Financial Risk Manager Part 1

Explanation

This is a one-sample z-test for a population mean with known standard deviation.

Step 1: Formulate Hypotheses

• Null hypothesis (H₀): μ = 120 (average IQ is 120)
• Alternative hypothesis (H₁): μ > 120 (average IQ is greater than 120)

This is a one-tailed test since we're testing if the mean is greater than 120.

Step 2: Test Statistic Calculation

Given:

• Sample mean (μ̄) = 125
• Population standard deviation (σ) = 20 (known)
• Sample size (n) = 50
• Hypothesized mean (μ₀) = 120

The test statistic formula for z-test:

$z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$

$z = \frac{125 - 120}{20/\sqrt{50}} = \frac{5}{20/7.071} = \frac{5}{2.828} = 1.768$

Step 3: Critical Value Approach

For a one-tailed test at 5% significance level (α = 0.05):

• Critical value = zₐ = 1.6449 (from standard normal distribution)
• Decision rule: Reject H₀ if z > 1.6449

Since 1.768 > 1.6449, we reject H₀.

Step 4: P-value Approach

P-value = P(Z > 1.768) = 1 - P(Z < 1.768) = 1 - 0.96147 = 0.03853 or 3.853%

Since p-value (0.03853) < α (0.05), we reject H₀.

Step 5: Conclusion

We have sufficient evidence at the 5% significance level to conclude that the average IQ of FRM candidates is greater than 120.

Why other options are incorrect:

• Option B (2.828): Incorrect test statistic calculation
• Option C (1.768; Fail to reject): Correct test statistic but wrong conclusion
• Option D (1.0606; Fail to reject): Incorrect test statistic and wrong conclusion