cha anthu, ndi sayansi yachilengedwe. M'nkhaniyi, tikambirana za kubwerera kwa mzere, momwe tingawerengere, ndikupereka zitsanzo zingapo za mavuto ndi mafotokozedwe kuti tithandize owerenga kumvetsetsa lingaliro ili mozama.
Kumvetsetsa Kugonjetsedwa kwa Linear
Kubwerezabwereza kwa mzere ndi njira yowunikira yomwe imagwiritsidwa ntchito poyesa ubale pakati pa chimodzi kapena zingapo zodziyimira pawokha (zoneneratu) ndi chosinthika chodalira (yankho). Kubwerezabwereza kosavuta kwa mzere kumaphatikizapo chosinthika chimodzi chodziyimira pawokha ndi chosinthika chimodzi chodalira, pomwe kubwerezabwereza kwa mzere wambiri kumaphatikizapo zosinthika zodziyimira pawokha zoposa chimodzi.
Equation ya mzere wosavuta wowongolera mzere ndi:
\[ Y = a + bX \]
Kumene:
– \( Y \) ndi chosinthika chodalira.
– \( X \) ndi chosinthika chodziyimira pawokha.
– \( a \) ndi intercept, yomwe ndi mtengo wa Y pamene X = 0.
– \( b \) ndi regression coefficient, ndiko kuti, kuchuluka kwa Y komwe kumasintha ngati X yasintha ndi unit imodzi.
Masitepe Ochepetsa Mizere
1. Kusonkhanitsa Deta: Choyamba, kusonkhanitsa deta kuti isanthulidwe.
2. Deta ya Chiwembu: Pangani chiwembu chogawa kuti muwone ngati pali ubale wolunjika pakati pa zosintha.
3. Werengani Regression Coefficient: Gwiritsani ntchito njira ya least squares kuti mudziwe mzere wabwino kwambiri.
4. Kuyesa Chitsanzo: Yesani kufunika kwa ma coefficients a regression ndi mayeso a t ndikuzindikira mtengo wa R-squared kuti muwone momwe chitsanzocho chikugwirizanirana bwino ndi deta.
Mafunso ndi Kukambirana Zitsanzo
Funso la Chitsanzo 1: Kubwereza Kosavuta kwa Linear
Funso:
Wofufuza akufuna kudziwa ubale womwe ulipo pakati pa chiwerengero cha maola ophunzirira (X) ndi zigoli za mayeso a ophunzira (Y). Deta yomwe yapezeka ndi iyi:
| Maola Ophunzirira (X) | Zigoli za Mayeso (Y) |
|———————–|——————–|
| 2 | 70 |
| 3 | 75 |
| 5 | 80 |
| 7 | 85 |
| 8 | 90 |
Pangani equation yolunjika yobwerera kuchokera ku deta iyi!
Kukambirana:
1. Kuwerengera Avereji:
\[
\bar{X} = \frac{2 + 3 + 5 + 7 + 8}{5} = 5
\]
\[
\bar{Y} = \frac{70 + 75 + 80 + 85 + 90}{5} = 80
\]
2. Kuwerengera Regression Coefficient \( b \):
\[
b = \frac{\sum (X_i – \bar{X})(Y_i – \bar{Y})}{\sum (X_i – \bar{X})^2}
\]
\[
\sum (X_i – \bar{X})(Y_i – \bar{Y}) = (2 – 5)(70 – 80) + (3 – 5)(75 – 80) + (5 – 5)(80 – 80) + (7 – 5)(85 – 80) + (8 – 5)(90 – 80)
\]
\[
= (-3)(-10) + (-2)(-5) + (0)(0) + (2)(5) + (3)(10) = 30 + 10 + 0 + 10 + 30 = 80
\]
\[
\sum (X_i – \bar{X})^2 = (2 – 5)^2 + (3 – 5)^2 + (5 – 5)^2 + (7 – 5)^2 + (8 – 5)^2
\]
\[
= 9 + 4 + 0 + 4 + 9 = 26
\]
\[
b = \frac{80}{26} \pafupifupi 3.08
\]
3. Kuwerengera Intercept \( a \):
\[
a = \bar{Y} – b\bar{X}
\]
\[
a = 80 – 3.08 \kuwirikiza 5 = 80 – 15.4 = 64.6
\]
4. Chiyerekezo cha Kubwerera M'mbuyo:
\[
Y = 64.6 + 3.08X
\]
Kotero, equation yolunjika ya deta ndi \( Y = 64.6 + 3.08X \). Izi zikutanthauza kuti ola lililonse lowonjezera la kuphunzira likuyembekezeka kuwonjezera chigoli cha mayeso ndi mapointi 3.08.
Chitsanzo Funso 2: Mayeso a Chitsanzo ndi Kutanthauzira
Funso:
Mukapitiliza ndi deta yomweyi, werengani mtengo wa R-squared (R²) kuti muyese momwe chitsanzocho chikugwirizanirana bwino ndi detayo. Komanso, yesani kufunika kwa regression coefficient \( b \).
Kukambirana:
1. Werengani Chiwerengero Chonse cha Masikweya (SST), Chiwerengero Chobwerezabwereza cha Masikweya (SSR), ndi Chiwerengero Cholakwika cha Masikweya (SSE):
\[
SST = \sum (Y_i – \bar{Y})^2
\]
\[
SST = (70 – 80)^2 + (75 – 80)^2 + (80 – 80)^2 + (85 – 80)^2 + (90 – 80)^2 = 100 + 25 + 0 + 25 + 100 = 250
\]
\[
SSR = \sum (\hat{Y}_i – \bar{Y})^2
\]
Kumene \( \hat{Y}_i \) ndi mtengo wonenedweratu wa equation ya regression:
\[
\hat{Y}_i = 64.6 + 3.08X_i
\]
\[
\chipewa{Y} = [67.76, 70.84, 76.0, 82.16, 85.24]
\]
\[
\bar{Y} = 80
\]
\[
SSR = (67.76 – 80)^2 + (70.84 – 80)^2 + (76.0 – 80)^2 + (82.16 – 80)^2 + (85.24 – 80)^2
\]
\[
SSR = (-12.24)^2 + (-9.16)^2 + (-4.0)^2 + 2.16^2 + 5.24^2 = 149.8
\]
2. Kuwerengera SSE:
\[
SSE = SST – SSR = 250 – 149.8 = 100.2
\]
3. Kuwerengera R-squared:
\[
R^2 = \frac{SSR}{SST} = \frac{149.8}{250} \pafupifupi 0.6
\]
Mtengo wa R-squared wa 0.6 umasonyeza kuti chitsanzo ichi chikufotokoza pafupifupi 60% ya kusiyana kwa deta. Izi zikusonyeza kuti mzere wobwerera ukugwirizana bwino ndi deta.
4. t-Test ya Kufunika kwa Koefficient \( b \):
\[
t = \frac{b}{SE(b)}
\]
\[
SE(b) = \sqrt{\frac{SSE}{n-2}} / \sqrt{\sum (X_i – \bar{X})^2}
\]
\[
SE(b) = \sqrt{\frac{100.2}{5-2}} / \sqrt{26}
\]
\[
SE(b) = \sqrt{33.4} / \sqrt{26} \pafupifupi 1.13
\]
\[
t = \frac{3.08}{1.13} \pafupifupi 2.73
\]
Ndi \( t-statistic \pafupifupi 2.73 \), ngati tigwiritsa ntchito muyezo wofanana pa kufunika (α = 0.05), timauyerekeza ndi tebulo la t. Mwachitsanzo, pa \( df = 3 \), critical \( t \) ndi pafupifupi 2.353. Kenako \( t-observed > t-critical \), zomwe zikusonyeza kuti coefficient ndi yofunika.
Mapeto
Munkhaniyi, tafotokoza mfundo zoyambira za linear regression, momwe tingawerengere regression coefficient ndi intercept, komanso momwe tingatanthauzire zotsatira pogwiritsa ntchito zitsanzo zamavuto. Kuchita mobwerezabwereza ndi ma data osiyanasiyana ndikofunikira kuti mukhale waluso pakugwiritsa ntchito njira iyi. Linear regression ndi chida chofunikira pakusanthula deta ndipo chingapereke chidziwitso chakuya pa ubale womwe ulipo pakati pa zosintha.
