Mekhoa ea Khakanyo Lipalopalong
Lipalopalo ke saense ea ho bokella, ho sekaseka le ho toloka lintlha, 'me e 'ngoe ea likarolo tsa eona tsa bohlokoa ke khakanyo. Khakanyo lipalopalong e bolela mokhoa oa ho fumana boleng bo hakanyetsoang ba parameter ea baahi ho latela tlhaiso-leseling e fumanoeng sampole. Mekhoa ea khakanyo e ka aroloa ka mefuta e 'meli e meholo: khakanyo ea lintlha le khakanyo ea nako. Sehloohong sena, re tla tšohla mekhoa e fapaneng ea khakanyo e sebelisoang hangata lipalopalong.
Kutloisiso ea Motheo ea Khakanyo
Pele re kena mekhoeng ea ho hakanya, ho bohlokoa ho utloisisa mantsoe a mang a motheo:
– Dipharamithara: Dibopeho tsa dipalo tsa baahi. Mohlala, karolelano ya baahi (µ), phapang ya baahi (σ²).
– Lipalopalo: Litšobotsi tsa lipalo tsa sampole. Mohlala, karolelano ea sampole (x̄), phapang ea sampole (s²).
Sepheo se seholo sa khakanyo ke ho etsa liqeto mabapi le liparamente tsa baahi ho latela lintlha tsa sampole . Ho na le mefuta e 'meli e meholo ea khakanyo lipalopalong:
1. Khakanyo ea Lintlha: E fana ka boleng bo le bong feela e le khakanyo ea parameter ea baahi.
2. Khakanyo ea Nako: E fana ka mefuta e fapaneng ea boleng e le khakanyo ea parameter ea baahi, ho kenyeletsoa le boemo bo itseng ba kholiseho.
Mokhoa oa ho Hakanya Lintlha
Khakanyo ea lintlha ke mokhoa oa ho fana ka nomoro e le 'ngoe e leng khakanyo e ntle ka ho fetisisa ea parameter ea baahi. Tse ling tsa likhakanyo tsa lintlha tse sebelisoang haholo ke:
1. Mohlala o Akaretsang
Mokhoa o bonolo le o tloaelehileng oa ho hakanya karolelano ea baahi ke ka karolelano ea sampole, e baloang ka tsela ena:
\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \]
moo \( x_i \) e leng tlhokomeliso ka 'ngoe sampoleng 'me \( n \) e le boholo ba sampole.
2. Mohlala o Bohareng
Mohareng oa sampole ke boleng bo bohareng ba data ea sampole e hlophisitsoeng. Ke khakanyo e matla hobane ha e angoe ke lintho tse sa tloaelehang.
3. Tekanyo ea Sampole
Ho hakanya karolo ea baahi, karolo ea sampole e sebelisoa, e baloang ka tsela ena:
\[ \hat{p} = \frac{x}{n} \]
moo \( x \) e leng palo ea katleho sampoleng le \( n \) e leng boholo ba sampole.
Mokhoa oa Khakanyo ea Nako
Likhakanyo tsa nako li fana ka mefuta e fapaneng ea boleng, boo ho lebeletsoeng hore bo koahele paramethara ea baahi ka boemo bo itseng ba kholiseho (mohlala, 95%). Likhakanyo tsa nako hangata li hlahisoa ka mokhoa oa Nako ea Kholiseho (CI).
1. Nako ea Boitšepo bakeng sa Palo ea Baahi
Haeba data ea mohlala e tsoa ho kabo e tloaelehileng kapa (n) e kgolo ka ho lekaneng (CLT e a sebetsa), karolelano ya boitshepo bakeng sa baahi (mu) ke:
\[ \bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}} \]
Moo:
– \( \bar{x} \) ke karolelano ea mohlala
– \( z_{\alpha/2} \) ke boleng ba z ba kabo e tloaelehileng e tsamaellanang le boemo ba kholiseho (mohlala, 1.96 bakeng sa 95%).
– (sigma) ke ho kheloha ho tloaelehileng baahi. Haeba \( \sigma \) e sa tsejwe, \( s \) (sampole ya ho kheloha ho tloaelehileng) e a sebediswa.
– \( n \) ke boholo ba sampole.
2. Nako ea Kholiseho bakeng sa Palo ea Baahi
Ho hakanya karolo ea baahi \( p \):
\[ \hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]
moo \( \hat{p} \) e leng karolo ea sampole le liparamente tse ling joalo ka ha ho hlalositsoe pejana.
Mekhoa e meng ea ho Hakanya
1. Mokhoa oa ho ba le monyetla o moholo (ML)
Mokhoa oa Maximum Likelihood ke mokhoa o sebelisoang ho fumana khakanyo e ntle ka ho fetisisa bakeng sa paramethara ea baahi \( \theta \) ka ho eketsa mosebetsi oa likelihood \( L(\theta) \). Mosebetsi oa likelihood ke monyetla oa ho fumana data e shebiloeng e fanoeng paramethara \( \theta \):
\[ L(\theta|x) = \prod_{i=1}^{n} f(x_i|\theta) \]
moo \( f(x_i|\theta) \) e leng mosebetsi wa ho ba le bokgoni ba ho ba le bokgoni (PDF) ba data. Kgakanyo e eketsang \( L(\theta) \) e bitswa kgakanyo ya bokgoni ba ho ba le bokgoni bo boholo (MLE).
2. Mokhoa oa Khakanyo ea Bayesian
Mokhoa oa Bayesian o nka liparamente e le diphetoho tse sa reroang 'me u sebelise kabo ea menyetla ho hakanya liparamente. Ho latela Theorem ea Bayes:
\[ P(\theta|x) = \frac{P(x|\theta) P(\theta)}{P(x)} \]
moo \( P(\theta|x) \) e leng kabo ya morao, \( P(x|\theta) \) e leng kgonahalo, \( P(\theta) \) e leng ya pele, mme \( P(x) \) e le margin ya kgonahalo. Dikhakanyo tsa Bayesian di itshetlehile haholo ka di-prior tse sebedisitsweng.
Tlhahlobo ea Khakanyo
Ho lekola mohakanyetsi oa lintlha, re tlameha ho hlahloba litšobotsi tsa hae:
– Toka/Leeme: Mohlahlobi \( \hat{\theta} \) ho thwe ha a na leeme haeba \( E[\hat{\theta}] = \theta \).
– Bokgoni: Mohlahlobi ea sebetsang hantle o na le phapang e nyane ka ho fetisisa har'a likhakanyo tsohle tse sa nke lehlakore.
– Ho tsitsa: Ho thoe khakanyo e tsitsitse haeba \( \hat{\theta} \) e atamela \( \theta \) ha boholo ba sampole \( n \) bo ntse bo eketseha.
Mehlala ea Ts'ebeliso
1. Khakanyo ea Moputso o Akaretsang
Phuputsong ea moruo, ho hakanngoa chelete e kenang ea baahi hangata ho etsoa. Bafuputsi ba nka sampole ea baahi 'me ba bala karolelano ea sampole e le khakanyo ea lintlha, 'me ba fana ka nako ea kholiseho ho bontša ho se tsitse ha khakanyo ena.
2. Khakanyo ea Palo ea Bakhethi
Phuputsong ea likhetho, mofuputsi a ka 'na a batla ho hakanya peresente ea bakgethi ba tšehetsang mokhethoa ea itseng. Karolelano ea mohlala \( \hat{p} \) ea ba arabelitseng ba tšehetsang mokhethoa eo e sebelisoa e le khakanyo ea lintlha. Nako ea kholiseho e ka fanoa ho bontša moeli oa phoso.
Qetello
Mekhoa ea ho hakanya e bapala karolo ea bohlokoa lipalo-palong hobane e lumella bafuputsi ho etsa liqeto mabapi le baahi ho latela lintlha tsa sampole. Mekhoa ea ho hakanya lintlha le nako e fana ka lisebelisoa tse matla bakeng sa sena, ka mekhoa e kang monyetla o moholo le khakanyo ea Bayesian e tebang ka botebo mathateng a lintlha. Ho sebelisa likhakanyo tse lokileng, tse sebetsang hantle le tse tsitsitseng ho netefatsa liphetho tse tšepahalang le tse nepahetseng tsa tlhahlobo ea data , ho nolofalletsa ho etsa liqeto tse betere masimong a kang moruo, mahlale a sechaba, bophelo bo botle le tse ling.
