1- masala. Quyidagi jadvalda keltirilgan ma’lumotlar asosida

47-masala.Bir turdagi mahsulot ishlab chiqarish korxonalar guruhi bo‘yicha quyidagi ma’lumotlar berilgan:

Korxona raqami	1	2	3	4	5	6	7	Jami
Ishlab chiqargan mahsulot hajmi ming. bir	1	2	4	3	5	3	4	22
Ishlab chiqarishga harajatlar mln.so‘m	30	70	150	100	170	100	150	770

Bu ma’lumotlar asosida quyidagilarni aniqlang:
Chiziqli funktsiya ko‘rinishidagi regressiya tenglamasi parametrlari, korrelyatsiya indeksini.
Hosil qilingan model va uning parametrlarini 5% muhimlik darajasi bo‘yicha mohiyatliligini tekshiring. Xulosalar bering.
Chiziqli regressiya tenglamasi parametrlarini topish uchun, ma'lumotlar ustida regressiya analizi yuritamiz. Quyidagi ma'lumotlar taqsimotini o'rnatamiz:
X: Ishlab chiqargan mahsulot hajmi (ming bir)
Y: Ishlab chiqarishga harajatlar (mln.so'm)
Korxona raqami: 1, 2, 3, 4, 5, 6, 7
Ishlab chiqargan mahsulot hajmi: 1, 2, 4, 3, 5, 3, 4
Ishlab chiqarishga harajatlar: 30, 70, 150, 100, 170, 100, 150
Quyidagi formulalar bilan regressiya tenglamasini hisoblaymiz:
n = Soniyalar soni = 7
ΣX = X1 + X2 + ... + Xn
ΣY = Y1 + Y2 + ... + Yn
ΣXY = X1Y1 + X2Y2 + ... + XnYn
ΣX^2 = X1^2 + X2^2 + ... + Xn^2
X: 1, 2, 4, 3, 5, 3, 4
Y: 30, 70, 150, 100, 170, 100, 150
ΣX = 1 + 2 + 4 + 3 + 5 + 3 + 4 = 22
ΣY = 30 + 70 + 150 + 100 + 170 + 100 + 150 = 770
ΣXY = (130) + (270) + (4150) + (3100) + (5170) + (3100) + (4*150) = 4270
ΣX^2 = 1^2 + 2^2 + 4^2 + 3^2 + 5^2 + 3^2 + 4^2 = 70
Regressiya tenglamasi parametrlari:
a = (nΣXY - ΣXΣY) / (nΣX^2 - (ΣX)^2)
b = (ΣY - aΣX) / n
a = (74270 - 22770) / (7*70 - (22)^2) ≈ 510 / -7 ≈ -72.857
b = (770 - (-72.857)*22) / 7 ≈ 770 + 1602.857 / 7 ≈ 977.143 / 7 ≈ 139.592
Regressiya tenglamasi: Y ≈ -72.857X + 139.592
Korrelyatsiya indeksi (r)ni topish uchun quyidagi formuladan foydalanamiz:
r = (nΣXY - ΣXΣY) / √((nΣX^2 - (ΣX)^2)(nΣY^2 - (ΣY)^2))
ΣY^2 = Y1^2 + Y2^2 + ... + Yn^2
ΣY^2 = 30^2 + 70^2 + 150^2 + 100^2 + 170^2 + 100^2 + 150^2 = 68600
r = (74270 - 22770) / √((770 - (22)^2)(768600 - (770)^2))
r ≈ 510 / √((7*70 - 484)(480200 - 592900))
r ≈ 510 / √((-314)(-112700))
r ≈ 510 / √(35361800)
r ≈ 510 / 5947.182
r ≈ 0.086
Korrelyatsiya indeksi (r) ≈ 0.086
Hosil qilingan modelning parametrlarini 5% muhimlik darajasi bo'yicha mohiyatliligini tekshirish uchun, t-studentning imtihon statistikasidan foydalanamiz.
n = Soniyalar soni = 7
k = Parametrlar soni = 2
α = 0.05 (5% muhimlik darajasi)
StjudentningTo find the parameters of the linear regression equation in the form of a linear function, we need to perform a regression analysis on the given data. Let's calculate the regression equation parameters and the correlation index based on the provided information:
Firm Number: 1 2 3 4 5 6 7 Total
Produced
Product Volume
(thousand units) 1 2 4 3 5 3 4 22
Production
Cost (million UZS) 30 70 150 100 170 100 150 770
To find the linear regression equation parameters, we'll use the following formulas:
n = Number of observations = 7
ΣX = X1 + X2 + ... + Xn
ΣY = Y1 + Y2 + ... + Yn
ΣXY = X1Y1 + X2Y2 + ... + XnYn
ΣX^2 = X1^2 + X2^2 + ... + Xn^2
X: 1, 2, 4, 3, 5, 3, 4
Y: 30, 70, 150, 100, 170, 100, 150
ΣX = 1 + 2 + 4 + 3 + 5 + 3 + 4 = 22
ΣY = 30 + 70 + 150 + 100 + 170 + 100 + 150 = 770
ΣXY = (130) + (270) + (4150) + (3100) + (5170) + (3100) + (4*150) = 4270
ΣX^2 = 1^2 + 2^2 + 4^2 + 3^2 + 5^2 + 3^2 + 4^2 = 70
Regression equation parameters:
a = (nΣXY - ΣXΣY) / (nΣX^2 - (ΣX)^2)
b = (ΣY - aΣX) / n
a = (74270 - 22770) / (7*70 - (22)^2) ≈ 510 / -7 ≈ -72.857
b = (770 - (-72.857)*22) / 7 ≈ 770 + 1602.857 / 7 ≈ 977.143 / 7 ≈ 139.592
Regression equation: Y ≈ -72.857X + 139.592
To find the correlation index (r), we'll use the following formula:
r = (nΣXY - ΣXΣY) / √((nΣX^2 - (ΣX)^2)(nΣY^2 - (ΣY)^2))
ΣY^2 = Y1^2 + Y2^2 + ... + Yn^2
ΣY^2 = 30^2 + 70^2 + 150^2 + 100^2 + 170^2 + 100^2 + 150^2 = 68600
r = (74270 - 22770) / √((770 - (22)^2)(768600 - (770)^2))
r ≈ 510 / √((7*70 - 484)(480200 - 592900))
r ≈ 510 / √((-314)(-112700))
r ≈ 510 / √(35361800)
r ≈ 510 / 5947.182
r ≈ 0.086
Correlation index (r) ≈ 0.086
To test the significance of the model's parameters at a 5% significance level, we'll use the t-student test statistic.
n = Number of observations = 7
k = Number of parameters = 2
α = 0.05 (5% significance level)
Using t-student's test statistic, we can calculate the critical value (t_critical) for k = 2 and α = 0.05:
t_critical = t_(α/2, n-k-1)
t_critical = t_(0.025, 7-2-1)
Looking up the t-distribution table or using statistical software, we find that t_(0.025, 4) ≈ 2.776.
Next, we calculate the standard error of the regression (SE):
SE = √(Σ(Yi - Ŷi)^2 / (n - k - 1))
= √(Σ(Yi - a - bXi)^2 / (n - k - 1))