He Kupu Whakataki ki ngā Tohatoha Tauira
I roto i te ao o ngā tatauranga, he mahi nui ngā tohatoha tauira, hei tūāpapa mō ngā tatauranga whakatau me te tātari raraunga. Ko te whāinga o tēnei tuhinga he whakamārama i te ariā o ngā tohatoha tauira, he whakamārama i tōna hiranga, me te tūhura i ōna āhuatanga me ōna momo.
He aha te Tohatoha Tauira?
Hei tīmatanga i tā tātou tūhuratanga o ngā tohatoha tauira, he mea nui kia mārama tuatahi ki ētahi kupu matua.
Ko te taupori e pā ana ki te huinga katoa o ngā tūemi, o ngā tāngata takitahi rānei e taea ai te kohikohi raraunga. Ina akohia ngā āhuatanga, ngā whanonga rānei o tētahi taupori, he maha ngā wā kāore e taea, e kore rānei e taea te tirotiro i ia mema. I konei, ka tino whai hua te tauira – he huinga iti o te taupori e whakaatu ana i te rahi, engari he rahi e taea te whakahaere.
Ko te tohatoha tauira (te tohatoha tauira rānei) ko te tohatoha tūponotanga o tētahi tatauranga kua hoatu (pēnei i te toharite, te rerekētanga rānei) i runga i tētahi tauira matapōkere. I ngā kupu māmā ake, ko te whanonga o te ine tatauranga mai i ngā tauira (hei tauira, ngā toharite tauira) ina tangohia ngā tauira maha mai i te taupori.
Te Hiranga o ngā Tohatoha Tauira
Tauanga Inferential
Ko te whakamahinga matua o ngā tohatoha tauira kei roto i ngā tatauranga whakatau, e whakatau ana i ngā tawhā taupori i runga i ngā tatauranga tauira. Hei tauira, mā te tohatoha tauira, ka taea e tātou te whakatau tata i ngā toharite taupori, ngā rerekētanga, me ngā ōwehenga, ā, ka taea ai e tātou te whakatau whai whakaaro mō te taupori katoa.
Te Ariā Herenga Waenga (CLT)
Ko te Kaupapa Here Matua tētahi o ngā mātāpono tino nui o te tatauranga. E kī ana, ina nui te rahi o te tauira, ka tata rite te tohatoha tauira o te toharite tauira ki te tohatoha noa, ahakoa te tohatoha o te taupori. He maha ngā tikanga tatauranga e tautokona ana e tēnei kaupapa, ā, e whakamārama ana i te whakamahinga o te tohatoha noa i roto i te whakamātautau whakapae me te whakatau tata i te wā whakawhirinaki.
Te Whakamatautau Whakaaro
I roto i te whakamātautau whakapae, ka taea e ngā tohatoha tauira te whakatau i te tūponotanga o te kitenga i tētahi tatauranga tauira i raro i tētahi whakapae kore. Mā te whakataurite i ngā tatauranga tauira ki te tohatoha e tumanakohia ana i raro i te whakapae kore, ka taea e tātou te whakatau mēnā ka whakakahoretia te whakapae kore, kāore rānei.
Nga Wae Whakaaetanga
Mā ngā tohatoha tauira ka āwhina i te hanga i ngā āputa whakapono, e whakarato ana i tētahi awhe e tūpono ai te tawhā taupori. Mā tēnei awhe, me te taumata whakapono (hei tauira, 95%), ka taea te ine i te tino pono.
Ngā Āhuatanga o ngā Tohatoha Tauira
He mea nui te mārama ki ngā āhuatanga matua o ngā tohatoha tauira hei whakamārama me te whakamahi whai hua i aua tohatoha.
Te Toharite o ngā Tohatoha Tauira
Ko te toharite o te tohatoha tauira o te toharite tauira (e tohuhia ana ko \( \mu_{\bar{x}} \)) he ōrite ki te toharite taupori ( \( \mu \) ). Mā te pāngarau, \( \mu_{\bar{x}} = \mu \). Ko te tikanga o tēnei āhuatanga he rite te uara e tumanakohia ana o te toharite tauira ki te toharite taupori.
Te Rerekētanga me te Hapa Paerewa
Ka hopukina te rerekētanga o ngā tohatoha tauira e te hapa paerewa (SE), e ine ana i te marara o ngā tatauranga tauira huri noa i te tawhā taupori. Mō te toharite tauira, ka tatauhia te hapa paerewa penei:
\[ SE_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]
ko \( \sigma \) te paerewa rerekētanga o te taupori, ā, ko \( n \) te rahi o te tauira. E tohu ana tēnei tātai he iti ake te hapa paerewa o ngā tauira nui ake, ā, ka piki ake te tika o te toharite tauira hei whakatau tata mō te toharite taupori.
Te Āhua o te Tohatoha
Ko te āhua o te tohatoha tauira e whakawhirinaki ana ki te rahi o te tauira me te tohatoha taupori. E ai ki te Central Limit Theorem, ko ngā rahi tauira nui ake ka puta he tohatoha tauira e tata ana ki te tohatoha noa, ahakoa te tohatoha taketake o te taupori.
Ngā Momo Tohatoha Tauira
Te Tohatoha Tauira o te Toharite
Ko te tohatoha tauira o te toharite pea te tohatoha tauira e kitea nuitia ana. E tohu ana i te tohatoha o ngā toharite tauira i tangohia mai i ngā tauira katoa o tētahi rahi motuhake i tangohia mai i te taupori. He mea nui tēnei tohatoha mō te whakatau tata i ngā toharite taupori me te hanga i ngā wā whakawhirinaki.
Te Tohatoha Tauira o te Ōwehenga
I te wā e mahi ana ki ngā raraunga kāwai, ka whakamahia e ngā kaitātari tatauranga te ōwehenga tauira. Ko te tohatoha tauira o te ōwehenga ko te tohatoha o ngā ōwehenga tauira i whiwhihia mai i ngā tauira rerekē. He mea nui tēnei tohatoha ki te whakatau tata i ngā ōwehenga taupori me te whakamātautau whakapae mō ngā ōwehenga.
Tohatoha-T
Ina kore e mōhiotia te paerewa rerekētanga o te taupori (\( \sigma \)) ā, he iti te rahi o te tauira, ka whakamahia e ngā tātai tatauranga te tohatoha-t. He rite te tohatoha-t ki te tohatoha noa engari he matotoru ake ngā hiku, e whakaatu ana i te pikinga o te rerekētanga nā te iti o te rahi o te tauira. I te tipu haere o te rahi o te tauira, ka hono te tohatoha-t ki te tohatoha noa.
Te Tohatoha Chi-tapawhā
E whakamahia ana te tohatoha chi-square i roto i ngā whakamātautau tū motuhake me ngā whakamātautau pai-o-te-taurite. He mea nui anō hoki mō te hanga i ngā wā whakawhirinaki mō ngā rerekētanga me ngā paerewa rerekētanga.
Tohatoha-F
E whakamahia ana te tohatoha-F hei whakatairite i ngā rerekētanga me te tātari i te rerekētanga (ANOVA). I ahu mai i te ōwehenga o ngā tohatoha chi-square e rua, ā, ka āwhina i te whakatau mena he tino rerekē ngā rerekētanga o ngā taupori e rua.
Tauira: Te Mārama ki ngā Tohatoha Tauira me te Horopaki Ao Tūturu
Me ruku tātou ki tētahi tauira mahi hei whakau i tō tātou māramatanga. Whakaarohia e hiahia ana tātou ki te whakatau tata i te teitei toharite o ngā tāne pakeke i roto i tētahi tāone. Kāore e taea te ine i ia tangata, nō reira ka tīpakohia e tātou he tauira o te 100 tāne. Ka tatauhia e tātou te toharite tauira (\( \bar{x} \)) kia 68 inihi, ā, ko te paerewa rerekētanga tauira (s) kia 3 inihi.
Mena ka tangohia e tātou he tauira maha mai i tēnei taupori, ka rerekē paku te toharite o ngā tauira mō ia take, ka hangaia he tohatoha tauira o ngā toharite tauira. E ai ki te Central Limit Theorem, mena he nui te rahi o tā tātou tauira, ka tata noa tēnei tohatoha.
Mā te whakamahi i ngā raraunga tauira, ka taea e tātou te whakatau tata i te toharite o te taupori, ā, ka hangaia he āputa whakapono 95%. Mēnā kāore i te mōhiotia te paerewa rerekētanga o te taupori, ka whakamahia e tātou te tohatoha-t. Ko te hapa paerewa ko:
\[ SE_{\bar{x}} = \frac{s}{\sqrt{n}} = \frac{3}{\sqrt{100}} = 0.3 \]
Me te 99 nekehanga herekore (n-1), ko te uara matua mai i te ripanga tohatoha-t mō te taumata whakapono 95% he tata ki te 1.984. Nō reira, ko te tawhā hapa (ME) ko:
\[ ME = 1.984 \whakareatia ki te 0.3 = 0.5952 \]
Nō reira, ko te āputa whakapono 95% mō te toharite o te taupori ko:
\[ (68 – 0.5952, 68 + 0.5952) = (67.4048, 68.5952) \]
Nō reira, e 95% tō mātou whakapono ko te teitei toharite o ngā tāne pakeke katoa o te tāone kei waenga i te 67.4048 inihi me te 68.5952 inihi.
Opaniraa
Ko ngā tohatoha tauira te tūāpapa o te whakatau tatauranga, e āhei ai tātou ki te whakatau whakaaro arorau mō ngā tawhā taupori i runga i ngā raraunga tauira. He mea nui te mārama ki ō rātou āhuatanga, momo, me ngā tono mō tetahi kaitātari raraunga, kairangahau rānei. Mai i te whakamātautau whakapae ki te hanga i ngā wā whakawhirinaki, he taputapu tino nui ngā tohatoha tauira mō te whakamaori me te tango māramatanga whai tikanga mai i ngā raraunga.
