Bài giảng Thống kê xã hội - Chương 6: Phân phối của tham số mẫu và khoảng tin cậy cho các tham số tổng thể cung cấp cho người học các kiến thức: Phân phối của trung bình mẫu, phân phối ủa tỷ lệ mẫu, ước lượng điểm, ước lượng khoảng. Mời các bạn cùng tham khảo.
❚❍➮◆● ❑➊ ❳❶ ❍❐■ ❍➴❈ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✣↕✐ ❤å❝ ❚❍❿◆● ▲❖◆● ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✉♥✐✈❡rs✐t②✲❧♦❣♦ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✶ ✴ ✷✾ ❈❤÷ì♥❣ ❱■ P❤➙♥ ♣❤è✐ ❝õ❛ ❝→❝ t❤❛♠ sè ♠➝✉ ✈➔ ❦❤♦↔♥❣ t✐♥ ❝➟② ❝❤♦ ❝→❝ t❤❛♠ sè tê♥❣ t❤➸ ✉♥✐✈❡rs✐t②✲❧♦❣♦ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✷ ✴ ✷✾ ◆ë✐ ❞✉♥❣ ❣✐↔♥❣ ❞↕② ✶ P❤➙♥ ♣❤è✐ ❝õ❛ t❤❛♠ sè ♠➝✉ P❤➙♥ ♣❤è✐ ❝õ❛ tr✉♥❣ ❜➻♥❤ ♠➝✉ P❤➙♥ ♣❤è✐ ❝õ❛ t✛ ❧➺ ìợ ữủ t số tờ t ìợ ữủ ìợ ữủ rst ộ ú ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✸ ✴ ✷✾ ◆ë✐ ❞✉♥❣ ❣✐↔♥❣ ❞↕② ✶ P❤➙♥ ♣❤è✐ ❝õ❛ t❤❛♠ sè ♠➝✉ P❤➙♥ ♣❤è✐ ❝õ❛ tr✉♥❣ ❜➻♥❤ P ố t ìợ ữủ t số tờ t ìợ ữủ ìợ ữủ ✉♥✐✈❡rs✐t②✲❧♦❣♦ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✸ ✴ ✷✾ ❚❤❛♠ sè ♠➝✉ ✈➔ t❤❛♠ sè tê♥❣ t❤➸ ❇➔✐ t♦→♥ ❚r♦♥❣ ♠ët ❝✉ë❝ ✤✐➲✉ tr❛ ❦❤↔♦ s→t t✐➳♥ ❤➔♥❤ tr➯♥ ♥❤â♠ ❝→❝ ♥❤➙♥ ✈✐➯♥ ✈➠♥ ♣❤á♥❣ ❝â ♠ët ❝➙✉ ❤ä✐ ❙❛✉ ❜❛♦ ❧➙✉ t❤➻ ❜↕♥ tr↔ ❧í✐ ❧↕✐ ❦❤✐ ♥❤➟♥ ✤÷đ❝ ❡✲♠❛✐❧❄✳ ◆➳✉ ♥❣÷í✐ t❛ ♠✉è♥ t➼♥❤ t❤í✐ ❣✐❛♥ tr✉♥❣ ❜➻♥❤ ♠➔ ❝→❝ ♥❤➙♥ ✈✐➯♥ ✈➠♥ ♣❤á♥❣ tr↔ ❧í✐ t❤÷ s❛✉ ❦❤✐ ♥❤➟♥ ✤÷đ❝ t❤➻ ❝✉ë❝ ✤✐➲✉ tr❛ ♣❤↔✐ t✐➳♥ ❤➔♥❤ tr➯♥ tê♥❣ t❤➸ ♥➔♦❄ ❑❤✐ t✐➳♥ ❤➔♥❤ ✤✐➲✉ tr❛ tr➯♥ tê♥❣ t❤➸✱ ✤↕✐ ❧÷đ♥❣ ❜↕♥ ❝➛♥ t➻♠ ❝â ①→❝ ✤à♥❤ ✤÷đ❝ ❦❤ỉ♥❣❄ ●✐↔ sû ♥❣÷í✐ t❛ q✉②➳t ✤à♥❤ t✐➳♥ ❤➔♥❤ ✤✐➲✉ tr❛ ♠➝✉ ✈➔ ❝ï ♠➝✉ ①→❝ ✤à♥❤ ❧➔ ✸✵✵ ♥❣÷í✐✳ ❍❛✐ ✤✐➲✉ tr❛ ✈✐➯♥ ❝ị♥❣ t✐➳♥ ❤➔♥❤ ❝❤å♥ ♠➝✉ ✈➔ ✤÷đ❝ t❤ỉ♥❣ t✐♥ tr➯♥ ❤❛✐ ♠➝✉✳ ❚❤❡♦ ❜↕♥✱ ❦➳t q✉↔ t➼♥❤ t♦→♥ tr➯♥ ❤❛✐ ♠➝✉ ❝â ♥❤÷ ♥❤❛✉ ❦❤æ♥❣❄ ◆➳✉ tø t❤æ♥❣ t✐♥ ❝õ❛ ♠➝✉✱ ❧➔♠ t❤➳ ♥➔♦ t➻♠ ✤÷đ❝ ❣✐→ trà ❝❤ó♥❣ t❛ ❝➛♥ ✉♥✐✈❡rs✐t②✲❧♦❣♦ tr➯♥ t♦➔♥ tê♥❣ t❤➸❄ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✹ ✴ ✷✾ ❚❤❛♠ sè ♠➝✉ ❚❤❛♠ sè ♠➝✉ ❧➔ ♠ët ❜✐➳♥ ợ ộ t t t ữủ ♠ët ❣✐→ trà ❝õ❛ t❤❛♠ sè ♠➝✉ ✤â✳ ❈→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝❤➾ t❤❛♠ sè tr✉♥❣ ❜➻♥❤ ♠➝✉✱ t✛ ❧➺ ữỡ s ữủt ữủ ỵ , ❙✳ ❧➔ ❳ , P ❚❤æ♥❣ t✐♥ tr➯♥ tê♥❣ t❤➸ t❤÷í♥❣ ❦❤ỉ♥❣ ❝â✱ ♥❣÷í✐ t❛ ❞ị♥❣ t❤ỉ♥❣ t✐♥ tr➯♥ ♠➝✉ ữợ tờ t ố õ ữỡ ❞ü ✤♦→♥ tèt✱ ♥❣÷í✐ t❛ ❝➛♥ ❜✐➳t t❤ỉ♥❣ t✐♥ ✈➲ ♣❤➙♥ ♣❤è✐ ❝õ❛ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝❤➾ t❤❛♠ sè ♠➝✉✳ ✣ë ❧➺❝❤ ❝❤✉➞♥ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝❤➾ t❤❛♠ sè ♠➝✉ ✤÷đ❝ ❣å✐ ❧➔ s❛✐ sè ❝❤✉➞♥✳ ✉♥✐✈❡rs✐t②✲❧♦❣♦ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✺ ✴ ✷✾ ◆ë✐ ❞✉♥❣ ❣✐↔♥❣ ❞↕② ✶ P❤➙♥ ♣❤è✐ ❝õ❛ t❤❛♠ sè ♠➝✉ P❤➙♥ ♣❤è✐ ❝õ❛ tr✉♥❣ ❜➻♥❤ ♠➝✉ P❤➙♥ ♣❤è✐ ❝õ❛ t✛ ❧➺ ìợ ữủ t số tờ t ìợ ữủ ìợ ữủ rst ộ ú ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✻ ✴ ✷✾ ❚r✉♥❣ ❜➻♥❤ ♠➝✉ ✣ë ❧➺❝❤ ❝❤✉➞♥ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝❤➾ tr✉♥❣ ❜➻♥❤ ♠➝✉ ✤÷đ❝ ❣å✐ ❧➔ s❛✐ sè ❝❤✉➞♥ ✭❙❊✮ ❝õ❛ tr✉♥❣ ❜➻♥❤ ♠➝✉✳ ●å✐ ❳ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝❤➾ tr✉♥❣ ❜➻♥❤ ❝õ❛ ♠ët ♠➝✉ ❝â ♥ q✉❛♥ s→t ✤÷đ❝ ❝❤å♥ tø ♠ët tê♥❣ t❤➸ ❝â tr✉♥❣ ❜➻♥❤ ❧➔ µ✱ ✤ë ❧➺❝❤ ❧➔ σ t❤➻✿ ♣ q ❊ ❳ ❂µ ❙❛✐ sè ❝❤✉➞♥ ❝õ❛ tr✉♥❣ ❜➻♥❤ ♠➝✉ ❙❊ ❧➔ ❄♥ σ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ Ñ ●✐→ trà tr✉♥❣ ❜➻♥❤ tr➯♥ ❝→❝ ♠➝✉ ❦❤→❝ ♥❤❛✉ ❧➔ ❦❤→❝ ♥❤❛✉✳ ◆❤÷♥❣ tr✉♥❣ ❜➻♥❤ ❝õ❛ t➜t ❝↔ ❝→❝ ❣✐→ trà t➼♥❤ ✤÷đ❝ tr➯♥ ♠➝✉ ❝❤➼♥❤ ❧➔ tr✉♥❣ ❜➻♥❤ t❤➟t ❝õ❛ tê♥❣ t❤➸✳ ●✐→ trà tr✉♥❣ ❜➻♥❤ ♠➝✉ ❜✐➳♥ t❤✐➯♥ ①✉♥❣ q✉❛♥❤ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❝õ❛ tê♥❣ t❤➸✳ ❑❤✐ ❝ï ♠➝✉ ❝➔♥❣ ❧ỵ♥ t❤➻ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ♠➝✉ ✧❝➔♥❣ ❣➛♥✧ ✉♥✐✈❡rs✐t②✲❧♦❣♦ ✈ỵ✐ tr✉♥❣ ❜➻♥❤ tê♥❣ t❤➸✳ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✼ ✴ ✷✾ ❚r✉♥❣ ❜➻♥❤ ♠➝✉ ✣ë ❧➺❝❤ ❝❤✉➞♥ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝❤➾ tr✉♥❣ ❜➻♥❤ ♠➝✉ ✤÷đ❝ ❣å✐ ❧➔ s❛✐ sè ❝❤✉➞♥ ✭❙❊✮ ❝õ❛ tr✉♥❣ ❜➻♥❤ ♠➝✉✳ ●å✐ ❳ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝❤➾ tr✉♥❣ ❜➻♥❤ ❝õ❛ ♠ët ♠➝✉ ❝â ♥ q✉❛♥ s→t ✤÷đ❝ ❝❤å♥ tø ♠ët tê♥❣ t❤➸ ❝â tr✉♥❣ ❜➻♥❤ ❧➔ µ✱ ✤ë ❧➺❝❤ ❧➔ σ t❤➻✿ ♣ q ❊ ❳ ❂µ ❙❛✐ sè ❝❤✉➞♥ ❝õ❛ tr✉♥❣ ❜➻♥❤ ♠➝✉ ❙❊ ❧➔ ❄♥ σ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ Đ ●✐→ trà tr✉♥❣ ❜➻♥❤ tr➯♥ ❝→❝ ♠➝✉ ❦❤→❝ ♥❤❛✉ ❧➔ ❦❤→❝ ♥❤❛✉✳ ◆❤÷♥❣ tr✉♥❣ ❜➻♥❤ ❝õ❛ t➜t ❝↔ ❝→❝ ❣✐→ trà t➼♥❤ ✤÷đ❝ tr➯♥ ♠➝✉ ❝❤➼♥❤ ❧➔ tr✉♥❣ ❜➻♥❤ t❤➟t ❝õ❛ tê♥❣ t❤➸✳ ●✐→ trà tr✉♥❣ ❜➻♥❤ ♠➝✉ ❜✐➳♥ t❤✐➯♥ ①✉♥❣ q✉❛♥❤ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❝õ❛ tê♥❣ t❤➸✳ ❑❤✐ ❝ï ♠➝✉ ❝➔♥❣ ❧ỵ♥ t❤➻ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ♠➝✉ ✧❝➔♥❣ ❣➛♥✧ ✉♥✐✈❡rs✐t②✲❧♦❣♦ ✈ỵ✐ tr✉♥❣ ❜➻♥❤ tê♥❣ t❤➸✳ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✼ ✴ ✷✾ P❤➙♥ ♣❤è✐ ❝õ❛ tr✉♥❣ ❜➻♥❤ ♠➝✉ ◆➳✉ tê♥❣ t❤➸ t✉➙♥ t❤❡♦ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ ợ tr t tr ♠➝✉ t✉➙♥ t❤❡♦ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥✳ ◆➳✉ ❝ï ♠➝✉ ❧➔ ❝ï ♠➝✉ ❧ỵ♥ ✭♥ ♣❤è✐ ❝❤✉➞♥✳ ➙ ✸✵✮ t❤➻ tr✉♥❣ ❜➻♥❤ ♠➝✉ ①➜♣ ①➾ ♣❤➙♥ ✉♥✐✈❡rs✐t②✲❧♦❣♦ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✽ ✴ ✷✾ ❱➼ ❞ö ❇➔✐ t♦→♥ ❚rå♥❣ ữủ ợ s t t❤❡♦ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ ✈ỵ✐ ✤ë ❧➺❝❤ ❝❤✉➞♥ ❧➔ ✵✳✶❦❣✳ ởt ỗ ♠ỵ✐ s✐♥❤ ✤õ ♥❣➔② t❤➜② tr✉♥❣ ❜➻♥❤ ❧➔ ✸❦❣✱ ✤ë ữợ ữủ ợ t ❝➟② ✾✺✪❝❤♦ trå♥❣ ❧÷đ♥❣ tr✉♥❣ ❜➻♥❤ ❝õ❛ ❝→❝ ❡♠ ❜➨ ❣→✐ ♠ỵ✐ s✐♥❤✳ ●✐↔✐✿ ❚❛ ❝â ① ③✵.✵✷✺ ✏, σ ✏, s ✏, ♥ ✏, ✶ ✁ α ✏ ✏ ✶.✾✻✳ ❑❤♦↔♥❣ t✐♥ ❝➟② ✾✺✪ ❝❤♦ trå♥❣ ❧÷đ♥❣ tr✉♥❣ ❜➻♥❤ ợ s rst ị ♥❣❤➽❛ ❝õ❛ ❦❤♦↔♥❣ t➻♠ ✤÷đ❝❄ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✷✷ ✴ ✷✾ ❱➼ ❞ư ❇➔✐ t♦→♥ ❚rå♥❣ ❧÷đ♥❣ ❝õ❛ ❝→❝ ❜➨ ❣→✐ ♠ỵ✐ s✐♥❤ ✤õ ♥❣➔② t✉➙♥ t❤❡♦ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ ✈ỵ✐ ✤ë ❧➺❝❤ ❝❤✉➞♥ ❧➔ ✵✳✶❦❣✳ ▲➜② ởt ỗ ợ s✐♥❤ ✤õ ♥❣➔② t❤➜② tr✉♥❣ ❜➻♥❤ ❧➔ ✸❦❣✱ ✤ë ❧➺❝❤ ữợ ữủ ợ t ✾✺✪❝❤♦ trå♥❣ ❧÷đ♥❣ tr✉♥❣ ❜➻♥❤ ❝õ❛ ❝→❝ ❡♠ ❜➨ ❣→✐ ♠ỵ✐ s✐♥❤✳ ●✐↔✐✿ ❚❛ ❝â ① ③✵.✵✷✺ ✏, σ ✏, s ✏, ♥ ✏, ✶ ✁ α ✏ ✏ ✶.✾✻✳ ❑❤♦↔♥❣ t✐♥ ❝➟② ✾✺✪ ❝❤♦ trå♥❣ ❧÷đ♥❣ tr✉♥❣ ❜➻♥❤ ❝→❝ ợ s rst ị ❝õ❛ ❦❤♦↔♥❣ t➻♠ ✤÷đ❝❄ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✷✷ ✴ ✷✾ ❱➼ ❞ư ❇➔✐ t♦→♥ ❚rå♥❣ ❧÷đ♥❣ ❝õ❛ ❝→❝ ❜➨ ❣→✐ ♠ỵ✐ s✐♥❤ ✤õ ♥❣➔② t✉➙♥ t❤❡♦ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ ✈ỵ✐ ✤ë ❧➺❝❤ ❝❤✉➞♥ ❧➔ ✵✳✶❦❣✳ ▲➜② ♠ët ỗ ợ s ✤õ ♥❣➔② t❤➜② tr✉♥❣ ❜➻♥❤ ❧➔ ✸❦❣✱ ✤ë ❧➺❝❤ ❧➔ ữợ ữủ ợ t trồ ữủ tr ợ s✐♥❤✳ ●✐↔✐✿ ❚❛ ❝â ① ③✵.✵✷✺ ✏, σ ✏, s ✏, ♥ ✏, ✶ ✁ α ✏ ✏ ✶.✾✻✳ ❑❤♦↔♥❣ t✐♥ ❝➟② ✾✺✪ ❝❤♦ trå♥❣ ❧÷đ♥❣ tr✉♥❣ ❜➻♥❤ ❝→❝ ❡♠ ợ s rst ị ❦❤♦↔♥❣ t➻♠ ✤÷đ❝❄ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✷✷ ìợ ữủ tr tờ t ❚ê♥❣ t❤➸ t✉➙♥ t❤❡♦ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥✱ ❝❤÷❛ ❜✐➳t ♣❤÷ì♥❣ s❛✐✱ ❝ï ♠➝✉ ♥❤ä✿ ✣➸ ①➙② ❞÷♥❣ ❦❤♦↔♥❣ t✐♥ ❝➟② ❝❤♦ tr✉♥❣ ❜➻♥❤ tê♥❣ t❤➸ tr÷í♥❣ ❤đ♣ ♥➔② t❛ ❝➛♥ sû ❞ö♥❣ ♣❤➙♥ ♣❤è✐ ❙t✉❞❡♥t✳ P❤➙♥ ♣❤è✐ ❙t✉❞❡♥t ❝â t➼♥❤ ❝❤➜t s❛✉✿ P❤➙♥ ♣❤è✐ ❙t✉❞❡♥t ❝â ♠ët t❤❛♠ sè ❣å✐ ❧➔ ❜➟❝ tü ❞♦✳ ❍å ♣❤➙♥ ♣❤è✐ ❙t✉❞❡♥t ❝â ❤➻♥❤ ❞→♥❣ ❣✐è♥❣ ♥❤÷ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ ❤â❛✿ ❝➙♥ ✤è✐ ❤➻♥❤ ❝❤✉ỉ♥❣✱ ✤è✐ ①ù♥❣ q✉❛ µ ✵✱ ❝â tr✉♥❣ ❜➻♥❤ ❧➔ ữ ữỡ s ợ ỡ ố õ ♥➯♥ ❤❛✐ ✤✉ỉ✐ ❞➔✐ ❤ì♥ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ ❤â❛ ✏ P❤÷ì♥❣ s❛✐ ❝õ❛ ♣❤➙♥ ♣❤è✐ ❙t✉❞❡♥t ♣❤ư t❤✉ë❝ ✈➔♦ ❜➟❝ tü ❞♦✲ t❤❛♠ sè ❝õ❛ ♣❤➙♥ ♣❤è✐ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ Ñ ❚❤è♥❣ ❦➯ ❳❍❍ ✉♥✐✈❡rs✐t②✲❧♦❣♦ ◆❣➔② t ìợ ữủ ❦❤♦↔♥❣ ❝❤♦ tr✉♥❣ ❜➻♥❤ tê♥❣ t❤➸ ❚ê♥❣ t❤➸ t✉➙♥ t❤❡♦ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥✱ ❝❤÷❛ ❜✐➳t ♣❤÷ì♥❣ s❛✐✱ ❝ï ♠➝✉ ♥❤ä✿ ✣➸ ①➙② ❞÷♥❣ ❦❤♦↔♥❣ t✐♥ ❝➟② ❝❤♦ tr✉♥❣ ❜➻♥❤ tê♥❣ t❤➸ tr÷í♥❣ ❤đ♣ ♥➔② t❛ ❝➛♥ sû ❞ư♥❣ ♣❤➙♥ ♣❤è✐ ❙t✉❞❡♥t✳ P❤➙♥ ♣❤è✐ ❙t✉❞❡♥t ❝â t➼♥❤ ❝❤➜t s❛✉✿ P❤➙♥ ♣❤è✐ ❙t✉❞❡♥t ❝â ♠ët t❤❛♠ sè ❣å✐ ❧➔ ❜➟❝ tü ❞♦✳ ❍å ♣❤➙♥ ♣❤è✐ ❙t✉❞❡♥t ❝â ❤➻♥❤ ❞→♥❣ ❣✐è♥❣ ♥❤÷ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ ❤â❛✿ ❝➙♥ ✤è✐ ❤➻♥❤ ❝❤✉æ♥❣✱ ✤è✐ ①ù♥❣ q✉❛ µ ✵✱ ❝â tr✉♥❣ ❜➻♥❤ ❧➔ ✵ ♥❤÷♥❣ ♣❤÷ì♥❣ s❛✐ ❧ỵ♥ ❤ì♥ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ ❤â❛ ♥➯♥ ❤❛✐ ✤✉ỉ✐ ❞➔✐ ❤ì♥ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ ❤â❛ ✏ P❤÷ì♥❣ s❛✐ ❝õ❛ ♣❤➙♥ ♣❤è✐ ❙t✉❞❡♥t ♣❤ö t❤✉ë❝ ✈➔♦ ❜➟❝ tü ❞♦✲ t❤❛♠ sè ❝õ❛ ♣❤➙♥ ♣❤è✐ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ Ñ ❚❤è♥❣ ❦➯ ❳❍❍ ✉♥✐✈❡rs✐t②✲❧♦❣♦ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✷✸ ✴ ✷✾ ●✐→ trà tα P❤➙♥ ♣❤è✐ ❙t✉❞❡♥t ợ tỹ ữủ ỵ ♣✵, ✶q✱ sè t♥,α ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ ❣✐→ trà s❛♦ ❝❤♦ ♣ → t♥,αq ✏ α✳ ❱ỵ✐ α P ❚♥ ✉♥✐✈❡rs✐t②✲❧♦❣♦ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✷✹ ìợ ữủ tr tờ t ❚ê♥❣ t❤➸ t✉➙♥ t❤❡♦ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥✱ ❝❤÷❛ ❜✐➳t ♣❤÷ì♥❣ s❛✐✱ ❝ï ♠➝✉ ♥❤ä✿ ♣ ✁ αq✶✵✵% ❝❤♦ tr✉♥❣ ❜➻♥❤ tê♥❣ t❤➸ ❧➔ ❑❤♦↔♥❣ t✐♥ ❝➟② ✶ ♣① ✁ t♥✁✶,α④✷ ✝ ❄s♥ , ① t♥✁✶,α④✷ ✝ ❄s♥ q ✉♥✐✈❡rs✐t②✲❧♦❣♦ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✷✺ ✴ ✷✾ ❱➼ ❞ư ❈❤✐➲✉ ❝❛♦ ❝õ❛ tr➫ ❡♠ ♠ỵ✐ s✐♥❤ t✉➙♥ t❤❡♦ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥✳ ❈❤å♥ ♥❣➝✉ ♥❤✐➯♥ ✷✵ tr➫ ❡♠ ♠ỵ✐ s✐♥❤ t↕✐ ♠ët ❜➺♥❤ ✈✐➺♥ t❤➻ t➼♥❤ ✤÷đ❝ ❝❤✐➲✉ ❝❛♦ tr✉♥❣ ❜➻♥❤ ❧➔ ✹✽✳✺❝♠✱ ✤ë ❧➺❝❤ ❧➔ ✵✳✷❝♠✳ ❚➻♠ ♠ët ÷ỵ❝ ❧÷đ♥❣ ❦❤♦↔♥❣ ✈ỵ✐ ✤ë t✐♥ ❝➟② ✾✺✪ ❝❤♦ ❝❤✐➲✉ ❝❛♦ tr✉♥❣ ❜➻♥❤ ❝õ❛ tr➫ ❡♠ ♠ỵ✐ s✐♥❤✳ ❇➔✐ t♦→♥✿ ●✐↔✐✿ ✏, s ✏, ♥ ✏, ✶ ✁ α ✏ t✵.✵✷✺,✶✾ ✏ ✷.✶ ❚❛ ❝â ① ❑❤♦↔♥❣ t✐♥ ❝➟② ❝❤♦ ❝❤✐➲✉ ❝❛♦ tr✉♥❣ ❜➻♥❤ ❝õ❛ ❝→❝ ❡♠ ❜➨ ♠ỵ✐ s✐♥❤ ❧➔✿ ✭✹✽✳✹✵✸✱ ✹✽✳✺✾✻✮✳ ✉♥✐✈❡rs✐t②✲❧♦❣♦ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✷✻ ✴ ✷✾ ❱➼ ❞ö ❈❤✐➲✉ ❝❛♦ ❝õ❛ tr➫ ❡♠ ♠ỵ✐ s✐♥❤ t✉➙♥ t❤❡♦ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥✳ ❈❤å♥ ♥❣➝✉ ♥❤✐➯♥ ✷✵ tr➫ ❡♠ ♠ỵ✐ s✐♥❤ t↕✐ ♠ët ❜➺♥❤ ✈✐➺♥ t❤➻ t➼♥❤ ✤÷đ❝ ❝❤✐➲✉ ❝❛♦ tr✉♥❣ ❜➻♥❤ ❧➔ ✹✽✳✺❝♠✱ ✤ë ❧➺❝❤ ởt ữợ ữủ ợ t ❝➟② ✾✺✪ ❝❤♦ ❝❤✐➲✉ ❝❛♦ tr✉♥❣ ❜➻♥❤ ❝õ❛ tr➫ ❡♠ ♠ỵ✐ s✐♥❤✳ ❇➔✐ t♦→♥✿ ●✐↔✐✿ ✏, s ✏, ♥ ✏, ✶ ✁ α ✏ t✵.✵✷✺,✶✾ ✏ ✷.✶ ❚❛ ❝â ① ❑❤♦↔♥❣ t✐♥ ❝➟② ❝❤♦ ❝❤✐➲✉ ❝❛♦ tr✉♥❣ ❜➻♥❤ ❝õ❛ ❝→❝ ❡♠ ❜➨ ♠ỵ✐ s✐♥❤ ❧➔✿ ✭✹✽✳✹✵✸✱ ✹✽✳✺✾✻✮✳ ✉♥✐✈❡rs✐t②✲❧♦❣♦ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✷✻ ✴ ✷✾ ❱➼ ❞ö ❈❤✐➲✉ ❝❛♦ ❝õ❛ tr➫ ❡♠ ♠ỵ✐ s✐♥❤ t✉➙♥ t❤❡♦ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥✳ ❈❤å♥ ♥❣➝✉ ♥❤✐➯♥ ✷✵ tr➫ ❡♠ ♠ỵ✐ s✐♥❤ t↕✐ ♠ët ❜➺♥❤ ✈✐➺♥ t❤➻ t➼♥❤ ✤÷đ❝ ❝❤✐➲✉ ❝❛♦ tr✉♥❣ ❜➻♥❤ ởt ữợ ữủ ❦❤♦↔♥❣ ✈ỵ✐ ✤ë t✐♥ ❝➟② ✾✺✪ ❝❤♦ ❝❤✐➲✉ ❝❛♦ tr✉♥❣ ❜➻♥❤ ❝õ❛ tr➫ ❡♠ ♠ỵ✐ s✐♥❤✳ ❇➔✐ t♦→♥✿ ●✐↔✐✿ ✏, s ✏, ♥ ✏, ✶ ✁ α ✏ t✵.✵✷✺,✶✾ ✏ ✷.✶ ❚❛ ❝â ① ❑❤♦↔♥❣ t✐♥ ❝➟② ❝❤♦ ❝❤✐➲✉ ❝❛♦ tr✉♥❣ ❜➻♥❤ ❝õ❛ ❝→❝ ❡♠ ❜➨ ♠ỵ✐ s✐♥❤ ❧➔✿ ✭✹✽✳✹✵✸✱ ✹✽✳✺✾✻✮✳ ✉♥✐✈❡rs✐t②✲❧♦❣♦ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✷✻ ✴ ✷✾ ❱➼ ❞ư ❈❤✐➲✉ ❝❛♦ ❝õ❛ tr➫ ❡♠ ♠ỵ✐ s✐♥❤ t✉➙♥ t❤❡♦ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥✳ ❈❤å♥ ♥❣➝✉ ♥❤✐➯♥ ✷✵ tr➫ ợ s t ởt t t ữủ ❝❤✐➲✉ ❝❛♦ tr✉♥❣ ❜➻♥❤ ❧➔ ✹✽✳✺❝♠✱ ✤ë ❧➺❝❤ ❧➔ ✵✳✷❝♠✳ ởt ữợ ữủ ợ t ❝❤♦ ❝❤✐➲✉ ❝❛♦ tr✉♥❣ ❜➻♥❤ ❝õ❛ tr➫ ❡♠ ♠ỵ✐ s✐♥❤✳ ❇➔✐ t♦→♥✿ ●✐↔✐✿ ✏, s ✏, ♥ ✏, ✶ ✁ α ✏ t✵.✵✷✺,✶✾ ✏ ✷.✶ ❚❛ ❝â ① ❑❤♦↔♥❣ t✐♥ ❝➟② ❝❤♦ ❝❤✐➲✉ ❝❛♦ tr✉♥❣ ❜➻♥❤ ❝õ❛ ❝→❝ ❡♠ ❜➨ ♠ỵ✐ s✐♥❤ ❧➔✿ ✭✹✽✳✹✵✸✱ ✹✽✳✺✾✻✮✳ ✉♥✐✈❡rs✐t②✲❧♦❣♦ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ìợ ữủ trà tr✉♥❣ ❜➻♥❤ ▼ët ✈➔✐ ♥❤➟♥ ①➨t ❦❤✐ t➻♠ ❦❤♦↔♥❣ t✐♥ ❝➟② ❝❤♦ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❚r÷í♥❣ ❤đ♣ ❜✐➳t σ ❧➔ ❤✐➳♠ ❣➦♣✳ ❑❤✐ ❜➟❝ tü ❞♦ ❝õ❛ ♣❤➙♥ ♣❤è✐ ❙t✉❞❡♥t ❧ỵ♥ t❤➻ ♣❤➙♥ ♣❤è✐ ♥➔② ①➜♣ ①➾ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥✳ ❱➻ ✈➟② tr➯♥ t❤ü❝ t➳✱ ❦❤✐ ❝ï ♠➝✉ ợ ữớ t ũ tr t, tr ổ tự t➼♥❤ ❦❤♦↔♥❣ t✐♥ ❝➟② ❝❤♦ ❣✐→ trà tr✉♥❣ ❜➻♥❤ t❤❛② ❝❤♦ ❣✐→ trà ③α④✷ ✉♥✐✈❡rs✐t②✲❧♦❣♦ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✷✼ ✴ ✷✾ ❑❤♦↔♥❣ t✐♥ ❝➟② ❝❤♦ t✛ ❧➺ tê♥❣ t❤➸✱❝ï ♠➝✉ ❧ỵ♥ ❇➔✐ t♦→♥ ●✐↔ sû t❛ ♠✉è♥ ÷ỵ❝ ❧÷đ♥❣ t✛ ❧➺ ❝õ❛ ✤➦❝ ✤✐➸♠ ❆ tr➯♥ tê♥❣ t❤➸ ♣ ✈ỵ✐ ✤ë t✐♥ ❝➟② ✶ α ✶✵✵%✱ t❛ t✐➳♥ ❤➔♥❤ ❝❤å♥ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ ✈ỵ✐ ❝ï ♠➝✉ ♥✱ t✛ ❧➺ ❝õ❛ ✤➦❝ ✤✐➸♠ ❆ ①✉➙t ❤✐➺♥ tr➯♥ ♠➝✉ ❧➔ ♣ ˆ✳ ♣ ✁ q ♣ ✁ ♣ˆq ➙ ✺ t❤➻✿ ◆➳✉ ❝ï ♠➝✉ ❧ỵ♥✱ tù❝ ❧➔ t❤ä❛ ♠➣♥ ♥♣ ˆ, ♥ ✶ ♣ ✁ αq✶✵✵% ❑❤♦↔♥❣ t✐♥ ❝➟② ❝❤♦ t✛ ❧➺ tê♥❣ t❤➸ ✈ỵ✐ ✤ë t✐♥ ❝➟② ✶ ♣♣ˆ ✁ ③α④✷ ❛ ✝ ♣♣✶ ✁ ♣q④♥, ♣ˆ ③ α ④✷ ❛ ✝ ♣♣✶ ✁ ♣q④♥q ✉♥✐✈❡rs✐t②✲❧♦❣♦ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✷✽ ✴ ✷✾ ❑❤♦↔♥❣ t✐♥ ❝➟② ❝❤♦ t✛ ❧➺ tê♥❣ t❤➸✱❝ï ♠➝✉ ❧ỵ♥ ❇➔✐ t♦→♥ sỷ t ố ữợ ữủ t ✤✐➸♠ ❆ tr➯♥ tê♥❣ t❤➸ ♣ ✈ỵ✐ ✤ë t✐♥ ❝➟② ✶ α ✶✵✵%✱ t❛ t✐➳♥ ❤➔♥❤ ❝❤å♥ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ ✈ỵ✐ ❝ï ♠➝✉ ♥✱ t✛ ❧➺ ❝õ❛ ✤➦❝ ✤✐➸♠ ❆ ①✉➙t ❤✐➺♥ tr➯♥ ♠➝✉ ❧➔ ♣ ˆ✳ ♣ ✁ q ♣ ✁ ♣ˆq ➙ ✺ t❤➻✿ ◆➳✉ ❝ï ♠➝✉ ❧ỵ♥✱ tù❝ ❧➔ t❤ä❛ ♠➣♥ ♥♣ ˆ, ♥ ✶ ♣ ✁ αq✶✵✵% ❑❤♦↔♥❣ t✐♥ ❝➟② ❝❤♦ t✛ ❧➺ tê♥❣ t❤➸ ✈ỵ✐ ✤ë t✐♥ ❝➟② ✶ ♣♣ˆ ✁ ③α④✷ ❛ ✝ ♣♣✶ ✁ ♣q④♥, ♣ˆ ③ α ④✷ ❛ ✝ ♣♣✶ ✁ ♣q④♥q ✉♥✐✈❡rs✐t②✲❧♦❣♦ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✷✽ ✴ ✷✾ ❱➼ ❞ö ❇➔✐ t♦→♥ ❈❤å♥ ♥❣➝✉ ♥❤✐➯♥ ✸✵✵ ♥❤➙♥ ✈✐➯♥ ✈➠♥ ♣❤á♥❣ ✈➔ t❤➜② r➡♥❣ ❝â ✾✵ ♥❣÷í✐ tr↔ ❧í✐ r➡♥❣ ❤å s➩ tr↔ ❧í✐ ♥❣❛② s❛✉ ❦❤✐ ♥❤➟♥ ✤÷đ❝ ❡✲♠❛✐❧✳ ❍➣② t➻♠ ❦❤♦↔♥❣ t✐♥ ❝➟② ✾✺✪ ❝❤♦ t✛ ❧➺ ♥❤ú♥❣ ♥❤➙♥ ✈✐➯♥ ✈➠♥ ♣❤á♥❣ tr↔ ❧í✐ ♥❣❛② ❦❤✐ ♥❤↕ ✤÷đ❝ ❡✲♠❛✐❧✳ ✉♥✐✈❡rs✐t②✲❧♦❣♦ ✣é ❚❤à ❚❤ó② ❍➡♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❚❤è♥❣ ❦➯ ❳❍❍ ◆❣➔② ✻ t❤→♥❣ ✶ ♥➠♠ ✷✵✶✹ ✷✾ ✴ ✷✾