Tauira o ngā Pātai Kōrero mō te Tikanga Tapawhā Iti rawa
Ko te Tikanga Tapawhā Iti (LEM) he tikanga tatauranga e whakamahia ana hei kimi i te rārangi taunga pai rawa atu e tino whai hua ana ki te matapae raraunga. He maha ngā wā ka whakamahia tēnei tikanga i roto i te tātari whakatautau rārangi hei kimi i te whanaungatanga i waenga i ngā taurangi motuhake me ngā taurangi whakawhirinaki. Ka kapi i tēnei tuhinga ngā ariā taketake o te tikanga tapawhā iti, me ngā tauira me ngā whakamārama taahiraa-i-te-taahiraa hei māramatanga hohonu ake mō te mahi a tēnei tikanga.
Ngā Ariā Taketake o te Tikanga Tapawhā Iti Rawa
Ko te whāinga o te tikanga tapawhā iti rawa ko te whakaiti i te tapeke o ngā tapawhā o ngā rerekētanga i waenga i ngā uara i kitea me ngā uara i matapaetia e te tauira whakatauira. Ka taea te tuhi i te whārite o tētahi rārangi whakatauira rārangi māmā penei:
\[ y = a + bx \]
Kei hea:
– Ko te taurangi whakawhirinaki ko \( y \),
– Ko te taurangi motuhake ko \( x \),
– Ko te \( a \) te haukoti (te uara o \( y \) ina \( x = 0 \)),
– Ko te \( b \) te pikinga o te rārangi (te pikinga, te tauwehenga taurite rānei).
Mā te tikanga tapawhā iti rawa ka whakatauhia ngā tawhā \( a \) me \( b \) e whakaiti ana i te mahi e whai ake nei:
\[ \text{SSE} = \sum_{i=1}^{n} (y_i – \hat{y_i})^2 \]
Ko SSE te Tapeke o ngā Hapa Tapawhā, ko \( y_i \) te uara tuturu, ā, ko \( \hat{y_i} = a + bx_i \) te uara matapae.
Ngā Hipanga Tikanga Tapawhā Iti Rawa
Hei whakamārama i te ariā, ka whakaotihia e mātou tētahi tauira rapanga e whakamahi ana i te tikanga tapawhā iti rawa.
Tauira raruraru
I runga i ngā raraunga e whai ake nei:
| x (Ngā Hāora Ako) | y (Kaute Whakamātautau) |
|———————–|———————–|
| 2 | 81 |
| 4 | 93 |
| 6 | 91 |
| 8 | 97 |
| 10 | 103 |
Whakatauhia te rārangi whakatauira rārangi e tino ōrite ana ki ngā raraunga.
Kōrero
1. Te Tātai i te Toharite o \( \bar{x} \) me \( \bar{y} \)
\[
\bar{x} = \frac{\tapeke x_i}{n} = \frac{2 + 4 + 6 + 8 + 10}{5} = 6
\]
\[
\bar{y} = \frac{\sum y_i}{n} = \frac{81 + 93 + 91 + 97 + 103}{5} = 93
\]
2. Te Tātai i te Tawhā \( b \) (Pīnakitanga)
Ka tatauhia te tawhā \( b \) mā te:
\[
b = \frac{\tapeke (x_i – \bar{x})(y_i – \bar{y})}{\tapeke (x_i – \bar{x})^2}
\]
Te tatau i ia wāhanga:
\[
tapeke (x_i – x})(y_i – y}) = (2-6)(81-93) + (4-6)(93-93) + (6-6)(91-93) + (8-6)(97-93) + (10-6)(103-93)
\]
\[
= (-4)(-12) + (-2)(0) + (0)(-2) + (2)(4) + (4)(10)
\]
\[
= 48 + 0 + 0 + 8 + 40 = 96
\]
\[
tapeke (x_i – \bar{x})^2 = (2-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2
\]
\[
= (-4)^2 + (-2)^2 + 0^2 + 2^2 + 4^2
\]
\[
= 16 + 4 + 0 + 4 + 16 = 40
\]
Nō reira:
\[
b = \frac{96}{40} = 2.4
\]
3. Te Tātai i te Tawhā \( a \) (Whakataunga)
Mā te whakamahi i te toharite o \( \bar{x} \) me \( \bar{y} \):
\[
a = \bar{y} – b\bar{x} = 93 – 2.4 \times 6 = 93 – 14.4 = 78.6
\]
4. Te Tuhi i te Whārite Raina Whakamuri
Mā ngā tawhā kua kitea, ka taea e tātou te tuhi i te whārite o te rārangi whakatauira:
\[
y = 78.6 + 2.4x
\]
Whakamārama me te Manatoko
Hei whakarite kia taurite tēnei rārangi whakatauira, ka taea e tātou te tatau i te uara-y i matapaetia (\(\hat{y}\)) mō ia x i roto i ngā raraunga tuatahi, me te tatau anō hoki i te Tapeke o ngā Hapa Tapawhā (SSE) hei whakamana i te tika o te matapae.
| x | y | \(\pōtae{y}\) | \((y – \pōtae{y})^2\) |
|—|—-|—————|——————–|
| 2 | 81 | 83.4 | (81-83.4)^2 = 5.76 |
| 4 | 93 | 88.2 | (93-88.2)^2 = 23.04|
| 6 | 91 | 93.0 | (91-93.0)^2 = 4.00 |
| 8 | 97 | 97.8 | (97-97.8)^2 = 0.64 |
|10 |103 |102.6 | (103-102.6)^2= 0.16|
SSS:
\[
SSE = 5.76 + 23.04 + 4.00 + 0.64 + 0.16 = 33.6
\]
Ki te iti noa te SSE, ka taea e tātou te whakatau he pai te rārangi whakatauira i whakaputaina e te tikanga tapawhā iti rawa mō tēnei raraunga.
Whakamutunga
He taputapu tātari tatauranga kaha te Tikanga Tapawhā Iti mō te whakatau i te rārangi tino pai mō tētahi huinga raraunga, me te whakaiti i te hapa matapae i runga i te tapawhā o ngā rerekētanga. Mā te whakamahi i ngā mahi o te tatau i te toharite, te whakatau tata i te pari me te haukoti, me te tuhi me te manatoko i te whārite rārangi whakatauira, ka taea e tātou te matapae tika i te uara o te taurangi whakawhirinaki mai i ngā taurangi motuhake.
He tino whai hua te māramatanga pai ki tēnei tikanga i roto i ngā mara pēnei i te ōhanga, te tatauranga koiora, te hangarau, me ngā pūtaiao pāpori, e whakamahia pinepine ai te tātari whakatauira. E whakaatu ana tēnei tuhinga, me ngā tauira tuturu, i te hiranga me te whai huatanga o tēnei tikanga i roto i te tātari raraunga.
