Financial Risk Manager Part 1
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A 95% confidence interval for β₁ is determined to be (20, 25). This means that:
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A
We can be 95% confident that the mean value of Y lies between 20 and 25 units.
B
We can be 95% confident that the value of X will increase by between 20 and 25 units for every one unit increase in Y.
C
We can be 95% confident that the value of Y will increase by between 20 and 25 units for every one unit increase in X.
D
At the 5% level of significance, we would not find evidence of a linear relationship between X and Y.
Explanation:
Explanation
In linear regression, β₁ represents the slope coefficient, which indicates the change in the dependent variable Y for a one-unit change in the independent variable X.
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Option A is incorrect because it refers to the mean value of Y, not the slope coefficient. The confidence interval for the mean response would be different from the confidence interval for the slope.
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Option B is incorrect because it reverses the relationship - it suggests that X changes with Y, but in regression, we typically model Y as a function of X.
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Option C is correct because a 95% confidence interval for β₁ = (20, 25) means we can be 95% confident that the true slope coefficient lies between 20 and 25. This means that for every one-unit increase in X, Y increases by between 20 and 25 units.
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Option D is incorrect because a confidence interval that does not contain zero (which (20, 25) does not) actually provides evidence of a linear relationship at the 5% significance level. If the interval contained zero, we would not find evidence of a linear relationship.
This question tests understanding of confidence intervals for regression coefficients and their interpretation in the context of linear relationships between variables.
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