b. Find two lines of regression from the following data

Steps :

1. assume mean
2. dx=x-(assumed mean of x) , dy= y – (assumed mean of y )
3. prepare an table and calculate summations of
   x , y , dx , dy , dx^2 , dy^2 , dxdy
4. Now calculate mean of x and y
   x mean = Σx/n and y mean = Σy/n
5. Now find regression coefficients bxy , byx
   
6. Calculate regression lines
   y on x
   y – ymean= byx(x- xmean)
   x on y
   x- xmean = bxy(y-ymean)
7. Now solve the question by putting in the equation

Solution of the above question :

To find the two lines of regression from the given data, we need to follow the steps you have mentioned.

Given data:
Age of Husband (x): 25, 22, 28, 26, 35, 20, 22, 40, 20, 18
Age of Wife (y): 18, 15, 20, 17, 22, 14, 16, 21, 15, 14

Step 1: Assume the mean values of x and y.
Let’s assume the mean values as x̄ = 26 and ȳ = 17.

Step 2: Prepare a table and calculate the summations of x, y, dx, dy, dx^2, dy^2, and dxdy.

Σx = 256, Σy = 172, Σdx = -4, Σdy = 2, Σdx^2 = 450, Σdy^2 = 78, Σdxdy = 172

calculate mean of x and y x=25.6 and y=17.2

Step 3: Calculate the regression coefficients bxy and byx.

bxy = (nΣ(dxdy) – Σdx Σdy) / (nΣ(dy^2) – (Σdy)^2)
= (10 × 172 – 4 × 2) / (10 × 78 – 4)
= 2.227

byx = (nΣ(dxdy) – Σdx Σdy) / (nΣ(dx^2) – (Σdx)^2)
= (10 × 172 – 4 × 2) / (10 × 450 – 16)
= 0.385

Step 4: Calculate the regression lines y on x and x on y.

y – ȳ = byx(x – x̄)
y – 17.2 = 0.385(x – 25.6)
y = 0.385x + 7.344

x – x̄ = bxy(y – ȳ)
x – 25.6 = 2.227(y – 17.2)
x = 2.227y – 12.704

Step 5: Solve the given questions using the regression equations.

(i) To find the age of the husband when the age of the wife is 19:
Substitute y = 19 in the second equation (x on y):
x = 2.227(19)-12.704
x = 29.601

Therefore, when the age of the wife is 19, the estimated age of the husband is approximately 30 years.

(ii) To find the age of the wife when the age of the husband is 30:
Substitute x = 30 in the first equation (y on x):
y = 0.385(30)+7.344 = 18.894

Therefore, when the age of the husband is 30, the estimated age of the wife is approximately 19 years.