To compute the range, sample variance, and sample standard deviation, let's break each calculation down:

Given data:

$410, 467, 415, 230$

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1. Range

$\text{Range} = \text{Maximum value} - \text{Minimum value}$ $\text{Range} = 467 - 230 = 237$ The range is $237.

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2. Sample Variance ($s^2$)

The formula for sample variance is: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$

Step 1: Compute the mean

$\bar{x} = \frac{410 + 467 + 415 + 230}{4} = \frac{1522}{4} = 380.5$

Step 2: Compute deviations from the mean and square them

$(410 - 380.5)^2 = (29.5)^2 = 870.25$ $(467 - 380.5)^2 = (86.5)^2 = 7482.25$ $(415 - 380.5)^2 = (34.5)^2 = 1190.25$ $(230 - 380.5)^2 = (-150.5)^2 = 22650.25$

Step 3: Sum the squared deviations

$870.25 + 7482.25 + 1190.25 + 22650.25 = 32193$

Step 4: Divide by $n - 1 = 4 - 1 = 3$

$s^2 = \frac{32193}{3} \approx 10731$

The sample variance ($s^2$) is $10,731 dollars squared.

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3. Sample Standard Deviation ($s$)

$s = \sqrt{10731} \approx 103.6$ The sample standard deviation is $104 (rounded to the nearest whole number).

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Final Answers:

• Range: $237
• Sample Variance: $10,731
• Sample Standard Deviation: $104

Do you want further clarifications on any part?

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Follow-up questions:

1. What is the importance of variance in real-world insurance claims analysis?
2. How is the standard deviation used to assess financial risks?
3. What would the range indicate about the crash repair costs?
4. How does removing the highest or lowest value affect the variance?
5. What is the impact of using population variance instead of sample variance?

Tip: The range is sensitive to outliers, but variance provides a more robust view of data dispersion.