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Answer: I
**Explanation:** This is an F-test for comparing two variances. The correct approach is: 1. **Hypothesis formulation**: Since we suspect the insurance industry earnings are MORE divergent (greater variance) than banking, we test: - H₀: σ₁² ≤ σ₂² (insurance variance ≤ banking variance) - H₁: σ₁² > σ₂² (insurance variance > banking variance) This is a right-tailed test. 2. **Test statistic calculation**: - s₁ = 4.8 (insurance standard deviation) - s₂ = 4.3 (banking standard deviation) - n₁ = 31 (insurance sample size) - n₂ = 41 (banking sample size) F-statistic = s₁² / s₂² = (4.8²) / (4.3²) = (23.04) / (18.49) = 1.2461 3. **Critical value and decision**: - Degrees of freedom: numerator df = n₁ - 1 = 30, denominator df = n₂ - 1 = 40 - At 5% significance level (α = 0.05), the critical F-value for F(30,40) is approximately 1.74 - Since calculated F (1.2461) < critical F (1.74), we fail to reject H₀ - Conclusion: Earnings are statistically not significantly different from one another **Matching with options:** - Option I has correct hypothesis (H₀: σ₁² ≤ σ₂² and H₁: σ₁² > σ₂²), correct test statistic (1.2461), and correct decision (not significant) - Option II has correct hypothesis but wrong test statistic (1.74) - Option III has wrong hypothesis (H₁: σ₁² ≤ σ₂²) and wrong decision - Option IV has wrong hypothesis (H₁: σ₁² < σ₂²) and wrong test statistic Therefore, Option I is correct.
Author: Nikitesh Somanthe
Justin Heinz, FRM, suspects that the earnings of the insurance industry are more divergent than those of the banking industry. In a bid to confirm his suspicion, Heinz collects data from a total of 31 insurance companies and establishes that the standard deviation of earnings across that industry is $4.8. Similarly, he collects data from 41 banks and establishes that the standard deviation of earnings across that industry is $4.3. Conduct a hypothesis test at the 5% level of significance to determine if the earnings of the insurance industry have a greater standard deviation than those of the banking industry.
| Choice | Hypothesis | Test statistic | Decision |
|---|---|---|---|
| I. | H₀: σ₁² ≤ σ₂² <br> H₁: σ₁² > σ₂² | 1.2461 | Earnings are statistically not significant from one another |
| II. | H₀: σ₁² ≤ σ₂² <br> H₁: σ₁² > σ₂² | 1.74 | Earnings are statistically not significant from one another |
| III. | H₀: σ₁² ≤ σ₂² <br> H₁: σ₁² ≤ σ₂² | 1.2461 | Earnings are statistically significant from one another |
| IV. | H₀: σ₁² ≤ σ₂² <br> H₁: σ₁² < σ₂² | 1.74 | Earnings are statistically not significant from one another |
A
I
B
II
C
III
D
IV
