1. Muammo: Berilgan statistik ma'lumotlar asosida tanlanma o'rtacha qiymat $X_T$, dispersiya $D_T$ va standart og'ish $C3$ ni hisoblash. 2. Formulalar va qoidalar: - Tanlanma o'rtacha qiymat: $$X_T = \frac{\sum n_i X_i}{\sum n_i}$$ - Tanlanma dispersiyasi: $$D_T = \frac{\sum n_i (X_i - X_T)^2}{\sum n_i}$$ - Standart og'ish: $$\sigma = \sqrt{D_T}$$ 3. Hisoblashlar: - $\sum n_i = 4 + 6 + 4 = 14$ - $\sum n_i X_i = 4 \times 15 + 6 \times 20 + 4 \times 25 = 60 + 120 + 100 = 280$ - Shunday qilib, $$X_T = \frac{280}{14} = 20$$ - Endi dispersiyani hisoblaymiz: $$\sum n_i (X_i - X_T)^2 = 4(15 - 20)^2 + 6(20 - 20)^2 + 4(25 - 20)^2 = 4 \times 25 + 6 \times 0 + 4 \times 25 = 100 + 0 + 100 = 200$$ - Dispersiya: $$D_T = \frac{200}{14} \approx 14.29$$ - Standart og'ish: $$\sigma = \sqrt{14.29} \approx 3.78$$ 4. Muammo: Berilgan $X$ va $Y$ qiymatlar asosida chiziqli regressiya tenglamasini $y = a_0 + a_1 x$ va korrelyatsiya koeffitsientini topish. 5. Formulalar: - Regressiya koeffitsientlari: $$a_1 = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$$ $$a_0 = \bar{y} - a_1 \bar{x}$$ - Korrelyatsiya koeffitsienti: $$r = \frac{n \sum xy - \sum x \sum y}{\sqrt{(n \sum x^2 - (\sum x)^2)(n \sum y^2 - (\sum y)^2)}}$$ 6. Hisoblashlar: - $n=5$ - $\sum x = 1 + 2 + 5 + 6 + 8 = 22$ - $\sum y = 2.5 + 3.1 + 4.1 + 3.5 + 4.2 = 17.4$ - $\sum x^2 = 1^2 + 2^2 + 5^2 + 6^2 + 8^2 = 1 + 4 + 25 + 36 + 64 = 130$ - $\sum y^2 = 2.5^2 + 3.1^2 + 4.1^2 + 3.5^2 + 4.2^2 = 6.25 + 9.61 + 16.81 + 12.25 + 17.64 = 62.56$ - $\sum xy = 1 \times 2.5 + 2 \times 3.1 + 5 \times 4.1 + 6 \times 3.5 + 8 \times 4.2 = 2.5 + 6.2 + 20.5 + 21 + 33.6 = 83.8$ - Hisoblaymiz $a_1$ ni: $$a_1 = \frac{5 \times 83.8 - 22 \times 17.4}{5 \times 130 - 22^2} = \frac{419 - 382.8}{650 - 484} = \frac{36.2}{166} \approx 0.2181$$ - Hisoblaymiz $a_0$ ni: $$\bar{x} = \frac{22}{5} = 4.4, \quad \bar{y} = \frac{17.4}{5} = 3.48$$ $$a_0 = 3.48 - 0.2181 \times 4.4 = 3.48 - 0.9596 = 2.5204$$ - Regressiya tenglamasi: $$y = 2.5204 + 0.2181 x$$ - Korrelyatsiya koeffitsienti $r$: $$r = \frac{5 \times 83.8 - 22 \times 17.4}{\sqrt{(5 \times 130 - 22^2)(5 \times 62.56 - 17.4^2)}} = \frac{36.2}{\sqrt{166 \times (312.8 - 302.76)}} = \frac{36.2}{\sqrt{166 \times 10.04}} = \frac{36.2}{\sqrt{1666.64}} = \frac{36.2}{40.83} \approx 0.8865$$ Natija: - Tanlanma o'rtacha qiymat $X_T = 20$ - Dispersiya $D_T \approx 14.29$ - Standart og'ish $\sigma \approx 3.78$ - Regressiya tenglamasi: $$y = 2.5204 + 0.2181 x$$ - Korrelyatsiya koeffitsienti: $$r \approx 0.8865$$