Tauira o tētahi Pātai Kōrero mō te Tohatoha Noa
Ko te tohatoha noa, e mōhiotia ana ko te tohatoha Gaussian, te tohatoha tūponotanga e whakamahia whānuitia ana i roto i ngā tatauranga. He āhua pere ōrite tēnei tohatoha, e tohu ana kei te whakaritehia ngā raraunga huri noa i te toharite, ā, he iti te tūponotanga o ngā pito rawa (ngā uara e tawhiti atu ana i te toharite).
I roto i tēnei tuhinga, ka matapakihia e mātou ngā tauira raruraru e pā ana ki te tohatoha noa me pēhea te whakaoti i aua raruraru. Ka tīmata mātou mā te whakauru i ētahi ariā taketake, kātahi ka neke atu ki ngā tauira uaua ake.
Ngā Kaupapa Taketake o te Tohatoha Noa
He tohatoha tonu te tohatoha noa me ngā tawhā e rua: te toharite me te paerewa rerekētanga (SD). Ko te toharite te mea e whakatau ana i te pokapū o te tohatoha, ko te paerewa rerekētanga te mea e whakatau ana i te whānui o te tohatoha.
Ngā āhuatanga nui o te tohatoha noa:
1. Āhua ōrite: He ōrite te tohatoha noa ki te toharite.
2. Ture Whakamātautau (Ture Whakamātautau):
– E tata ana ki te 68% o ngā raraunga kei roto i te kotahi paerewa rerekētanga o te toharite.
– E tata ana ki te 95% o ngā raraunga kei roto i ngā paerewa rerekētanga e rua o te toharite.
– E tata ana ki te 99.7% o ngā raraunga kei roto i ngā paerewa rerekētanga e toru o te toharite.
Ngā Pātai Tauira me te Kōrero
Tauira Pātai 1: Te Tatau i te Kaute-Z
Pātai: Ko te toharite o te kaute mō tētahi whakamātautau he 70, ā, ko te paerewa rerekētanga he 10. Ka whiwhi te tauira i te kaute he 80. He aha te kaute-Z a te tauira?
Otinga:
Ko te kaute-Z he ine mō te maha o ngā paerewa rerekētanga o tētahi uara mai i te toharite.
Tātai kaute-Z:
\[ Z = \frac{X – \mu}{\sigma} \]
Dimana:
– Ko te uara i kitea ko \( X \).
– Ko te toharite te \( \mu \).
– Ko te paerewa rerekētanga ko \( \sigma \).
E mōhiotia ana:
– \( X = 80 \)
– \( \mu = 70 \)
– \( \tau = 10 \)
Te whakamahinga o te tātai:
\[ Z = \frac{80 – 70}{10} = 1 \]
Nō reira, ko te kaute-Z a te ākonga he 1, ko te tikanga ko te kaute 80 he kotahi te paerewa rerekētanga kei runga ake i te toharite.
Tauira Pātai 2: Te Tūponotanga o tētahi Uara
Pātai: I roto i tētahi tohatoha noa me te toharite o te 100 me te paerewa rerekētanga o te 15, he aha te tūponotanga o te kitea o tētahi uara i raro i te 85?
Otinga:
Ngā kaupae:
1. Tātaihia te kaute-Z mō te uara \( X = 85 \):
\[ Z = \frac{85 – 100}{15} = \frac{-15}{15} = -1 \]
2. Whakamahia he ripanga-Z, he tātaitai tatauranga rānei hei kimi i te tūponotanga e rite ana ki te kaute-Z o te -1. I roto i te ripanga-Z, ko te tūponotanga o te kaute-Z o te -1 he tata ki te 0.1587.
Nō reira, ko te tūponotanga o te kitea o tētahi uara i raro i te 85 he 0.1587, arā, 15.87%.
Tauira Pātai 3: Te Whakamahi i ngā Ture Pūtaiao
Pātai: E mōhiotia ana ko te tohatoha o ngā kaute whakamātautau pāngarau i roto i ngā kura e whai ana i te tohatoha noa me te toharite o te 75 me te paerewa rerekētanga o te 8. He aha te ōwehenga o ngā ākonga i whiwhi i waenga i te 67 me te 83?
Otinga:
Ngā Hipanga:
1. Tātaihia te kaute-Z mō ngā uara 67 me te 83:
\[ Z_{67} = \frac{67 – 75}{8} = \frac{-8}{8} = -1 \]
\[ Z_{83} = \frac{83 – 75}{8} = \frac{8}{8} = 1 \]
2. E ai ki ngā ture whakamātautau, ko ngā uara i waenga i te -1 SD me te +1 SD mai i te toharite e kapi ana i te 68% o te taupori.
Nō reira, ko te ōwehenga o ngā ākonga i whiwhi i waenga i te 67 me te 83 he tata ki te 68%.
Tauira Pātai 4: Te Tatau i ngā Uara mai i ngā Ōrau
Pātai: Mena ko te teitei toharite o ngā tāne pakeke i roto i tētahi whenua he 175 cm, ā, ko te paerewa rerekētanga he 7 cm, he aha te teitei kei te 90 ōrau?
Otinga:
Ngā Hipanga:
1. Kimihia te kaute-Z e rite ana ki te ōrau 90. I runga i te ripanga-Z, ko te kaute-Z e tata ana ki te 0.9000 he tata ki te 1.28.
2. Whakamahia te tātai hei tatau i te uara o \( X \):
\[ X = \mu + Z \times \sigma \]
\[ X = 175 + 1.28 \whakanuia te 7 \]
\[ X = 175 + 8.96 \]
\[ X = 183.96 \]
Nō reira, ko te teitei i te ōrau 90 kei te 183.96 henemita te roa.
Tauira Pātai 5: Te Tūponotanga o tētahi Wā
Pātai: I te mea he tohatoha noa te tohatoha o ngā taumaha o te pēpi hou me te toharite o te 3.5 kg me te paerewa rerekētanga o te 0.5 kg, he aha te tūponotanga ka taumaha te pēpi i waenga i te 3 kg me te 4 kg?
Otinga:
Ngā Hipanga:
1. Tātaihia te kaute-Z mō ngā uara 3 kg me te 4 kg:
\[ Z_{3} = \frac{3 – 3.5}{0.5} = \frac{-0.5}{0.5} = -1 \]
\[ Z_{4} = \frac{4 – 3.5}{0.5} = \frac{0.5}{0.5} = 1 \]
2. Ko te tūponotanga mō te kaute-Z i waenga i te -1 me te 1 i runga i te ripanga Z he tata ki te 0.6826, arā, 68.26%.
Nō reira, ko te tūponotanga o te taumaha o tētahi pēpi i waenga i te 3 kg me te 4 kg he tata ki te 68.26%.
Whakamutunga
He ariā taketake te tohatoha noa i roto i ngā tatauranga, he mea nui, ā, he maha ngā whakamahinga o te ao tūturu. I roto i tēnei tuhinga, kua whakamāramahia e mātou ngā ariā taketake o te tohatoha noa, ā, kua whakaotihia ētahi tauira hei whakahōhonu ake i tō mātou māramatanga.
Ehara i te mea he mea nui te mārama ki te tohatoha noa mō ngā tatauranga anake, engari mō ngā momo mara mahi anō hoki pērā i te hinengaro, te ōhanga, me ētahi atu pūtaiao pāpori. Mēnā he nui te mahi, ka māmā ake te whakaoti rapanga tohatoha noa, ā, ka āwhina i te whakatau kaupapa e ahu mai ana i ngā raraunga.
