Ngā Tauira Pātai e Matapaki ana i te Rerekētanga me te Paerewa Rerekētanga o ngā Raraunga Rōpū
Pendahuluan
I roto i ngā tatauranga, ko te rerekētanga me te paerewa rerekētanga he ine tatauranga e rua e tino hira ana mō te mārama ki te horapa, te horapa rānei, o ngā raraunga mai i te toharite. Ka ine te rerekētanga i te tawhiti o te horapa o ngā raraunga mai i te toharite, ko te paerewa rerekētanga ia ko te pūtake tapawhā o te rerekētanga, e whakarato ana i te ine e rite ana ki ngā waeine o ngā raraunga taketake.
Whakamāramatanga
– Rerekētanga (σ², S² rānei): Ko te toharite o ngā tapawhā o ngā rerekētanga i waenga i ia uara raraunga me te toharite o ngā raraunga.
– Te Paerewa Rerekētanga (σ, S rānei): Ko te pūtake tapawhā o te rerekētanga.
Tātai mō te Rerekētanga me te Paerewa Rerekētanga o ngā Raraunga Rōpū
Mō ngā raraunga rōpū, ka whakamahia e mātou te auau o ngā raraunga i roto i ia akomanga. Anei te tātai:
Ngā momo rerekē
\[ S^2 = \frac{ \tapeke f_i \left( x_i – \bar{x} \right)^2 }{ N-1 } \]
Paerewa Rerekētanga
\[ S = \sqrt{S^2} \]
kāore i te mana:
– \( f_i \) = te auau o ia akomanga.
– \( x_i \) = waenganui o ia akomanga.
– \( \bar{x} \) = te toharite o ngā raraunga rōpū.
– \( N \) = te tapeke o ngā raraunga.
Ngā Pātai Tauira me te Kōrero
Me kī kei a tātou ngā raraunga taumaha mō tētahi rōpū tāngata kua whakarōpūtia ki ngā akomanga.
| Wā Taumaha (kg) | Auautanga (f) |
|————————|—————–|
| 50 – 54 | 2 |
| 55 – 59 | 5 |
| 60 – 64 | 8 |
| 65 – 69 | 7 |
| 70 – 74 | 3 |
Ko te taahiraa tuatahi ko te whakatau i te pūwāhi waenga o ia akomanga ( \( x_i \) ) kātahi ka tatau i te toharite (\( \bar{x} \)).
1. Te Tātai i te Waenganui ( \( x_i \) )
\[ \text{Waenganui} = \frac{\text{Tepe o raro} + \text{Tepe o runga}}{2} \]
| Wā Taumaha (kg) | Auautanga (f) | Waenganui ( \( x_i \) ) |
|———————|—————–|————————|
| 50 – 54 | 2 | 52 |
| 55 – 59 | 5 | 57 |
| 60 – 64 | 8 | 62 |
| 65 – 69 | 7 | 67 |
| 70 – 74 | 3 | 72 |
2. Te Tātai i te Toharite ( \( \bar{x} \) )
\[ \bar{x} = \frac{ \tapeke f_i x_i }{ N } \]
Te tapeke o ngā raraunga \( N \):
\[ N = 2 + 5 + 8 + 7 + 3 = 25 \]
\[ \tapeke f_i x_i = (2 \times 52) + (5 \times 57) + (8 \times 62) + (7 \times 67) + (3 \times 72) \]
\[ = 104 + 285 + 496 + 469 + 216 = 1570 \]
Nō reira, ko te toharite (\( \bar{x} \)):
\[ \bar{x} = \frac{ 1570 }{ 25 } = 62.8 \]
3. Te Tātai i te Rerekētanga ( \( S^2 \) )
Me tatau tātou \( \sum f_i ( x_i – \bar{x} )^2 \):
\[
\begin{align}
(x_i – \bar{x})^2: & (52 – 62.8)^2 = 118.84
& (57 – 62.8)^2 = 33.64
& (62 – 62.8)^2 = 0.64
& (67 – 62.8)^2 = 17.64
& (72 – 62.8)^2 = 84.64
\end{whakahāngai}
\]
Te whakarea mā te auau:
\[
\begin{align}
f_i (x_i – \bar{x})^2: & 2 \times 118.84 = 237.68 \\
& 5 \whakareatia ki te 33.64 = 168.2 \\
& 8 \whakareatia ki te 0.64 = 5.12 \\
& 7 \whakareatia ki te 17.64 = 123.48 \\
& 3 \whakareatia 84.64 = 253.92
\end{whakahāngai}
\]
\[
\tapeke f_i (x_i – \bar{x})^2 = 237.68 + 168.2 + 5.12 + 123.48 + 253.92 = 788.4
\]
Ka taea e tātou te tatau i te rerekētanga (\( S^2 \)):
\[ S^2 = \frac{ 788.4 }{ 25 – 1 } = \frac{ 788.4 }{ 24 } \approx 32.85 \]
4. Te Tātai i te Paerewa Rerekētanga ( \( S \) )
Paerewa rerekētanga ( \( S \)):
\[ S = \sqrt{ S^2 } \]
\[ S = \sqrt{ 32.85 } \approx 5.73 \]
Whakamutunga
Mai i ngā raraunga tauira i runga ake nei, kei a tātou:
– Te tatau i te taumaha tinana toharite: 62.8 kg
– Te tatau i te rerekētanga: 32.85 kg²
– Te tatau i te paerewa rerekētanga: 5.73 kg
Ko te whakamāramatanga o te paerewa rerekētanga ko te toharite rerekētanga o ngā raraunga taumaha mai i te toharite he tata ki te 5.73 kg. E tohu ana tēnei i te horapa o ngā raraunga e pā ana ki te toharite, ā, ka taea te whakatau i te rerekētanga o ā tātou raraunga.
He mea nui te māramatanga hōhonu ki te rerekētanga me te paerewa rerekētanga, inā koa mō te hunga e mahi ana i roto i ngā tatauranga, rangahau, me te whakamātautau, i a rātou e ngana ana ki te mārama ki ngā raraunga i roto i te āhua o ngā rōpū, o ngā tohatoha rānei. Mā te mōhio ki te tatau me te whakamārama i ēnei inenga e rua ka āwhina i te whakatau pai ake i runga i ngā raraunga kei te ringaringa.
