An analyst obtained the following linear regression relationship between 2 variables, X and Y: $ Y = \alpha + \beta_1X $ where $\alpha = 0.45$ and $\beta = 0.8823$. He proceeded to construct a 2-sided 95% confidence interval for the slope coefficient ($\beta_1$) and obtained the following interval: $ \beta = 0.8823 \pm 0.2147 $ Suppose the analyst decided to test the hypothesis $H_0: \beta_1 = 1$ vs $H_a: \beta_1 \ne 1$ at 5% significance, what would be the inference? | Financial Risk Manager Part 1 Quiz

An analyst obtained the following linear regression relationship between 2 variables, X and Y:

Y = \alpha + \beta_1X

where $\alpha = 0.45$ and $\beta = 0.8823$ . He proceeded to construct a 2-sided 95% confidence interval for the slope coefficient ( $\beta_1$ ) and obtained the following interval:

\beta = 0.8823 \pm 0.2147

Suppose the analyst decided to test the hypothesis $H_0: \beta_1 = 1$ vs $H_a: \beta_1 \ne 1$ at 5% significance, what would be the inference?

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Explanation:

Explanation

The 95% confidence interval for the slope coefficient is:

$\beta_1 = 0.8823 \pm 0.2147$

This gives us the interval:

Lower bound: 0.8823 - 0.2147 = 0.6676
Upper bound: 0.8823 + 0.2147 = 1.0970

The interval [0.6676, 1.0970] contains the value 1.

Key Concept: When testing a hypothesis at significance level α, if the (1-α)% confidence interval contains the hypothesized value, we fail to reject the null hypothesis.

Significance level: 5%
Confidence level: 95%
Hypothesized value: β₁ = 1
Since 1 is within the 95% confidence interval [0.6676, 1.0970], we do not reject H₀

This means there is insufficient evidence to conclude that the slope coefficient is statistically different from 1 at the 5% significance level._

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