Piv txwv ntawm Cov Lus Nug Sib Tham txog Kev Faib Tawm Ib Txwm
Qhov kev faib tawm ib txwm muaj, tseem hu ua Gaussian distribution, yog qhov kev faib tawm qhov muaj feem ntau siv hauv kev suav lej. Qhov kev faib tawm no muaj lub tswb zoo li lub tswb, qhia tias cov ntaub ntawv tau teeb tsa ib puag ncig qhov nruab nrab thiab qhov muaj feem ntau ntawm qhov kawg (tus nqi deb ntawm qhov nruab nrab) yog qis.
Hauv tsab xov xwm no, peb yuav tham txog ntau yam teeb meem piv txwv uas cuam tshuam nrog kev faib tawm ib txwm muaj thiab yuav daws lawv li cas. Peb yuav pib los ntawm kev qhia qee lub ntsiab lus yooj yim thiab tom qab ntawd txav mus rau cov piv txwv nyuaj dua.
Cov Ntsiab Lus ntawm Kev Faib Tawm Ib Txwm
Qhov kev faib tawm ib txwm muaj yog kev faib tawm tas mus li nrog ob qho kev ntsuas: qhov nruab nrab thiab qhov kev hloov pauv tus qauv (SD). Qhov nruab nrab txiav txim siab qhov chaw nruab nrab ntawm kev faib tawm, thaum qhov kev hloov pauv tus qauv txiav txim siab qhov dav ntawm kev faib tawm.
Cov yam ntxwv tseem ceeb ntawm kev faib tawm ib txwm muaj:
1. Symmetry: Qhov kev faib tawm ib txwm muaj yog symmetrical txog qhov nruab nrab.
2. Txoj Cai Siv Los Ntawm Kev Paub (Txoj Cai Siv Los Ntawm Kev Paub):
- Kwv yees li ntawm 68% ntawm cov ntaub ntawv nyob hauv ib qho kev hloov pauv ntawm qhov nruab nrab.
- Kwv yees li ntawm 95% ntawm cov ntaub ntawv nyob hauv ob qhov kev hloov pauv ntawm tus qauv nruab nrab.
- Kwv yees li ntawm 99.7% ntawm cov ntaub ntawv nyob hauv peb qhov kev hloov pauv ntawm tus qauv nruab nrab.
Cov Lus Nug Piv Txwv thiab Kev Sib Tham
Piv txwv lus nug 1: Xam tus Z-Score
Lo lus nug: Ib qho kev xeem muaj qhov qhab nia nruab nrab ntawm 70 nrog rau qhov sib txawv ntawm 10. Ib tug tub ntxhais kawm tau qhab nia ntawm 80. Tus tub ntxhais kawm qhov qhab nia Z yog dab tsi?
Kev daws teeb meem:
Tus qhab nia Z yog qhov ntsuas ntawm pes tsawg tus qauv sib txawv ntawm tus nqi los ntawm qhov nruab nrab.
Cov qauv Z-qhab nia:
\[Z = \frac{X – \mu}{\sigma} \]
Qhov twg:
- \( X \) yog tus nqi uas pom.
– \( \mu \) yog qhov nruab nrab.
- \( \sigma \) yog qhov sib txawv ntawm tus qauv.
Nws paub tias:
– \( X = 80 \)
– \( \mu = 70 \)
– \( \sigma = 10 \)
Kev siv cov mis:
\[Z = \frac{80–70}{10} = 1 \]
Yog li, tus tub ntxhais kawm qhov Z-qhab nia yog 1, uas txhais tau hais tias qhov qhab nia ntawm 80 yog ib qho kev hloov pauv tus qauv saum toj no qhov nruab nrab.
Piv txwv lus nug 2: Qhov muaj feem yuav muaj ib qho nqi
Lus Nug: Hauv kev faib tawm ib txwm muaj nrog qhov nruab nrab ntawm 100 thiab qhov sib txawv ntawm 15, qhov muaj feem yuav nrhiav tau tus nqi hauv qab 85 yog dab tsi?
Kev daws teeb meem:
Cov kauj ruam:
1. Xam tus qhab nia Z rau tus nqi \( X = 85 \):
\[Z = \frac{85–100}{15} = \frac{-15}{15} = -1 \]
2. Siv lub rooj Z lossis lub tshuab xam zauv los nrhiav qhov feem pua uas sib raug rau qhov qhab nia Z ntawm -1. Hauv lub rooj Z, qhov feem pua ntawm qhov qhab nia Z ntawm -1 yog kwv yees li 0.1587.
Yog li, qhov muaj feem yuav nrhiav tau tus nqi qis dua 85 yog 0.1587 lossis 15.87%.
Piv txwv lus nug 3: Siv cov cai Empirical
Lo lus nug: Paub tias qhov kev faib tawm ntawm cov qhab nia xeem lej hauv tsev kawm ntawv ua raws li kev faib tawm ib txwm nrog qhov nruab nrab ntawm 75 thiab qhov sib txawv ntawm 8. Feem pua ntawm cov tub ntxhais kawm tau qhab nia ntawm 67 thiab 83 yog dab tsi?
Kev daws teeb meem:
Cov Kauj Ruam:
1. Xam qhov Z-qhab nia rau cov nqi 67 thiab 83:
\[Z_{67} = \frac{67–75}{8} = \frac{-8}{8} = -1 \]
\[Z_{83} = \frac{83–75}{8} = \frac{8}{8} = 1 \]
2. Raws li cov cai ntawm kev sim, cov nqi ntawm -1 SD thiab +1 SD los ntawm qhov nruab nrab npog txog li 68% ntawm cov pej xeem.
Yog li, feem pua ntawm cov tub ntxhais kawm uas tau qhab nia ntawm 67 thiab 83 yog li ntawm 68%.
Piv txwv lus nug 4: Xam cov nqi los ntawm feem pua
Lo lus nug: Yog tias qhov siab nruab nrab ntawm cov txiv neej laus hauv ib lub tebchaws yog 175 cm nrog qhov sib txawv ntawm 7 cm, qhov siab twg ntawm 90th percentile?
Kev daws teeb meem:
Cov Kauj Ruam:
1. Nrhiav qhov qhab nia Z uas sib raug rau 90th percentile. Raws li lub rooj Z, qhov qhab nia Z ze tshaj plaws rau 0.9000 yog kwv yees li 1.28.
2. Siv cov mis los xam tus nqi ntawm \( X \):
\[ X = \mu + Z \times \sigma \]
\[ X = 175 + 1.28 \times 7 \]
\[ X = 175 + 8.96 \]
\[ X = 183.96 \]
Yog li, qhov siab hauv 90th percentile yog nyob ib ncig ntawm 183.96 cm.
Piv txwv lus nug 5: Qhov muaj feem yuav muaj qee lub sijhawm
Lus Nug: Muab qhov kev faib tawm ntawm qhov hnyav ntawm cov menyuam mos tshiab ua raws li kev faib tawm ib txwm muaj nrog qhov nruab nrab ntawm 3.5 kg thiab qhov sib txawv ntawm 0.5 kg, qhov feem pua uas tus menyuam hnyav ntawm 3 kg thiab 4 kg yog dab tsi?
Kev daws teeb meem:
Cov Kauj Ruam:
1. Xam qhov Z-qhab nia rau cov nqi 3 kg thiab 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. Qhov feem pua rau Z-qhab nia ntawm -1 thiab 1 raws li lub rooj Z yog kwv yees li 0.6826 lossis 68.26%.
Yog li, qhov muaj feem yuav muaj menyuam yaus hnyav ntawm 3 kg thiab 4 kg yog li ntawm 68.26%.
Xaus
Qhov kev faib tawm ib txwm yog lub tswv yim tseem ceeb hauv kev suav lej uas tseem ceeb thiab muaj ntau yam kev siv hauv ntiaj teb tiag. Hauv tsab xov xwm no, peb tau piav qhia txog cov ntsiab lus tseem ceeb ntawm kev faib tawm ib txwm thiab daws ntau qhov piv txwv kom peb nkag siab tob dua.
Kev nkag siab txog kev faib tawm ib txwm tsis yog qhov tseem ceeb rau kev suav lej xwb tab sis kuj tseem ceeb rau ntau yam kev ua haujlwm xws li kev puas siab puas ntsws, kev lag luam, thiab lwm yam kev kawm txog zej zog. Yog tias muaj kev xyaum txaus, kev daws teeb meem kev faib tawm ib txwm tuaj yeem ua rau nkag siab yooj yim dua thiab pab txiav txim siab raws li cov ntaub ntawv.
