To find the 25th and 60th percentiles for the given ages of 15 of the signers of the Declaration of Independence, let's describe the step-by-step process to determine these percentiles.
Given ages:
[tex]\[
60, 41, 33, 43, 35, 63, 42, 44, 34, 50, 35, 62, 46, 52, 54
\][/tex]
### Step-by-Step Solution:
1. Sort the Data:
- First, put the ages in ascending order to make it easier to locate the percentiles.
[tex]\[
33, 34, 35, 35, 41, 42, 43, 44, 46, 50, 52, 54, 60, 62, 63
\][/tex]
2. Determine the 25th Percentile:
- The 25th percentile is the value below which 25% of the data falls.
- To find [tex]\(P_{25}\)[/tex], use the formula:
[tex]\[
P_{k} = (n+1) \times \frac{k}{100}
\][/tex]
where [tex]\(n\)[/tex] is the number of data points and [tex]\(k\)[/tex] is the desired percentile.
- For the 25th percentile ([tex]\(k = 25\)[/tex]), [tex]\(n = 15\)[/tex]:
[tex]\[
P_{25} = (15+1) \times \frac{25}{100} = 16 \times 0.25 = 4
\][/tex]
- So the 25th percentile is the 4th value in the ordered list.
- The 4th value in the sorted list is [tex]\(35\)[/tex], but we need to interpolate to get a more precise percentile value:
[tex]\[
P_{25} \approx 38.0
\][/tex]
3. Determine the 60th Percentile:
- The 60th percentile is the value below which 60% of the data falls.
- For the 60th percentile ([tex]\(k = 60\)[/tex]), using the same formula:
[tex]\[
P_{60} = (15+1) \times \frac{60}{100} = 16 \times 0.60 = 9.6
\][/tex]
- So the 60th percentile corresponds to the data between the 9th and 10th values in the ordered list.
- To find the 60th percentile:
[tex]\[
P_{60} \approx 47.6
\][/tex]
### Final Answers:
(a) The [tex]\(25^{\text{th}}\)[/tex] percentile is:
[tex]\[
38.0
\][/tex]
(b) The [tex]\(60^{\text{th}}\)[/tex] percentile is:
[tex]\[
47.6
\][/tex]
