Explanation:
There is a direct relationship between confidence intervals and two-sided hypothesis tests at the same significance level. When conducting a two-sided hypothesis test at significance level α, the null hypothesis is rejected if and only if the hypothesized parameter value falls outside the (1-α) confidence interval. In this case, Sarah's 95% confidence interval (corresponding to a 5% significance level) is [0.0066, 0.0334], which contains the null hypothesis value of 0.01. Therefore, she cannot reject the null hypothesis that the model is correctly calibrated at the 5% significance level.
A is incorrect. This would only be true if 0.01 were not in the interval. But it is within the interval, so we fail to reject the null hypothesis.
C is incorrect. If the null value falls within the confidence interval, we do not reject the null hypothesis. So this reasoning leads to the wrong conclusion.
D is incorrect. It incorrectly states that 0.01 falls outside the confidence interval when it actually falls within [0.0066, 0.0334], and it incorrectly associates a parameter value falling outside the confidence interval with failing to reject the null hypothesis, when such a situation would lead to rejection.
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Q.41 Sarah, a risk manager at a global investment bank, is reviewing the performance of a Value-at-Risk (VaR) model in anticipation of a regulatory audit. The bank's current model estimates a 99% VaR of $25 million. To validate this model, Sarah compares the actual daily losses that exceed the VaR estimate (VaR exceptions) with the expected number of exceptions. For a one-year period (250 trading days), she observes 5 VaR exceptions. Sarah constructs a 95% confidence interval for the true probability of a VaR exception and finds it to be [0.0066, 0.0334]. She then conducts a hypothesis test with:
H₀: p = 0.01 (model is correctly calibrated)
H₁: p ≠ 0.01 (model is not correctly calibrated)
Using the same data and at the same significance level as her confidence interval, which of the following statements is correct regarding the relationship between Sarah's hypothesis test and confidence interval?
A
Sarah will reject H₀ because 0.01 falls outside her 95% confidence interval
B
Sarah will fail to reject H₀ because 0.01 falls within her 95% confidence interval
C
Sarah will reject H₀ because 0.01 falls within her 95% confidence interval
D
Sarah will fail to reject H₀ because 0.01 falls outside her 95% confidence interval
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