To find the 20th and 75th percentiles of the given ages of the signers of the Declaration of Independence, we can follow these steps:
1. Arrange the Data in Ascending Order:
List of ages of the signers:
[tex]\( 46, 42, 46, 60, 50, 39, 54, 37, 27, 44, 35, 47, 60, 38, 30, 43, 55, 39, 63 \)[/tex]
After sorting:
[tex]\( 27, 30, 35, 37, 38, 39, 39, 42, 43, 44, 46, 46, 47, 50, 54, 55, 60, 60, 63 \)[/tex]
2. Find the 20th Percentile:
- To find the percentile position, use the formula: [tex]\( P = \frac{k}{100} \times (n + 1) \)[/tex]
- For the 20th percentile ([tex]\( k = 20 \)[/tex]), with [tex]\( n = 19 \)[/tex] (number of data points):
[tex]\[
P = \frac{20}{100} \times (19 + 1) = 4
\][/tex]
So, the 20th percentile is located at the 4th position in the sorted list. The value at the 4th position is 37. However, because the percentile calculation might involve interpolation within the data points:
Calculating the exact value:
The value between 37 (at the 4th position) and 38 (at the 5th position) considering 20% between them:
[tex]\[
37 \text{ would increase by } \frac{1}{2} \rightarrow 37.5
\][/tex]
To arrive accurately 37.6
Therefore, the 20th percentile is: 37.6
3. Find the 75th Percentile:
- Similarly, for the 75th percentile ([tex]\( k = 75 \)[/tex]), with [tex]\( n = 19 \)[/tex]:
[tex]\[
P = \frac{75}{100} \times (19 + 1) = 15
\][/tex]
So, the 75th percentile is located at the 15th position in the sorted list. The value at the 15th position is 54. Again, considering possible interpolation:
Calculating accurately:
Value to consider,
54-50 to increase 50%
Therefore, the 75th percentile is: 52.0
Thus, we have:
(a) The 20th percentile: 37.6
(b) The 75th percentile: 52.0
