1- masala. Quyidagi jadvalda keltirilgan ma’lumotlar asosida

24- Masala. Berilgan ma’lumotlar asosida xususiy korrelyatsiya koeffitsientini hisoblang va tahlil qiling. Iqtisodiy tahlilni amalga oshirib, xulosa bering.

Y	5	8	10	11
X1	2	5	6	10
X2	3	7	9	14

Korrelyatsiya koeffitsientini hisoblash uchun, berilgan ma'lumotlardan foydalanamiz:
Y: 5, 8, 10, 11
X1: 2, 5, 6, 10
X2: 3, 7, 9, 14

1. 
   O'rtacha qiymatlarni hisoblaymiz:
   Y_mean = (5 + 8 + 10 + 11) / 4 = 8.5
   X1_mean = (2 + 5 + 6 + 10) / 4 = 5.75
   X2_mean = (3 + 7 + 9 + 14) / 4 = 8.25
   
2. 
   Bog'liqlikning kuchini hisoblaymiz:
   Cov(X1, Y) = Σ((X1 - X1_mean) * (Y - Y_mean)) / n
   = ((2 - 5.75) * (5 - 8.5) + (5 - 5.75) * (8 - 8.5) + (6 - 5.75) * (10 - 8.5) + (10 - 5.75) * (11 - 8.5)) / 4
   = (-3.75 * -3.5 + -0.75 * -0.5 + 0.25 * 1.5 + 4.25 * 2.5) / 4
   = (13.125 + 0.375 + 0.375 + 10.625) / 4
   = 24.5 / 4
   = 6.125
   

Cov(X2, Y) = Σ((X2 - X2_mean) * (Y - Y_mean)) / n
= ((3 - 8.25) * (5 - 8.5) + (7 - 8.25) * (8 - 8.5) + (9 - 8.25) * (10 - 8.5) + (14 - 8.25) * (11 - 8.5)) / 4
= (-5.25 * -3.5 + -1.25 * -0.5 + 0.75 * 1.5 + 5.75 * 2.5) / 4
= (18.375 + 0.625 + 1.125 + 14.375) / 4
= 34.5 / 4
= 8.625

3. 
   Bog'liqlikning tomoshasini hisoblaymiz:
   Var(X1) = Σ((X1 - X1_mean)^2) / n
   = ((2 - 5.75)^2 + (5 - 5.75)^2 + (6 - 5.75)^2 + (10 - 5.75)^2) / 4
   = (3.75^2 + 0.75^2 + 0.25^2 + 4.25^2) / 4
   = (14.0625 + 0.5625 + 0.0625 + 18.0625) / 4
   = 33.75 / 4
   = 8.4375
   

Var(X2) = Σ((X2 - X2_mean)^2) / n
= ((3 - 8.25)^2 + (7 - 8.25)^2 + (9 - 8.25)^2 + (14 - 8.25)^2) / 4
= (5.25^2 + 1.25^2 + 0.75^2 + 5.75^2) / 4
= (27.5625 + 1.5625 + 0.5625 + 33.0625) / 4
= 62.75 / 4
= 15.6875

4. 
   Korrelyatsiya koeffitsientini hisoblaymiz:
   Corr(X1, Y) = Cov(X1, Y) / sqrt(Var(X1) * Var(Y))
   = 6.125 / sqrt(8.4375 * Var(Y))
   = 6.125 / sqrt(8.4375 * ((5 - 8.5)^2 + (8 - 8.5)^2 + (10 - 8.5)^2 + (11 - 8.5)^2) / 4)
   = 6.125 / sqrt(8.4375 * ((-3.Korrelyatsiya koeffitsientini hisoblash uchun, berilgan ma'lumotlardan foydalanamiz:
   

Y: 5, 8, 10, 11
X1: 2, 5, 6, 10
X2: 3, 7, 9, 14

1. 
   O'rtacha qiymatlarni hisoblaymiz:
   Y_mean = (5 + 8 + 10 + 11) / 4 = 8.5
   X1_mean = (2 + 5 + 6 + 10) / 4 = 5.75
   X2_mean = (3 + 7 + 9 + 14) / 4 = 8.25
   
2. 
   Bog'liqlikning kuchini hisoblaymiz:
   Cov(X1, Y) = Σ((X1 - X1_mean) * (Y - Y_mean)) / n
   = ((2 - 5.75) * (5 - 8.5) + (5 - 5.75) * (8 - 8.5) + (6 - 5.75) * (10 - 8.5) + (10 - 5.75) * (11 - 8.5)) / 4
   = (-3.75 * -3.5 + -0.75 * -0.5 + 0.25 * 1.5 + 4.25 * 2.5) / 4
   = (13.125 + 0.375 + 0.375 + 10.625) / 4
   = 24.5 / 4
   = 6.125
   

Cov(X2, Y) = Σ((X2 - X2_mean) * (Y - Y_mean)) / n
= ((3 - 8.25) * (5 - 8.5) + (7 - 8.25) * (8 - 8.5) + (9 - 8.25) * (10 - 8.5) + (14 - 8.25) * (11 - 8.5)) / 4
= (-5.25 * -3.5 + -1.25 * -0.5 + 0.75 * 1.5 + 5.75 * 2.5) / 4
= (18.375 + 0.625 + 1.125 + 14.375) / 4
= 34.5 / 4
= 8.625

3. 
   Bog'liqlikning tomoshasini hisoblaymiz:
   Var(X1) = Σ((X1 - X1_mean)^2) / n
   = ((2 - 5.75)^2 + (5 - 5.75)^2 + (6 - 5.75)^2 + (10 - 5.75)^2) / 4
   = (3.75^2 + 0.75^2 + 0.25^2 + 4.25^2) / 4
   = (14.0625 + 0.5625 + 0.0625 + 18.0625) / 4
   = 33.75 / 4
   = 8.4375
   

Var(X2) = Σ((X2 - X2_mean)^2) / n
= ((3 - 8.25)^2 + (7 - 8.25)^2 + (9 - 8.25)^2 + (14 - 8.25)^2) / 4
= (5.25^2 + 1.25^2 + 0.75^2 + 5.75^2) / 4
= (27.5625 + 1.5625 + 0.5625 + 33.0625) / 4
= 62.75 / 4
= 15.6875

4. 
   Korrelyatsiya koeffitsientini hisoblaymiz:
   Corr(X1, Y) = Cov(X1, Y) / sqrt(Var(X1) * Var(Y))
   = 6.125 / sqrt(8.4375 * Var(Y))
   = 6.125 / sqrt(8.4375 * ((5 - 8.5)^2 + (8 - 8.5)^2 + (10 - 8.5)^2 + (11 - 8.5)^2) / 4)
   = 6.125 / sqrt(8.4375 * ((-3.