Answer :
To find the 24th percentile, [tex]\( P_{24} \)[/tex], from the given data set, we can follow these steps:
1. Sort the Data: Although the data is already sorted in ascending order, this step is important for finding percentiles. The sorted data is:
[tex]\[
10.7, 11.4, 11.7, 11.8, 12.7, 12.9, 13.9, 14.5, 17.4, 18.5,
19, 20.5, 25, 26.2, 26.3, 27.4, 28.9, 29, 29.1, 29.2,
31.2, 32.3, 33.6, 33.7, 35, 35.5, 35.8, 37.4, 37.9, 39.1,
39.5, 40.3, 44.1, 44.9, 47.8, 47.9
\][/tex]
2. Calculate the Index for the 24th Percentile:
- The formula to find the index for the [tex]\( P_{k} \)[/tex] percentile is:
[tex]\[
\text{Index} = \left(\frac{k}{100}\right) \times (n + 1)
\][/tex]
Here, [tex]\( k = 24 \)[/tex] and [tex]\( n = 36 \)[/tex] (the total number of data points).
- Plug in the values:
[tex]\[
\text{Index} = \frac{24}{100} \times (36 + 1) = \frac{24}{100} \times 37 = 8.88
\][/tex]
3. Determine the Position:
- The index 8.88 suggests looking between the 8th and 9th data points.
- In a sorted list, the 8th data point is 14.5, and the 9th data point is 17.4.
4. Apply Linear Interpolation:
- Since the index is not a whole number, we use the fractional part to interpolate between the 8th and 9th values.
- The fractional part is [tex]\( 0.88 \)[/tex].
The formula for linear interpolation is:
[tex]\[
P_{24} = \text{Lower Value} + \text{Fractional Part} \times (\text{Upper Value} - \text{Lower Value})
\][/tex]
- Substitute the known values:
[tex]\[
P_{24} = 14.5 + 0.88 \times (17.4 - 14.5)
\][/tex]
- Calculate:
[tex]\[
P_{24} = 14.5 + 0.88 \times 2.9 = 14.5 + 2.552 = 17.052
\][/tex]
5. Round the Result:
- The 24th percentile, [tex]\( P_{24} \)[/tex], is approximately [tex]\( 17.052 \)[/tex].
So, the 24th percentile of the data set is approximately 17.052.
1. Sort the Data: Although the data is already sorted in ascending order, this step is important for finding percentiles. The sorted data is:
[tex]\[
10.7, 11.4, 11.7, 11.8, 12.7, 12.9, 13.9, 14.5, 17.4, 18.5,
19, 20.5, 25, 26.2, 26.3, 27.4, 28.9, 29, 29.1, 29.2,
31.2, 32.3, 33.6, 33.7, 35, 35.5, 35.8, 37.4, 37.9, 39.1,
39.5, 40.3, 44.1, 44.9, 47.8, 47.9
\][/tex]
2. Calculate the Index for the 24th Percentile:
- The formula to find the index for the [tex]\( P_{k} \)[/tex] percentile is:
[tex]\[
\text{Index} = \left(\frac{k}{100}\right) \times (n + 1)
\][/tex]
Here, [tex]\( k = 24 \)[/tex] and [tex]\( n = 36 \)[/tex] (the total number of data points).
- Plug in the values:
[tex]\[
\text{Index} = \frac{24}{100} \times (36 + 1) = \frac{24}{100} \times 37 = 8.88
\][/tex]
3. Determine the Position:
- The index 8.88 suggests looking between the 8th and 9th data points.
- In a sorted list, the 8th data point is 14.5, and the 9th data point is 17.4.
4. Apply Linear Interpolation:
- Since the index is not a whole number, we use the fractional part to interpolate between the 8th and 9th values.
- The fractional part is [tex]\( 0.88 \)[/tex].
The formula for linear interpolation is:
[tex]\[
P_{24} = \text{Lower Value} + \text{Fractional Part} \times (\text{Upper Value} - \text{Lower Value})
\][/tex]
- Substitute the known values:
[tex]\[
P_{24} = 14.5 + 0.88 \times (17.4 - 14.5)
\][/tex]
- Calculate:
[tex]\[
P_{24} = 14.5 + 0.88 \times 2.9 = 14.5 + 2.552 = 17.052
\][/tex]
5. Round the Result:
- The 24th percentile, [tex]\( P_{24} \)[/tex], is approximately [tex]\( 17.052 \)[/tex].
So, the 24th percentile of the data set is approximately 17.052.
