Financial Risk Manager Part 1
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On Tuesday, an insurance company receives a total of 10 claims for automobile policies. After the first-round assessment, it's found that the mean claim amount of the 10 claims is $426 while the standard deviation is 112. On Tuesday, the chief claims analyst authorizes the removal of one of the claims for $545 from the list on grounds that it's fraught with fraud. Compute the standard deviation for the remaining set of 9 claims.
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Community
NNikitesh
A
110.2
B
12145.2
C
421.8
D
420
Explanation:
Step-by-step solution:
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Calculate total sum of original 10 claims: Σx = mean × n = 426 × 10 = 4,260
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Calculate total sum after removing fraudulent claim: New Σx = 4,260 - 545 = 3,715
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Calculate new mean for 9 claims: New mean = 3,715 ÷ 9 = 412.777... ≈ 412.8
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Calculate sum of squares for original 10 claims: Using formula: s² = 1/(n-1)[Σx² - nẍ²] 112² = 1/9[Σx² - 10 × 426²] 12,544 = 1/9[Σx² - 1,814,760] Σx² = 12,544 × 9 + 1,814,760 = 112,896 + 1,814,760 = 1,927,656
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Calculate sum of squares after removing fraudulent claim: New Σx² = 1,927,656 - 545² = 1,927,656 - 297,025 = 1,630,631
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Calculate variance for remaining 9 claims: s² = 1/(9-1)[1,630,631 - 9 × 412.8²] s² = 1/8[1,630,631 - 9 × 170,403.84] s² = 1/8[1,630,631 - 1,533,634.56] s² = 1/8[96,996.44] = 12,124.555
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Calculate standard deviation: s = √12,124.555 = 110.2
Key formulas used:
- Sample variance: s² = 1/(n-1)[Σx² - nẍ²]
- Standard deviation: s = √s²
Note: The slight difference in variance calculation (12,145.2 vs 12,124.555) is due to rounding of the mean to 412.8. Using the exact mean of 412.777... gives the exact variance of 12,145.2 as shown in the original solution.
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