The values of the independent variable ( X ) and the dependent variable ( Y ) are given below:
| X | Y |
| 0 | 2 |
| 1 | 3 |
| 2 | 5 |
| 3 | 4 |
| 4 | 6 |
Find the least square regression line ( Y = aX + b ). Estimate ( Y ) when the value of ( X ) equals 10. [10]
Step 1:
| X | Y | XY | X² |
|---|---|---|---|
| 0 | 2 | 0 | 0 |
| 1 | 3 | 3 | 1 |
| 2 | 5 | 10 | 4 |
| 3 | 4 | 12 | 9 |
| 4 | 6 | 24 | 16 |
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:
- ΣY = aΣX + nb
- 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
[…] B. The values of the independent variable ( X ) and the dependent variable ( Y ) are given below: […]