1- masala. Quyidagi jadvalda keltirilgan ma’lumotlar asosida

Yil	2012	2013	2014	2015	2016	2017	2018	2019	2020
2021
Y	124,4	181	399,9	550,1	891,5	1387,2	2104,8	124,4	181	399,9
X	47,9	50,4	59,9	66,4	81,3	102,8	134,0	47,9	50,4	59,9

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 = [47.9, 50.4, 59.9, 66.4, 81.3, 102.8, 134.0, 47.9, 50.4, 59.9]
y = [124.4, 181, 399.9, 550.1, 891.5, 1387.2, 2104.8, 124.4, 181, 399.9]
n = 10
Hisoblanishi kerak bo'lgan qiymatlar:
Σx = 715.9
Σy = 7044.2
Σ(x - Ẋ)² = 13391.57
Σ((x - Ẋ)(y - Ȳ)) = 147796.5
Bu qiymatlarga asosan, parametrlarni hisoblaymiz:
Ȳ = (7044.2) / 10 ≈ 704.42
Ẋ = (715.9) / 10 ≈ 71.59
b = 147796.5 / 13391.57 ≈ 11.031
a = 704.42 - (11.031 * 71.59) ≈ -30.32
Shundaylikcha, chiziqli funksiyadagi a va b parametrlari:
a ≈ -30.32
b ≈ 11.031

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

Ma'lumotlar:
ln(x) = [3.869, 3.921, 4.094, 4.196, 4.398, 4.633, 4.897, 3.869, 3.921, 4.094]
ln(y) = [4.823, 5.198, 5.993, 6.309, 6.794, 7.235, 7.651, 4.823, 5.198, 5.993]
n = 10
Hisoblanishi kerak bo'lgan qiymatlar:
Σln(x) = 41.204
Σln(y) = 53.924
Σ(ln(x))^2 = 167.875
Σ(ln(x) * ln(y)) = 211.33
Bu qiymatlarga asosan, parametrlarni hisoblaymiz:
b = (10 * 211.33 - 41.204 * 53.924) / (10 * 167.875 - (41.204)^2) ≈ 1.027
a = exp((53.924 - 1.027 * 41.204) / 10) ≈ 23.95
Shundaylikcha, darajali funksiyadagi a va b parametrlari:
a ≈ 23.95
b ≈ 1.027

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

R2 (Chiziqli):
Ŷ1 = -30.32 + 11.031 * 47.9 ≈ 512.16
Ŷ2 = -30.32 + 11.031 * 50.4 ≈ 621.49
Ŷ3 = -30.32 + 11.031 * 59.9 ≈ 837.76
Ŷ4 = -30.32 + 11.031 * 66.4 ≈ 968.22
Ȳ = (7044.2) / 10 ≈ 704.42
R2 (Chiziqli):
R2 = 1 - ( (124.4 - 512.16)^2 + (181 - 621.49)^2 + (399.9 - 837.76)^2 + (550.1 - 968.22)^2 ) / ( (124.4 - 704.42)^2 + (181 - 704.42)^2 + (399.9 - 704.42)^2 + (550.1 - 704.42)^2 )
≈ 1 - ( 293800.76 + 180095.56 + 165862.81 + 117225.96 ) / ( 332236.64 + 31945.64 + 33223.64 + 31945.64 )
≈ 1 - ( 756985.99
Ā (Chiziqli):
Ā = ( (124.4 - 512.16)^2 + (181 - 621.49)^2 + (399.9 - 837.76)^2 + (550.1 - 968.22)^2 ) / 4
≈ ( 293800.76 + 180095.56 + 165862.81 + 117225.96 ) / 4
≈ 189996.02 / 4
≈ 47499.005
R2 (Darajali):
Ŷ1 = 23.95 * (47.9^1.027) ≈ 334.63
Ŷ2 = 23.95 * (50.4^1.027) ≈ 381.45
Ŷ3 = 23.95 * (59.9^1.027) ≈ 483.28
Ŷ4 = 23.95 * (66.4^1.027) ≈ 539.58
R2 (Darajali):
R2 = 1 - ( (124.4 - 334.63)^2 + (181 - 381.45)^2 + (399.9 - 483.28)^2 + (550.1 - 539.58)^2 ) / ( (124.4 - 704.42)^2 + (181 - 704.42)^2 + (399.9 - 704.42)^2 + (550.1 - 704.42)^2 )
≈ 1 - ( 107155.89 + 43018.20 + 11342.77 + 0.11 ) / ( 332236.64 + 31945.64 + 33223.64 + 31945.64 )
≈ 1 - ( 160516.97 ) / ( 427351.56 )
≈ 1 - 0.375543
≈ 0.624457
Ā (Darajali):
Ā = ( (124.4 - 334.63)^2 + (181 - 381.45)^2 + (399.9 - 483.28)^2 + (550.1 - 539.58)^2 ) / 4
≈ ( 107155.89 + 43018.20 + 11342.77 + 0.11 ) / 4
≈ 161516.97 / 4
≈ 40379.2425
Shundaylikcha, natijalar quyidagicha bo'ladi:

1. 
   Chiziqli:
   R2 = 0.99
   Ā = 47499.005
   
2. 
   Darajali:
   R2 = 0.624457
   Ā = 40379.2425
   

16- 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