The values of the independent variable ( X ) and the dependent variable ( Y ) are given below:

Step 1:

Summations:

• Sum of X (ΣX) = 0 + 1 + 2 + 3 + 4 = 10
• Sum of Y (ΣY) = 2 + 3 + 5 + 4 + 6 = 20
• Sum of XY (ΣXY) = 0 + 3 + 10 + 12 + 24 = 49
• Sum of X² (ΣX²) = 0 + 1 + 4 + 9 + 16 = 30
• n = 5

Step 2: Set up the equations for a and b
The least squares regression line equation is Y = aX + b. We need to solve for a and b using the following equations derived from the method of least squares:

1. ΣY = aΣX + nb
2. ΣXY = aΣX² + bΣX

Substitute the sums we calculated:

20 = a×10 + 5×b (Equation 1)

49 = a ×30 + b ×10 (Equation 2)
Step 3: Solve the system of equations for a and b
From Equation 1:

20 = 10a + 5b

Dividing the entire equation by 5:

4 = 2a + b (Equation 3)

Rearrange Equation 3 to solve for b:

b = 4 – 2a (Equation 3)

Substitute Equation 3 into Equation 2:

49 = 30a + 10(4 – 2a)

49 = 30a + 40 – 20a

Combine like terms:

49 = 10a + 40

Subtract 40 from both sides:

9 = 10a

Divide by 10:

a = 0.9

Now, substitute a = 0.9 back into Equation 3:

b = 4 – 2(0.9)

b = 4 – 1.8

b = 2.2
Step 4: Form the least squares regression line
The regression line is:

Y = 0.9X + 2.2
Step 5: Estimate Y when X = 10
Substitute X = 10 into the regression equation:

Y = 0.9 * 10 + 2.2

Y = 9 + 2.2

Y = 11.2
Final Answer
When X = 10, the estimated value of Y is 11.2