Answer :
To find the 44th percentile, [tex]\( P_{44} \)[/tex], from the given data, you can follow these steps:
1. List the Data in Ascending Order: The data provided is already sorted in ascending order, which is:
10.3, 11.3, 11.6, 12.1, 14.1, 15.2, 15.8, 16.1, 16.9, 17, 17.8, 18.1, 19.6, 20.6, 21, 21.4, 23.4, 25.8, 27, 29.3, 31.9, 32.2, 36.9, 37.5, 37.9, 40.2, 41.1, 41.6, 42, 43.9, 45.5, 48.2, 49.9
2. Determine the Total Number of Observations: There are 33 data points.
3. Calculate the Position of the 44th Percentile:
To find the position in the data set, use the formula:
[tex]\[
\text{Position} = \frac{44}{100} \times (n + 1)
\][/tex]
Where [tex]\( n \)[/tex] is the number of data points.
[tex]\[
\text{Position} = 0.44 \times (33 + 1) = 0.44 \times 34 = 14.96
\][/tex]
4. Locate the 44th Percentile:
Since the position [tex]\( 14.96 \)[/tex] is not a whole number, we need to use interpolation:
- Find the values at the integer parts of the calculated position. These positions are 14 and 15 in the sorted data.
- The 14th data point is 20.6, and the 15th data point is 21.
- The fractional part of the position is 0.96.
5. Interpolate to Find [tex]\( P_{44} \)[/tex]:
Use the interpolation formula:
[tex]\[
P_{44} = \text{value at position 14} + (\text{fractional part}) \times (\text{value at position 15} - \text{value at position 14})
\][/tex]
[tex]\[
P_{44} = 20.6 + 0.96 \times (21 - 20.6)
\][/tex]
[tex]\[
P_{44} = 20.6 + 0.96 \times 0.4
\][/tex]
[tex]\[
P_{44} = 20.6 + 0.384
\][/tex]
[tex]\[
P_{44} = 20.984
\][/tex]
Thus, the 44th percentile [tex]\( P_{44} \)[/tex] of the data set is approximately 20.984.
1. List the Data in Ascending Order: The data provided is already sorted in ascending order, which is:
10.3, 11.3, 11.6, 12.1, 14.1, 15.2, 15.8, 16.1, 16.9, 17, 17.8, 18.1, 19.6, 20.6, 21, 21.4, 23.4, 25.8, 27, 29.3, 31.9, 32.2, 36.9, 37.5, 37.9, 40.2, 41.1, 41.6, 42, 43.9, 45.5, 48.2, 49.9
2. Determine the Total Number of Observations: There are 33 data points.
3. Calculate the Position of the 44th Percentile:
To find the position in the data set, use the formula:
[tex]\[
\text{Position} = \frac{44}{100} \times (n + 1)
\][/tex]
Where [tex]\( n \)[/tex] is the number of data points.
[tex]\[
\text{Position} = 0.44 \times (33 + 1) = 0.44 \times 34 = 14.96
\][/tex]
4. Locate the 44th Percentile:
Since the position [tex]\( 14.96 \)[/tex] is not a whole number, we need to use interpolation:
- Find the values at the integer parts of the calculated position. These positions are 14 and 15 in the sorted data.
- The 14th data point is 20.6, and the 15th data point is 21.
- The fractional part of the position is 0.96.
5. Interpolate to Find [tex]\( P_{44} \)[/tex]:
Use the interpolation formula:
[tex]\[
P_{44} = \text{value at position 14} + (\text{fractional part}) \times (\text{value at position 15} - \text{value at position 14})
\][/tex]
[tex]\[
P_{44} = 20.6 + 0.96 \times (21 - 20.6)
\][/tex]
[tex]\[
P_{44} = 20.6 + 0.96 \times 0.4
\][/tex]
[tex]\[
P_{44} = 20.6 + 0.384
\][/tex]
[tex]\[
P_{44} = 20.984
\][/tex]
Thus, the 44th percentile [tex]\( P_{44} \)[/tex] of the data set is approximately 20.984.
