Tauira Pātai me te Kōrero mō te Mahi Tohatoha Noa
Ko te tohatoha noa, e mōhiotia ana ko te tohatoha Gaussian, tētahi o ngā tohatoha tūponotanga tino taketake i roto i ngā tatauranga me te tātari raraunga. Ko tēnei tohatoha he āhua pere ōrite e karapoti ana i te toharite, me te horapa raraunga e whakaata ana i te paerewa rerekētanga o ngā uara e karapoti ana. Ko te tohatoha noa te pūtake mō ngā ariā maha i roto i ngā tatauranga whakatau, ā, e whakamahia whānuitia ana i roto i ngā momo mara, tae atu ki te ōhanga, te hinengaro, me ngā pūtaiao pāpori.
I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira raruraru me ā rātou otinga kia pai ake ai te mārama ki te mahi tohatoha noa.
Ngā Ariā Taketake o te Tohatoha Noa
E rua ngā tawhā matua e whakaahuatia ana te tohatoha noa:
1. Toharite (μ): Te toharite o te huinga raraunga.
2. Paerewa Rerekētanga (σ): E ine ana i te horapa o ngā raraunga huri noa i te toharite.
Ko te mahi tūponotanga kiato o te tohatoha noa ko:
\[ f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} } \]
Anei ētahi mahi matua mō te whakaoti rapanga mā te whakamahi i te tohatoha noa:
1. Te whakatau i te uara Z: Ko te uara Z he ine i te tawhiti o ngā raraunga mai i te toharite i roto i ngā waeine paerewa rerekētanga, ā, ka tatauhia mā te whakamahi i te tātai:
\[ Z = \frac{X – \mu}{\sigma} \]
2. Te Whakamahi i te Ripanga Z: Ka whakamahia te ripanga Z, te ripanga tohatoha noa paerewa rānei, hei kimi i te tūponotanga, i te ōrau rānei o ngā raraunga kei raro iho, kei runga ake rānei i tētahi uara Z.
Ngā Tauira Pātai me te Kōrero mō te Tohatoha Noa
Pātai 1
Ko te toharite o te kaute pāngarau o tētahi akomanga he 70, ā, ko te paerewa rerekētanga he 10. Mēnā he tohatoha noa ngā kaute whakamātautau, e hia ōrau o ngā ākonga i whiwhi i te neke atu i te 85?
Kōrero:
1. Te whakatau i te kaute-Z: Tuatahi, tatauhia te kaute-Z mō X = 85.
\[ Z = \frac{X – \mu}{\sigma} = \frac{85 – 70}{10} = 1.5 \]
2. Te Tirohanga ki te Ripanga Z: Ka rapua e mātou te uara tūponotanga mō Z = 1.5 mai i te ripanga Z. Ko te uara tūponotanga mō Z = 1.5 ko 0.9332. Ko te tikanga o tēnei ko te 93.32% o ngā uara kei raro iho i te Z = 1.5.
3. Te Tatau i te Ōrau: Nā te mea e hiahiatia ana te ōrau o ngā ākonga i whiwhi i te neke atu i te 85, ka tatauhia e mātou te 1 – 0.9332 = 0.0668.
Nō reira, 6.68% o ngā ākonga i whiwhi i te neke atu i te 85.
Pātai 2
Ko te teitei o ngā tāne pakeke i tētahi whenua e whai ana i tētahi tohatoha noa me te toharite o te 175 cm me te paerewa rerekētanga o te 6 cm. Tātaihia te ōrau o ngā tāne kei waenganui i te 170 cm me te 180 cm te teitei.
Kōrero:
1. Whakatauhia te kaute-Z mō te 170 cm:
\[ Z_{170} = \frac{170 – 175}{6} = – \frac{5}{6} \approx -0.83 \]
2. Whakatauhia te kaute-Z mō te 180 cm:
\[ Z_{180} = \frac{180 – 175}{6} \approx 0.83 \]
3. Tirohia te Ripanga Z:
– Ko te tūponotanga mō Z = -0.83 ko 0.2033.
– Ko te tūponotanga mō Z = 0.83 ko 0.7967.
4. Te Tatau i te Ōrau:
– Ko te tūponotanga o te teitei i waenga i te 170 cm me te 180 cm ko 0.7967 – 0.2033 = 0.5934.
– Nō reira, 59.34% o ngā tāne he 170 henimita me te 180 henimita te roa.
Pātai 3
He whakamātautau IQ e whakamahi ana i te tohatoha noa me te toharite o te 100 me te paerewa rerekētanga o te 15. He aha te kaute kei roto i te 85 ōrau?
Kōrero:
1. Te kimi i te uara Z mō te ōrau 85: Mai i te ripanga Z, mā te whakamahi rānei i tētahi tātaitai, ka rite te ōrau 85 ki a Z = 1.04.
2. Te tatau i te kaute IQ:
\[ X = Z\sigma + \mu \]
\[ X = 1.04 \whakareatia ki te 15 + 100 \]
\[ X = 15.6 + 100 \]
\[ X = 115.6 \]
Nō reira, ko te kaute IQ kei roto i te ōrau 85 kei te 115.6 pea.
Pātai 4
Mena e mōhiotia ana ko te kaute toharite o ngā hua whakamātautau hinengaro o ngā ākonga kura tuarua he 65 me te paerewa rerekētanga o te 12, he aha te kaute kei roto i te 25 ōrau?
Kōrero:
1. Te kimi i te uara Z mō te ōrau 25: Mai i te ripanga Z, mā te whakamahi rānei i tētahi tātaitai, ko te Z mō te ōrau 25 he tata ki te -0.674.
2. Te tatau i ngā kaute whakamātautau:
\[ X = Z\sigma + \mu \]
\[ X = -0.674 \whakareatia ki te 12 + 65 \]
\[ X = -8.088 + 65 \]
\[ X \tata ki te 56.912 \]
Nō reira, ko te uara kei roto i te 25 ōrau kei te 56.912 pea.
Whakamutunga
He ariā nui te tohatoha noa i roto i ngā tatauranga e āhei ai tātou ki te tātari me te mārama ki ngā raraunga mai i te tirohanga tūponotanga. Mā te whakamahi i te huarahi tohatoha noa, ka taea e tātou te tatau i ngā ōrau, te whakatau i ngā uara motuhake i runga i ngā ōrau, me te whakarite i ngā raraunga ki te toharite.
Ehara i te mea he mea whai hua te whakaoti rapanga me te tohatoha noa mō ngā whakamātautau me ngā rangahau mātauranga anake, engari he tono mahi anō hoki i roto i ngā mara tūturu pērā i te hinengaro, te pakihi, me ngā pūtaiao pāpori. Mā roto i ngā tauira me ngā kōrero i runga ake nei, ko te tumanako ka pai ake tō māramatanga ki te mahi tohatoha noa me pēhea te whakamahi i roto i ngā horopaki rerekē.
