Ngā Tauira Pātai me te Kōrero mō te Whakarerekētanga Raina
Ko te whakatauira rārangi he tikanga tatauranga e whakamahia ana hei whakatau i te whanaungatanga i waenga i ngā taurangi e rua, neke atu rānei. He whānuitia te whakamahinga o tēnei tikanga i roto i ngā momo mara, tae atu ki te ōhanga, te pakihi, ngā pūtaiao pāpori, me ngā pūtaiao taiao. I roto i tēnei tuhinga, ka matapakihia e mātou te whakatauira rārangi, me pēhea te tatau, ā, ka whakaratohia he tauira raruraru me ngā whakamārama hei āwhina i ngā kaipānui ki te mārama hōhonu ki tēnei ariā.
Te Mārama ki te Whakamuritanga Raina
Ko te whakatauira rārangi he tikanga tātari e whakamahia ana hei whakatauira i te whanaungatanga i waenga i tētahi, i ētahi rānei o ngā taurangi motuhake (ngā matapae) me tētahi taurangi whakawhirinaki (te urupare). Ko te whakatauira rārangi māmā e whai wāhi ana ki tētahi taurangi motuhake me tētahi taurangi whakawhirinaki, ko te whakatauira rārangi maha e whai wāhi ana ki ngā taurangi motuhake neke atu i te kotahi.
Ko te whārite o tētahi rārangi whakatauira rārangi māmā ko:
\[ Y = a + bX \]
Kei hea:
– Ko te taurangi whakawhirinaki ko \( Y \).
– Ko te taurangi motuhake ko \( X \).
– Ko te \( a \) te haukoti, koinei te uara o Y ina ko X = 0.
– Ko te tauwehenga whakatauira te \( b \), arā, te nui o te panonitanga o Y mēnā ka huri a X mā te kotahi waeine.
Ngā Hipanga Whakamuri Raina
1. Kohikohi Raraunga: Tuatahi, kohia ngā raraunga hei tātaritanga.
2. Tātaitanga Raraunga: Waihangahia he tātaitanga marara hei tirotiro mēnā he whanaungatanga rārangi i waenga i ngā taurangi.
3. Tātaihia te Tauwehenga Whakamuri: Whakamahia te tikanga tapawhā iti rawa hei whakatau i te rārangi pai rawa atu.
4. Te Whakamātautau i te Tauira: Whakamātauria te hiranga o ngā taunga whakatauira me te whakamātautau-t, ka whakatau i te uara tapawhā-R kia kite ai i te pai o te taunga o te tauira ki ngā raraunga.
Ngā Pātai Tauira me te Kōrero
Tauira Pātai 1: Whakamuritanga Raina Māmā
Pātai:
E hiahia ana tētahi kairangahau ki te mōhio ki te whanaungatanga i waenga i te maha o ngā hāora ako (X) me ngā kaute whakamātautau a ngā ākonga (Y). Ko ngā raraunga i kohia koia ēnei:
| Ngā Hāora Ako (X) | Kaute Whakamātautau (Y) |
|———————–|———————–|
| 2 | 70 |
| 3 | 75 |
| 5 | 80 |
| 7 | 85 |
| 8 | 90 |
Hangaia he whārite whakatauira rārangi mai i ēnei raraunga!
Kōrero:
1. Te Tātai i te Toharite:
\[
\bar{X} = \frac{2 + 3 + 5 + 7 + 8}{5} = 5
\]
\[
\pa{Y} = \frac{70 + 75 + 80 + 85 + 90}{5} = 80
\]
2. Te Tātai i te Tauwehenga Whakamuri \( b \):
\[
b = \frac{\tapeke (X_i – \bar{X})(Y_i – \bar{Y})}{\tapeke (X_i – \bar{X})^2}
\]
\[
Tapeke (X_i – X})(Y_i – Y}) = (2 – 5)(70 – 80) + (3 – 5)(75 – 80) + (5 – 5)(80 – 80) + (7 – 5)(85 – 80) + (8 – 5)(90 – 80)
\]
\[
= (-3)(-10) + (-2)(-5) + (0)(0) + (2)(5) + (3)(10) = 30 + 10 + 0 + 10 + 30 = 80
\]
\[
Tapeke (X_i – X})^2 = (2 – 5)^2 + (3 – 5)^2 + (5 – 5)^2 + (7 – 5)^2 + (8 – 5)^2
\]
\[
= 9 + 4 + 0 + 4 + 9 = 26
\]
\[
b = \frac{80}{26} \approx 3.08
\]
3. Te Tātai i te Āraitanga \( a \):
\[
a = \bar{Y} – b\bar{X}
\]
\[
a = 80 – 3.08 \whakareatia ki te 5 = 80 – 15.4 = 64.6
\]
4. Whārite Whakamuri:
\[
Y = 64.6 + 3.08X
\]
Nō reira, ko te whārite whakatauira rārangi mō ngā raraunga ko \( Y = 64.6 + 3.08X \). Ko te tikanga o tēnei, e matapaetia ana ka piki ake te kaute whakamātautau mā te 3.08 ngā tohu mō ia haora ako tāpiri.
Tauira Pātai 2: Whakamātautau Tauira me te Whakamārama
Pātai:
Me te whakamahi tonu i ngā raraunga ōrite, tatauhia te uara R-tapawhā (R²) hei ine i te pai o te taunga o te tauira ki ngā raraunga. Whakamātauria hoki te hiranga o te tauwehenga whakatauira \( b \).
Kōrero:
1. Tātaihia te Tapeke o ngā Tapawhā (SST), te Tapeke Whakamuri o ngā Tapawhā (SSR), me te Tapeke Hapa o ngā Tapawhā (SSE):
\[
SST = \tapeke (Y_i – \bar{Y})^2
\]
\[
SST = (70 – 80)^2 + (75 – 80)^2 + (80 – 80)^2 + (85 – 80)^2 + (90 – 80)^2 = 100 + 25 + 0 + 25 + 100 = 250
\]
\[
SSR = \tapeke (\hat{Y}_i – \bar{Y})^2
\]
Ko \( \hat{Y}_i \) te uara matapae o te whārite whakatauira:
\[
\hat{Y}_i = 64.6 + 3.08X_i
\]
\[
\potae{Y} = [67.76, 70.84, 76.0, 82.16, 85.24]
\]
\[
\bar{Y} = 80
\]
\[
SSR = (67.76 – 80)^2 + (70.84 – 80)^2 + (76.0 – 80)^2 + (82.16 – 80)^2 + (85.24 – 80)^2
\]
\[
SSR = (-12.24)^2 + (-9.16)^2 + (-4.0)^2 + 2.16^2 + 5.24^2 = 149.8
\]
2. Te tatau i te SSE:
\[
SSE = SST – SSR = 250 – 149.8 = 100.2
\]
3. Te tatau i te R-tapawhā:
\[
R^2 = \frac{SSR}{SST} = \frac{149.8}{250} \approx 0.6
\]
Ko te uara R-tapawhā o te 0.6 e tohu ana kei te whakamāramahia e tēnei tauira te 60% o te rerekētanga o ngā raraunga. E tohu ana tēnei kei te pai te uru o te rārangi whakatauira ki ngā raraunga.
4. Whakamātautau-t mō te Hiranga o te Tauwehenga \( b \):
\[
t = \frac{b}{SE(b)}
\]
\[
SE(b) = \sqrt{\frac{SSE}{n-2}} / \sqrt{\sum (X_i – \bar{X})^2}
\]
\[
SE(b) = \sqrt{\frac{100.2}{5-2}} / \sqrt{26}
\]
\[
SE(b) = \sqrt{33.4} / \sqrt{26} \approx 1.13
\]
\[
t = \frac{3.08}{1.13} \approx 2.73
\]
Ki te whakamahi i te \(t-statistic \approx 2.73 \), mēnā ka whakamahia he paepae noa mō te hiranga (α = 0.05), ka whakaritea ki te ripanga-t. Hei tauira, mō \(df = 3 \), ko te hiranga \(t \) he tata ki te 2.353. Kātahi ka \(t-observed > t-hiranga \), e tohu ana he hiranga te tauwehenga.
Whakamutunga
I roto i tēnei tuhinga, kua hipokina e mātou ngā kaupapa matua o te whakatauira rārangi, me pēhea te tatau i te tauwehenga whakatauira me te ārai, me pēhea te whakamārama i ngā hua mā te whakamahi i ngā tauira rapanga. He mea nui te whakaharatau auau ki ngā huinga raraunga rerekē kia matatau ai koe ki te whakamahi i tēnei tikanga. He taputapu whai hua te whakatauira rārangi i roto i te tātari raraunga, ā, ka taea te whakarato i ngā māramatanga hōhonu ki ngā whanaungatanga i waenga i ngā taurangi.
