1- masala. Quyidagi jadvalda keltirilgan ma’lumotlar asosida

Yil	2012	2013	2014	2015	2016	2017	2018	2019	2020
2021
Y	3255,3	4083,3	4615,8	5978,3	7538,8	9304,9	11310,7	3255,3	4083,3	4615,8
X	184,0	220,0	243,2	302,4	370,3	447,1	534,3	184,0	220,0	243,2

Topshiriq

Y ning X ga ga bog‘liqligini tavsiflash uchun quyidagi funktsiyalar parametrlarini hisoblang:

1. 
   Chiziqli;
   
2. 
   Darajali;
   
3. 
   Har bir aniqlangan modelni R², o‘rtacha aproksimatsiya hatoligi – Ā bilan baholang.
   

Berilgan ma'lumotlar asosida funktsiyalar parametrlarini hisoblaylik:

1. 
   Chiziqli funksiya:
   y = a + bx
   

Ma'lumotlar:
x = [184.0, 220.0, 243.2, 302.4, 370.3, 447.1, 534.3, 184.0, 220.0, 243.2]
y = [3255.3, 4083.3, 4615.8, 5978.3, 7538.8, 9304.9, 11310.7, 3255.3, 4083.3, 4615.8]
n = 10
Hisoblanishi kerak bo'lgan qiymatlar:
Σx = 2740.5
Σy = 65455.5
Σ(x - Ẋ)² = 345740.61
Σ((x - Ẋ)(y - Ȳ)) = 7496540.23
Bu qiymatlarga asosan, parametrlarni hisoblaymiz:
Ȳ = (65455.5) / 10 = 6545.55
Ẋ = (2740.5) / 10 = 274.05
b = 7496540.23 / 345740.61 = 21.68
a = 6545.55 - (21.68 * 274.05) = 8925.40
Shundaylikcha, chiziqli funksiyadagi a va b parametrlari:
a ≈ 8925.40
b ≈ 21.68

2. 
   Darajali funksiya:
   y = a * x^b
   

Ma'lumotlar:
ln(x) = [5.214, 5.393, 5.496, 5.712, 5.915, 6.103, 6.282, 5.214, 5.393, 5.496]
ln(y) = [8.087, 8.316, 8.437, 8.696, 8.928, 9.137, 9.332, 8.087, 8.316, 8.437]
n = 10
Hisoblanishi kerak bo'lgan qiymatlar:
Σln(x) = 56.232
Σln(y) = 83.178
Σ(ln(x))^2 = 312.687
Σ(ln(x) * ln(y)) = 461.245
Bu qiymatlarga asosan, parametrlarni hisoblaymiz:
b = (10 * 461.245 - 56.232 * 83.178) / (10 * 312.687 - (56.232)^2) = 1.285
a = exp((83.178 - 1.285 * 56.232) / 10) = 3013.84
Shundaylikcha, darajali funksiyadagi a va b parametrlari:
a ≈ 3013.84
b ≈ 1.285

3. 
   R2 koeffitsienti va o'rtacha aproksimatsiya hatoligi Ā ni hisoblaymiz:
   

R2 (Chiziqli):
Ŷ1 = 8925.40 + 21.68 * 184.0 ≈ 13563.79
Ŷ2 = 8925.40 + 21.68 * 220.0 ≈ 15398.12
Ŷ3 = 8925.40 + 21.68 * 243.2 ≈ 16525.18
Ŷ4 = 8925.40 + 21.68 * 302.4 ≈ 18848.39
Ȳ = (65455.5) / 10 = 6545.55
R2 (Chiziqli):
R2 = 1 - ( (3255.3 - 13563.79)^2 + (4083.3 - 15398.12)^2 + (4615.8 - 16525.18)^2 + (5978.3 - 18848.39)^2 ) / ( (3255.3 - 6545.55)^2 + (4083.3 - 6545.55)^2 + (4615.8 - 6545.55)^2 + (5978.3 - 6545.55)^2 )
≈ 1 - ( 83224245.11 + 74377740.53 + 514Kechirasiz, men ma'lumotlarni to'g'ri hisoblash uchun kesib turib qoldim. Men uni to'g'ri hisoblab chiqaraman.

1. 
   Chiziqli funksiya:
   y = a + bx
   

Ma'lumotlar:
x = [184.0, 220.0, 243.2, 302.4, 370.3, 447.1, 534.3, 184.0, 220.0, 243.2]
y = [3255.3, 4083.3, 4615.8, 5978.3, 7538.8, 9304.9, 11310.7, 3255.3, 4083.3, 4615.8]
n = 10
Hisoblanishi kerak bo'lgan qiymatlar:
Σx = 2740.5
Σy = 65455.5
Σ(x - Ẋ)² = 345740.61
Σ((x - Ẋ)(y - Ȳ)) = 7496540.23
Bu qiymatlarga asosan, parametrlarni hisoblaymiz:
Ẋ = Σx / n = 2740.5 / 10 = 274.05
Ȳ = Σy / n = 65455.5 / 10 = 6545.55
b = Σ((x - Ẋ)(y - Ȳ)) / Σ(x - Ẋ)² = 7496540.23 / 345740.61 ≈ 21.68
a = Ȳ - bẊ = 6545.55 - 21.68 * 274.05 ≈ 6598.75
Shundaylikcha, chiziqli funksiyadagi a va b parametrlari:
a ≈ 6598.75
b ≈ 21.68

2. 
   Darajali funksiya:
   y = a * x^b
   

Ma'lumotlar:
ln(x) = [5.214, 5.393, 5.496, 5.712, 5.915, 6.103, 6.282, 5.214, 5.393, 5.496]
ln(y) = [8.087, 8.316, 8.437, 8.696, 8.928, 9.137, 9.332, 8.087, 8.316, 8.437]
n = 10
Hisoblanishi kerak bo'lgan qiymatlar:
Σln(x) = 56.232
Σln(y) = 83.178
Σ(ln(x))^2 = 312.687
Σ(ln(x) * ln(y)) = 461.245
Bu qiymatlarga asosan, parametrlarni hisoblaymiz:
b = (n * Σ(ln(x) * ln(y)) - Σln(x) * Σln(y)) / (n * Σ(ln(x))^2 - (Σln(x))^2) = (10 * 461.245 - 56.232 * 83.178) / (10 * 312.687 - 56.232^2) ≈ 1.285
a = exp((Σln(y) - b * Σln(x)) / n) = exp((83.178 - 1.285 * 56.232) / 10) ≈ 3013.84
Shundaylikcha, darajali funksiyadagi a va b parametrlari:
a ≈ 3013.84
b ≈ 1.285

3. 
   R2 koeffitsienti va o'rtacha aproksimatsiya hatoligi Ā ni hisoblaymiz:
   

R2 (Chiziqli):
Ŷ1 = 6598.75 + 21.68 * 184.0 ≈ 10724.47
Ŷ2 = 6598.75 + 21.68 * 220.0 ≈ 12005.15
Ŷ3 = 6598.75 + 21.68 * 243.2 ≈ 12911.66
Ŷ4 = 6598.75 + 21.68 * 302.4 ≈ 14777.59
R2 (Darajali):
Ŷ1 = 3013.84 * 184.0^1.285 ≈ 10198.07


18- Masala. O‘zbekiston Rspublikasi aholisining daromadi – Y ming so‘mni tashkil etib undan X ming so‘mini jamg‘arma sifatida bankka omonat sifatida qo‘yishadi. Ko‘rsatkich qiymatlarining yillar davomida o‘zgarishi quyidagi jadvalda berilgan