Cov Lus Nug Piv Txwv thiab Kev Sib Tham Txog Kev Ua Haujlwm Faib Tawm Ib Txwm
Qhov kev faib tawm ib txwm muaj, tseem hu ua Gaussian kev faib tawm, yog ib qho ntawm cov kev faib tawm qhov muaj feem cuam tshuam tseem ceeb tshaj plaws hauv kev suav lej thiab kev tshuaj xyuas cov ntaub ntawv. Qhov kev faib tawm no yog tus cwj pwm los ntawm cov kab nkhaus symmetrical nyob ib puag ncig qhov nruab nrab, nrog rau cov ntaub ntawv kis tau qhia txog qhov sib txawv ntawm cov nqi nyob ib puag ncig nws. Qhov kev faib tawm ib txwm muaj yog lub hauv paus rau ntau lub tswv yim hauv kev suav lej thiab siv dav hauv ntau qhov chaw, suav nrog kev lag luam, kev puas siab puas ntsws, thiab kev sib raug zoo hauv zej zog.
Hauv tsab xov xwm no, peb yuav tham txog qee qhov teeb meem piv txwv thiab lawv cov kev daws teeb meem kom nkag siab zoo dua txog kev faib tawm ib txwm muaj.
Cov Ntsiab Lus Tseem Ceeb ntawm Kev Faib Tawm Ib Txwm
Qhov kev faib tawm ib txwm muaj yog piav qhia los ntawm ob qho kev ntsuas tseem ceeb:
1. Nruab Nrab (μ): Qhov nruab nrab ntawm cov ntaub ntawv teeb tsa.
2. Kev Sib Txawv Txheem (σ): Ntsuas seb cov ntaub ntawv nthuav dav li cas nyob ib puag ncig qhov nruab nrab.
Qhov kev ua haujlwm ntawm qhov muaj feem cuam tshuam ntawm kev faib tawm ib txwm yog:
\[ f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} } \]
Nov yog qee cov kauj ruam yooj yim hauv kev daws cov teeb meem siv kev faib tawm ib txwm muaj:
1. Txheeb xyuas tus nqi Z: Tus nqi Z yog qhov ntsuas seb cov ntaub ntawv nyob deb npaum li cas ntawm qhov nruab nrab hauv cov chav ntsuas kev hloov pauv tus qauv thiab suav nrog siv cov mis:
\[Z = \frac{X – \mu}{\sigma} \]
2. Siv lub rooj Z: Lub rooj Z lossis lub rooj faib tawm ib txwm yog siv los nrhiav qhov feem pua lossis feem pua ntawm cov ntaub ntawv uas qis dua lossis siab dua tus nqi Z.
Piv txwv cov lus nug thiab kev sib tham txog kev faib tawm ib txwm muaj
Lo lus nug thib 1
Ib chav kawm muaj qhov qhab nia xeem lej nruab nrab ntawm 70 nrog rau qhov sib txawv ntawm 10. Yog tias cov qhab nia xeem raug faib ua feem ntau, muaj pes tsawg feem pua ntawm cov tub ntxhais kawm tau qhab nia ntau dua 85?
Kev Sib Tham:
1. Txheeb xyuas qhov qhab nia Z: Ua ntej, xam qhov qhab nia Z rau X = 85.
\[ Z = \frac{X – \mu}{\sigma} = \frac{85 – 70}{10} = 1.5 \]
2. Saib ntawm Z Rooj: Peb nrhiav tus nqi feem pua rau Z = 1.5 los ntawm Z rooj. Tus nqi feem pua rau Z = 1.5 yog 0.9332. Qhov no txhais tau tias 93.32% ntawm cov nqi yog qis dua Z = 1.5.
3. Xam Feem Pua: Vim peb xav tau feem pua ntawm cov tub ntxhais kawm uas tau qhab nia ntau dua 85, peb xam 1 - 0.9332 = 0.0668.
Yog li ntawd, 6.68% ntawm cov tub ntxhais kawm tau qhab nia ntau dua 85.
Lo lus nug thib 2
Qhov siab ntawm cov txiv neej laus hauv ib lub teb chaws ua raws li kev faib tawm ib txwm muaj nrog qhov nruab nrab ntawm 175 cm thiab qhov sib txawv ntawm 6 cm. Txheeb xyuas feem pua ntawm cov txiv neej uas muaj qhov siab ntawm 170 cm thiab 180 cm.
Kev Sib Tham:
1. Txheeb xyuas qhov qhab nia Z rau 170 cm:
\[ Z_{170} = \frac{170 – 175}{6} = – \frac{5}{6} \approx -0.83 \]
2. Txheeb xyuas qhov qhab nia Z rau 180 cm:
\[ Z_{180} = \frac{180 – 175}{6} \kwv yees li 0.83 \]
3. Saib Rooj Z:
– Qhov feem pua rau Z = -0.83 yog 0.2033.
- Qhov feem pua rau Z = 0.83 yog 0.7967.
4. Xam Feem Pua:
– Qhov feem pua ntawm qhov siab ntawm 170 cm thiab 180 cm yog 0.7967 – 0.2033 = 0.5934.
- Yog li, 59.34% ntawm cov txiv neej muaj qhov siab ntawm 170 cm thiab 180 cm.
Lo lus nug thib 3
Ib qho kev xeem IQ siv qhov kev faib tawm ib txwm muaj nrog qhov nruab nrab ntawm 100 thiab qhov sib txawv ntawm tus qauv ntawm 15. Cov qhab nia twg poob rau hauv 85th percentile?
Kev Sib Tham:
1. Nrhiav tus nqi Z rau 85th percentile: Los ntawm lub rooj Z lossis siv lub tshuab xam zauv, 85th percentile sib raug rau Z = 1.04.
2. Xam qhov qhab nia IQ:
\[ X = Z\sigma + \mu \]
\[ X = 1.04 \times 15 + 100 \]
\[ X = 15.6 + 100 \]
\[ X = 115.6 \]
Yog li, qhov qhab nia IQ uas poob rau hauv 85th feem pua yog nyob ib puag ncig 115.6.
Lo lus nug thib 4
Yog tias paub tias qhov qhab nia nruab nrab ntawm cov tub ntxhais kawm ntawv theem siab cov qhab nia xeem kev paub yog 65 nrog rau qhov sib txawv ntawm 12, qhov qhab nia twg nyob rau hauv 25th feem pua?
Kev Sib Tham:
1. Nrhiav tus nqi Z rau feem pua 25: Los ntawm lub rooj Z lossis siv lub tshuab xam zauv, Z rau feem pua 25 yog kwv yees li -0.674.
2. Xam cov qhab nia xeem:
\[ X = Z\sigma + \mu \]
\[ X = -0.674 \times 12 + 65 \]
\[ X = -8.088 + 65 \]
\[ X \kwv yees li 56.912 \]
Yog li, tus nqi uas nyob hauv 25th percentile yog nyob ib ncig ntawm 56.912.
Xaus
Qhov kev faib tawm ib txwm yog lub tswv yim tseem ceeb hauv kev suav lej uas tso cai rau peb los tshuaj xyuas thiab nkag siab cov ntaub ntawv los ntawm qhov pom ntawm qhov muaj feem yuav tshwm sim. Siv txoj hauv kev faib tawm ib txwm, peb tuaj yeem xam feem pua, txiav txim siab cov nqi tshwj xeeb raws li feem pua, thiab piv cov ntaub ntawv rau qhov nruab nrab.
Kev daws cov teeb meem nrog kev faib tawm ib txwm tsis yog tsuas yog muaj txiaj ntsig zoo rau kev xeem thiab kev tshawb fawb hauv kev kawm xwb, tab sis kuj muaj kev siv tau hauv ntau qhov chaw hauv lub neej tiag tiag xws li kev puas siab puas ntsws, kev lag luam, thiab kev sib raug zoo hauv zej zog. Los ntawm cov piv txwv thiab kev sib tham saum toj no, peb vam tias koj yuav nkag siab zoo dua txog kev ua haujlwm faib tawm ib txwm thiab yuav siv nws li cas hauv ntau qhov xwm txheej.
