Financial Risk Manager Part 1

Financial Risk Manager Part 1

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A random sample of 50 FRM exam candidates was found to have an average IQ of 125. The standard deviation among candidates is known (approximately 20). Assuming that IQs follow a normal distribution, carry out a statistical test (5% significance level) to determine whether the average IQ of FRM candidates is greater than 120. Compute the test statistic and give a conclusion.

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Explanation:

Explanation

Hypothesis Formulation

  • H₀: μ = 120 (null hypothesis - average IQ is 120)
  • H₁: μ > 120 (alternative hypothesis - average IQ is greater than 120)
  • This is a one-tailed test since we're testing for an increase only

Test Statistic Calculation

Given:

  • Sample mean (x̄) = 125
  • Population mean (μ) = 120
  • Population standard deviation (σ) = 20
  • Sample size (n) = 50

Test statistic formula: z=(xˉ−μ)(σ/n)z = \frac{(\bar{x} - \mu)}{(\sigma / \sqrt{n})}

Calculation: z=(125−120)(20/50)=5(20/7.071)=52.828=1.768z = \frac{(125 - 120)}{(20 / \sqrt{50})} = \frac{5}{(20 / 7.071)} = \frac{5}{2.828} = 1.768

Decision Making

Critical Value Approach:

  • For 5% significance level (one-tailed), critical z-value = 1.6449
  • Since 1.768 > 1.6449, we reject H₀

P-value Approach:

  • P(Z > 1.768) = 1 - P(Z < 1.768) = 1 - 0.96147 = 0.03853 (3.853%)
  • Since p-value (0.03853) < significance level (0.05), we reject H₀

Conclusion

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

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