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 | XÂ² |
---|---|---|---|

0 | 2 | 0 | 0 |

1 | 3 | 3 | 1 |

2 | 5 | 10 | 4 |

3 | 4 | 12 | 9 |

4 | 6 | 24 | 16 |

Summations:

- Sum of X (Î£X) = 0 + 1 + 2 + 3 + 4 = 10
- Sum of Y (Î£Y) = 2 + 3 + 5 + 4 + 6 = 20
- Sum of XY (Î£XY) = 0 + 3 + 10 + 12 + 24 = 49
- Sum of XÂ² (Î£XÂ²) = 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:

- Î£Y = aÎ£X + nb
- Î£XY = aÎ£XÂ² + bÎ£X

Substitute the sums we calculated:

20 = aÃ—10 + 5Ã—b (Equation 1)

49 = a Ã—30 + b Ã—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

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