Question
e. Compute the Gini index for the cases \( L(x)=x^{p} \) and \( p=1.1,1,3,2,3,4 \). The Gini index for \( p=1.1 \) is 0.048 . (Type an integer or decimal rounded to three decimal places as needed.) The Gini index for \( p=1.3 \) is 0.13 . (Type an integer or decimal rounded to three decimal places as needed.) The Gini index for \( p=2 \) is 0.333 . (Type an integer or decimal rounded to three decimal places as needed.) The Gini index for \( p=3 \) is 0.5 . (Type an integer or decimal rounded to three decimal places as needed.) The Gini index for \( p=4 \) is \( \square \). (Type an integer or decimal rounded to three decimal places as needed.)
Ask by Campos Wright. in the United States
Jan 27,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The Gini index for \( p=4 \) is 0.600.
Solution
To compute the Gini index for \( L(x) = x^p \), we use the following formula:
\[
\text{Gini Index} = 1 - \frac{2}{p + 1}
\]
Let's apply this formula for \( p = 4 \):
\[
\text{Gini Index} = 1 - \frac{2}{4 + 1} = 1 - \frac{2}{5} = 1 - 0.4 = 0.6
\]
Rounded to three decimal places, the Gini index for \( p = 4 \) is:
\[
0.600
\]
**Answer:**
0.600
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Simplify this solution Extra Insights
To compute the Gini index for \( L(x)=x^{p} \) when \( p=4 \), you can use the formula for the Gini coefficient, which typically involves calculating the area between the Lorenz curve and the line of perfect equality. For the case of \( p=4 \), the expression can be calculated and will be typically noted to follow the pattern found in previous values: After using the relevant formula or reference values calculated for similar cases, the Gini index for \( p=4 \) is approximately 0.666.
