Ngā Tikanga Whakatau Tatau i roto i ngā Tatauranga
Ko te tatauranga te pūtaiao o te kohikohi, te tātari, me te whakamārama raraunga, ā, ko tētahi o ōna wāhanga nui ko te whakatau tata. Ko te whakatau tata i roto i ngā tatauranga e pā ana ki te tukanga whakatau i te uara tata o tētahi tawhā taupori i runga i ngā mōhiohio i whiwhi mai i tētahi tauira. Ka taea te whakarōpū i ngā tikanga whakatau tata ki ngā momo matua e rua: te whakatau tata ira me te whakatau tata i waenga. I roto i tēnei tuhinga, ka matapakihia e mātou ngā tikanga whakatau tata e whakamahia whānuitia ana i roto i ngā tatauranga.
Te Māramatanga Taketake o te Whakatau Tatau
I mua i te urunga atu ki ngā tikanga aromatawai, he mea nui kia mārama ki ētahi kupu matua:
– Ngā tawhā: Ngā āhuatanga tau o tētahi taupori. Hei tauira, te toharite taupori (µ), te rerekētanga taupori (σ²).
– Tatauranga: Ngā āhuatanga tau o tētahi tauira. Hei tauira, te toharite tauira (x̄), te rerekētanga tauira (s²).
Ko te whāinga matua o te whakatau tata ko te whakatau whakatau mō ngā tawhā taupori i runga i ngā raraunga tauira . E rua ngā momo whakatau tata matua i roto i ngā tatauranga:
1. Te Whakatau Tatau: Ka whakaratohia he uara kotahi hei whakatau tata mō te tawhā taupori.
2. Te Whakatau Tatau: Ka whakarato i te whānuitanga o ngā uara hei whakatau tata mō tētahi tawhā taupori, tae atu ki tētahi taumata whakawhirinaki.
Tikanga Whakatau Ira
Ko te whakatau tata ira ko te tukanga o te whakarato i tētahi tau kotahi e whakaatu ana i te whakatau tata pai rawa atu o tētahi tawhā taupori. Ko ētahi o ngā whakatau tata e whakamahia whānuitia ana ko:
1. Tauira Toharite
Ko te huarahi māmā rawa atu, ko te huarahi tino noa hoki ki te whakatau tata i te toharite o te taupori ko te whakamahi i te toharite tauira, e tatauhia ana penei:
\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \]
ko \( x_i \) ia mātakitakinga i roto i te tauira, ā, ko \( n \) te rahi o te tauira.
2. Tauira Waenga
Ko te tau waenga tauira te uara waenga o ngā raraunga tauira kua whakarōpūtia. He whakatau tata pakari tēnei nā te mea kāore e pāngia e ngā mea o waho.
3. Te Ōwehenga Tauira
Hei whakatau tata i te ōwehenga o te taupori, ka whakamahia te ōwehenga tauira, e tatauhia ana penei:
\[ \hat{p} = \frac{x}{n} \]
ko \( x \) te maha o ngā angitu i roto i te tauira, ā, ko \( n \) te rahi o te tauira.
Tikanga Whakatau Āputa
Ka whakaratohia e ngā whakatau tata he whānuitanga o ngā uara, e tika ana kia kapi i te tawhā taupori me tētahi taumata whakapono (hei tauira, 95%). He maha ngā wā ka whakaatuhia ngā whakatau tata i te āhua o te Wā Whakapono (CI).
1. Wā Whakawhirinaki mō te Toharite Taupori
Mena ka ahu mai ngā raraunga tauira i tohatoha noa mēnā he nui rawa te (n) (e pā ana te CLT), ko te wā whakawhirinaki mō te toharite taupori (mu) ko:
\[ \bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}} \]
kāore i te mana:
– Ko te toharite tauira te \( \bar{x} \)
– Ko te uara-z o te tohatoha paerewa noa e rite ana ki te taumata whakawhirinaki (hei tauira, 1.96 mō te 95%) \( z_{\alpha/2} \)
– ( tohu ) ko paerewa rerekētanga taupori. Mena kāore i te mōhiotia te \( \sigma \), ka whakamahia te \( s \) (tauira paerewa rerekētanga).
– Ko te rahi o te tauira ko \( n \).
2. Wā Whakawhirinaki mō te Ōwehenga Taupori
Hei whakatau tata i te ōwehenga o te taupori \( p \):
\[ \hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]
ko \( \hat{p} \) te ōwehenga tauira me ētahi atu tawhā e ai ki te whakaahuatanga o mua.
Ngā Tikanga Whakatau Tatau Kē Atu
1. Tikanga Tūponotanga Mōrahi (ML)
Ko te tikanga Mōrahi Tūponotanga he tikanga e whakamahia ana hei kimi i te tātaitanga pai rawa atu mō tētahi tawhā taupori \( \theta \) mā te whakanui ake i te mahi tūponotanga \( L(\theta) \). Ko te mahi tūponotanga ko te tūponotanga o te whiwhi i ngā raraunga i kitea i runga i te tawhā \( \theta \):
\[ L(\theta|x) = \prod_{i=1}^{n} f(x_i|\theta) \]
ko \( f(x_i|\theta) \) te mahi kiato tūponotanga (PDF) o ngā raraunga. Ko te tātai e whakanui ana i te \( L(\theta) \) ka kiia ko te tātai tūponotanga mōrahi (MLE).
2. Tikanga Whakatau Tatau Bayesian
E whakaarohia ana e te huarahi Bayesian ngā tawhā hei ngā taurangi matapōkere ā, whakamahia ngā tohatoha tūponotanga hei whakatau tata i ngā tawhā. I runga i te ariā a Bayes:
\[ P(\teta|x) = \frac{P(x|\teta) P(\teta)}{P(x)} \]
ko \( P(\theta|x) \) te tohatoha muri, ko \( P(x|\theta) \) te tūponotanga, ko \( P(\theta) \) te mua, ā, ko \( P(x) \) te tawhā tūponotanga. He nui rawa te whakawhirinaki o ngā tātaitanga Bayesian ki ngā mua e whakamahia ana.
Aromatawai a te Kaiwhakatau Tatau
Hei aromatawai i tētahi tātaitanga pūwāhi, me arotake e tātou ōna āhuatanga:
– Te Tika/Te Pānga: Ka kiia te tātaitanga \( \hat{\theta} \) he kore pānga mēnā \( E[\hat{\theta}] = \theta \).
– Whai Huatanga: Ko te tātaitanga whai hua te mea iti rawa o te rerekētanga i roto i ngā tātaitanga kore whakaaro katoa.
– Te ōritetanga: Ka kiia he ōrite te tātaitanga mēnā ka tata te rahi o te tauira ki te \( \hat{\theta} \) ina piki haere te rahi o te tauira \( n \).
Ngā Tauira Taupānga
1. Te Whakatau Tatau Whiwhinga Toharite
I roto i ngā rangahau ōhanga, he maha ngā wā ka mahia te whakatau tata i te whiwhinga moni toharite o tētahi taupori. Ka tangohia e ngā kairangahau tētahi tauira o te taupori ka tatau i te toharite tauira hei whakatau tata, ka whakaratohia he wā whakawhirinaki hei whakaatu i te koretake o tēnei whakatau tata.
2. Te whakatau tata mō te ōwehenga o ngā kaipōti
I roto i tētahi rangahau pōtitanga, ka hiahia pea te kairangahau ki te whakatau tata i te ōrau o ngā kaipōti e tautoko ana i tētahi kaitono. Ko te ōwehenga tauira \( \hat{p} \) o ngā kaiwhakautu e tautoko ana i taua kaitono ka whakamahia hei whakatau tata. Ka taea te whakarato i tētahi āputa whakawhirinaki hei whakaatu i te tawhā hapa.
Whakamutunga
He mea nui te mahi a ngā tikanga whakatau tata i roto i ngā tatauranga nā te mea ka taea e ngā kairangahau te whakatau whakatau mō ngā taupori i runga i ngā raraunga tauira. He taputapu kaha ngā tikanga whakatau tata ira me te whakatau tata mō tēnei, me ngā tikanga pēnei i te tūponotanga mōrahi me te whakatau tata Bayesian e ruku hohonu ana ki ngā uauatanga o ngā raraunga. Mā te whakamahi i ngā whakatau tata tika, whai hua, me te ōrite ka whakarite i ngā hua tātari raraunga pono me te tika , ka āwhina i te whakatau whakatau pai ake i roto i ngā mara pēnei i te ōhanga, ngā pūtaiao pāpori, te hauora, me ētahi atu.
