Financial Risk Manager Part 1

Explanation

Let's analyze each statement:

Statement I: "Decreasing the significance level will decrease the probability of failing to reject a false null"

• The significance level (α) is the probability of rejecting the null hypothesis when it is true (Type I error).
• Decreasing α makes it harder to reject the null hypothesis.
• The probability of failing to reject a false null hypothesis is β (Type II error).
• When α decreases, β increases because we're being more conservative about rejecting the null.
• Therefore, decreasing the significance level will increase (not decrease) the probability of failing to reject a false null.

Statement II: "Decreasing the significance level will increase the power of the test"

• Power of a test = 1 - β = Probability of correctly rejecting a false null hypothesis.
• When α decreases, β increases (as explained above).
• If β increases, then power (1 - β) decreases.
• Therefore, decreasing the significance level will decrease (not increase) the power of the test.

Conclusion: Both statements are incorrect.

Key Relationships:

• α (significance level) = P(Type I error)
• β = P(Type II error) = P(failing to reject a false null)
• Power = 1 - β
• When α decreases → β increases → Power decreases