ISS PREVIOUS YEAR 2016 PAPER-2 SOLUTION SET-A Q.NO. 10

ISS PREVIOUS YEAR 2016 PAPER-2 SOLUTION SET-A

ISS PREVIOUS YEAR 2016 PAPER-2 SOLUTION SET-A

 Question 10 :

Which of the following statements are correct in order to obtain the hypothesis H₀: β₁ = 0 for the model:

yᵢ = β₀ + β₁xᵢ + εᵢ

Statements:

1. yᵢ ~ N(β₀ + β₁xᵢ, σ²)

2. β̂₁ ~ N(β₁, σ² / Σ(xᵢ − x̄)²)

3. (n − 2)σ̂² / σ² ~ χ²(n − 2)

4. β̂₁ and σ̂² are not independent

Options:(a)1, 2 and 3(b) 1 and 2 only(c) 1, 2 and 4(d) 3 and 4

Solution

To test the hypothesis H₀: β₁ = 0, we rely on the classical assumptions of the simple linear regression model.

Statement 1

yᵢ ~ N(β₀ + β₁xᵢ, σ²)

This is the normality assumption of the error term. It ensures that statistical tests (like t-test) are valid. Correct

Statement 2

β̂₁ ~ N(β₁, σ² / Σ(xᵢ − x̄)²)

This gives the sampling distribution of the OLS estimator β̂₁, which is essential for hypothesis testing. Correct

Statement 3

(n − 2)σ̂² / σ² ~ χ²(n − 2)

This result is used to estimate variance and construct the t-statistic. Correct

Statement 4

β̂₁ and σ̂² are not independent

In classical linear regression with normal errors, β̂₁ and σ̂² are actually independent. Incorrect

Final Answer

The correct statements are:1, 2, and 3

ISS PREVIOUS YEAR 2016 PAPER-2 SOLUTION SET-A

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