1- masala. Quyidagi jadvalda keltirilgan ma’lumotlar asosida

46-masala.Respublikada aholi jon boshiga to‘g‘ri kelgan yalpi ichki maxsulot qiymati, ming so‘m (raqamlar shartli):

Yillar	2015	2016	2017	2018	2019	2020	2021	2022
YaIM	243,7	253,6	272,3	306,3	394,6	445,5	544,0	652,0

Berilgan ma’lumotlar asosida:
1) To‘rt yillik sirg‘anchiq o‘rtachalarni aniqlang;
2) Analitik tekislang;
3) Approksimatsiya o‘rtacha xatosini aniqlang;
4) Avtokorrelyatsiya mavjud yoki mavjud emasligini tekshiring.

1. 
   To'rt yillik sirg'an chiqish o'rtachalarini hisoblash uchun, berilgan ma'lumotlarni yillar bo'yicha yig'indisini hisoblaymiz va o'rta arifmetikani hisoblaymiz:
   

Yillar: 2015, 2016, 2017, 2018
YaIM: 243.7, 253.6, 272.3, 306.3
Yillar bo'yicha yig'indi: 243.7 + 253.6 + 272.3 + 306.3 = 1076.9
To'rt yillik sirg'an chiqish o'rtachasi: 1076.9 / 4 = 269.225

2. 
   Analitik tekislanish:
   

Analitik tekislanishni hisoblash uchun, har bir yilning YaIM qiymatini o'rtacha qiymatdan ayiramiz va natijani kvadratga oshiramiz. Keyin, olishgan kvadrat qiymatlarini yig'indisini hisoblaymiz va n - 1 ga bo'lib bo'lagan holda kvadrat o'rtachasini hisoblaymiz.
Yillar: 2015, 2016, 2017, 2018, 2019, 2020, 2021, 2022
YaIM: 243.7, 253.6, 272.3, 306.3, 394.6, 445.5, 544.0, 652.0
Ortacha qiymat: 269.225
Analitik tekislanishlar:
(243.7 - 269.225)^2 ≈ 649.296
(253.6 - 269.225)^2 ≈ 242.050
(272.3 - 269.225)^2 ≈ 9.472
(306.3 - 269.225)^2 ≈ 1371.856
(394.6 - 269.225)^2 ≈ 15646.849
(445.5 - 269.225)^2 ≈ 30967.600
(544.0 - 269.225)^2 ≈ 30481.376
(652.0 - 269.225)^2 ≈ 66083.776
Yig'indi: 649.296 + 242.050 + 9.472 + 1371.856 + 15646.849 + 30967.600 + 30481.376 + 66083.776 ≈ 159452.225
Kvadrat o'rtacha: √(159452.225 / (8 - 1)) ≈ 228.458

3. 
   Approksimatsiya o'rtacha xatosi:
   

Approksimatsiya o'rtacha xatosini hisoblash uchun, o'rtacha qiymatlarni YaIM qiymatlaridan ayirib absolut qiymatlarni hisoblaymiz. Keyin, absolut qiymatlarning o'rtachasini hisoblaymiz.
Yillar: 2015, 2016, 2017, 2018, 2019, 2020, 2021, 2022
YaIM: 243.7, 253.6, 272.3, 306.3, 394.6, 445.5, 544.0, 652.0
Ortacha qiymat: 269.225
Approksimatsiya o'rtacha xatosi:
|243.7 - 269.225| ≈ 25.525
|253.6 - 269.225| ≈ 15.625
|272.3 - 269.225| ≈ 3.075
|306.3 - 269.225| ≈ 37.075
|394.6 - 269.225| ≈ 125.375
|445.5 - 269.225| ≈ 176.275
|544.0 - 269.225| ≈ 274.775
|652.0 - 269.225| ≈ 382.775
O'rtacha qiymati: (25.525 + 15.625 + 3.075 + 37.075 + 125.375 + 176.275 + 274.775 + 382.775) / 8 ≈ 116.775

4. 
   Avtokorrelyatsiya4) Avtokorrelyatsiya mavjudligini tekshirish uchun, berilgan ma'lumotlar ustida avtokorrelyatsiya funksiyasini hisoblaymiz. Avtokorrelyatsiya funksiyasi, ma'lumotlar o'rtachasidan qanday qancha aniq aloqani ifodalaydi.
   

Avtokorrelyatsiya funksiyasini hisoblash uchun quyidagi formuladan foydalanamiz:
r_k = (Σ[(Y_t - Y_ort) * (Y_t+k - Y_ort)]) / (Σ[(Y_t - Y_ort)^2])
Bu yerda Y_t ma'lumotlar qatoridagi t-ta elementni, Y_ort esa ma'lumotlar o'rtachasini ifodalaydi. k avtokorrelyatsiya uzluksini ifodalayadi.
Yillar: 2015, 2016, 2017, 2018, 2019, 2020, 2021, 2022
YaIM: 243.7, 253.6, 272.3, 306.3, 394.6, 445.5, 544.0, 652.0
Ortacha qiymat: 269.225
Avtokorrelyatsiya mavjudligini tekshirish uchun k = 1 dan 6 gacha qaraladi:
k = 1:
r_1 = ((243.7 - 269.225) * (253.6 - 269.225) + (253.6 - 269.225) * (272.3 - 269.225) + (272.3 - 269.225) * (306.3 - 269.225) + (306.3 - 269.225) * (394.6 - 269.225) + (394.6 - 269.225) * (445.5 - 269.225) + (445.5 - 269.225) * (544.0 - 269.225) + (544.0 - 269.225) * (652.0 - 269.225)) / ((243.7 - 269.225)^2 + (253.6 - 269.225)^2 + (272.3 - 269.225)^2 + (306.3 - 269.225)^2 + (394.6 - 269.225)^2 + (445.5 - 269.225)^2 + (544.0 - 269.225)^2 + (652.0 - 269.225)^2)
r_1 ≈ 0.967
k = 2:
r_2 = ((243.7 - 269.225) * (272.3 - 269.225) + (253.6 - 269.225) * (306.3 - 269.225) + (272.3 - 269.225) * (394.6 - 269.225) + (306.3 - 269.225) * (445.5 - 269.225) + (394.6 - 269.225) * (544.0 - 269.225) + (445.5 - 269.225) * (652.0 - 269.225)) / ((243.7 - 269.225)^2 + (253.6 - 269.225)^2 + (272.3 - 269.225)^2 + (306.3 - 269.225)^2 + (394.6 - 269.225)^2 + (445.5 - 269.225)^2 + (544.0 - 269.225)^2 + (652.0 - 269.225)^2)
r_2 ≈ 0.922
k = 3:
r_3 = ((243.7 - 269.225) * (306.3 - 269.225) + (253.6 - 269.225) * (394.6 - 269.225) + (272.3 - 269.225) * (445.5 - 269.225) + (306.3 - 269.225) * (544.0 - 269.225) + (394.6 - 269.225) * (652.0 - 269.225)) / ((243.7 - 269.225)^2 + (253.6 - 269.225)^2 + (272.3 - 269.225)^2 + (306.3 - 269.225)^2 + (394.6 - 269.