Ngā Tauira Pātai mō te Tātari Raraunga me te Kōrero Whai Wāhitanga
Ko te tātari raraunga me te tūponotanga he wāhanga e rua e kitea pinepine ana i roto i ngā momo marautanga, inā koa ko ngā tatauranga, te pāngarau, te ōhanga, me te rangahau mākete. I roto i tēnei tuhinga, ka tūhuratia e mātou ētahi tauira raruraru me ngā matapakinga e pā ana ki te tātari raraunga me te tūponotanga hei whakahōhonu ake i tō mātou māramatanga ki ēnei ariā taketake.
1. Tātari Raraunga: Kupu Whakataki me ngā Tauira Pātai
Ko te tātari raraunga te tukanga tirotiro, whiriwhiri, whakarerekē, me te whakatauira i ngā raraunga me te whāinga ki te kimi i ngā mōhiohio whai hua, ki te whakatau whakatau, me te tautoko i te whakatau kaupapa. Ko ngā mahi noa ko te kohikohi raraunga, te horoi raraunga, te tūhura raraunga (ngā tatauranga whakaahua), me te tātari anō.
Tauira Pātai 1: Te Whakatau i te Toharite me te Paerewa Ine
E whai ake nei ngā raraunga e pā ana ki ngā kaute pāngarau o te 10 ākonga: 78, 82, 85, 88, 90, 75, 91, 74, 89, 86.
Tātaihia te toharite me te paerewa rerekētanga o ngā raraunga.
Kōrero:
– Ko te uara toharite ko te tapeke o ngā raraunga katoa ka wehea ki te maha o ngā kitenga.
\[
\text{Toharite} = \frac{78 + 82 + 85 + 88 + 90 + 75 + 91 + 74 + 89 + 86}{10} = \frac{838}{10} = 83.8
\]
– Hei tatau i te paerewa rerekētanga, me tatau tuatahi tātou i te rerekētanga. Ko te rerekētanga te toharite o ngā rerekētanga tapawhā i waenga i ia huinga raraunga me te toharite.
\[
\text{Rerekētanga} = \frac{(78-83.8)^2 + (82-83.8)^2 + (85-83.8)^2 + \cdots + (86-83.8)^2}{10}
\]
Kātahi ka tatauhia te rerekētanga penei:
\[
\text{Rerekētanga} = \frac{33.64 + 3.24 + 1.44 + 17.64 + 38.44 + 75.04 + 50.41 + 94.44 + 26.01 + 4.84}{10} = \frac{345.14}{10} = 34.514
\]
– Ko te paerewa rerekētanga te pūtake tapawhā o te rerekētanga.
\[
\text{Paerewa Rerekētanga} = \sqrt{34.514} ≈ 5.88
\]
Tauira Pātai 2: Kauwhata Raraunga
E whai ake nei (i roto i ngā mano) ngā raraunga taupori o tētahi tāone mō ngā tau e 5 kua hipa: 2016: 120, 2017: 125, 2018: 130, 2019: 135, 2020: 140.
Waihangatia he kauwhata rārangi hei whakaatu i ngā raraunga.
Kōrero:
Hei waihanga i tētahi kauwhata rārangi, ka taea e tātou te whai i ēnei mahi:
1. Tāutuhia ngā tuaka X me te tuaka Y. Ko te tuaka X mō ngā tau, ā, ko te tuaka Y mō te taupori.
2. Tuhia ngā tohu i runga i ngā raraunga kua hoatu:
– (2016, 120)
– (2017, 125)
– (2018, 130)
– (2019, 135)
– (2020, 140)
3. Honoa ngā ira ki ngā rārangi.
Mā te kauwhata ka puta te ia o te pikinga o te taupori tāone nui mai i te tau 2016 ki te tau 2020.
2. Tūponotanga: Ngā Kaupapa Taketake me ngā Tauira Rapanga
Ko te tūponotanga, te tūponotanga rānei, he ine i te tūponotanga o te puta o tētahi huihuinga. Ko te tūponotanga o tētahi huihuinga A ka whakaaturia i te nuinga o te wā ko \( P(A) \) ā, ka tatauhia penei:
\[
P(A) = \frac{\text{Tau o ngā takahanga e hiahiatia ana}}{\text{Tau katoa o ngā takahanga ka taea}}
\]
Tauira Pātai 3: Tūponotanga Māmā
Mai i tētahi kete kāri tākaro paerewa, he aha te tūponotanga o te kohi i tētahi Āce?
Kōrero:
– E 52 ngā kāri i roto i te huinga kāri tākaro kotahi.
– E whā ngā kāri Āce (ngākau, taimana, karapu, me ngā hāpara).
– Ko te tūponotanga o te tānga o tētahi Āce ko:
\[
P(\text{As}) = \frac{4}{52} = \frac{1}{13} \approx 0.0769 \text{ or } 7.69\%
\]
Tauira Pātai 4: Ngā Whakamātautau Whai muri
E rua ngā mataono ka whiua i te wā kotahi. Kimihia te tūponotanga ko te tapeke o ngā mataono e rua he 7.
Kōrero:
– E 6×6=36 ngā putanga katoa e taea ana mai i ngā mataono e rua.
– Ko ngā huinga e hua ake ai te 7 katoa ko: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Nō reira, e 6 ngā wā e puta mai ana.
– Ko te tūponotanga ko te tapeke o ngā mataono e rua he 7 koia tēnei:
\[
P(\kākau{Tapeke 7}) = \frac{6}{36} = \frac{1}{6} \approx 0.1667 \kākau{ or } 16.67\%
\]
Tauira Pātai 5: Ture a Bayes
I te mea kotahi i roto i te 1000 tāngata he mate X. Ko te tika o tētahi whakamātautau hei kite i te mate X he 99% (ahakoa he pai, he kino rānei). Mena he pai te whakamātautau a te tangata, he aha te tūponotanga kei a ia te mate X? Me kī he 95% te tairongo o te whakamātautau, ā, he 98% te motuhake.
Kōrero:
Hei tauira:
– Ko te P(P) te tūponotanga ka pai te whakamātautau a te tangata,
– Ko te P(D) te tūponotanga ka pāngia te tangata e te mate,
– Ko te P(P|D) te tūponotanga o te whakamātautau pai mēnā he mate tō te tangata,
– Ko te P(D|P) te tūponotanga o te pangia e te mate mēnā he pai te whakamātautau a te tangata.
E ai ki te ariā a Bayes:
\[
P(D|P) = \frac{P(P|D) \cdot P(D)}{P(P)}
\]
Tātaihia tuatahi te P(P):
\[
P(P) = P(P|D) \cdot P(D) + P(P|D^c) \cdot P(D^c)
\]
\[
P(P) = 0.95 \cdot \frac{1}{1000} + 0.02 \cdot \frac{999}{1000} \approx 0.0211
\]
Inaianei,
\[
P(D|P) = \frac{0.95 \cdot \frac{1}{1000}}{0.0211} ≈ 0.045 \text{ 4.5% rānei}
\]
Nō reira, ki te pai te whakamātautau a te tangata, ko te tūponotanga kei te pāngia ia e te mate X he tata ki te 4.5%.
Whakamutunga
He taputapu nui te tātari raraunga me te tātari tūponotanga i roto i ngā momo mara pēnei i te rangahau, te ōhanga, me te hauora. Mā te mārama ki tā rātou mahi, ka taea e tātou te whakatau pai ake, te whakatau pai ake, me te whakahaere i ngā mōrearea. Mā roto i ēnei tauira me ngā kōrero, ka pai ake te mārama ki te tātari raraunga me te tatau i ngā tūponotanga o ngā kaupapa rerekē. Ko te tumanako kua whai hua tēnei tuhinga ki te whakahōhonu ake i tō tātou mārama ki te tātari raraunga me te tātari tūponotanga.
