Answer :
To calculate the 85th percentile of the given data set, we follow a systematic procedure traditionally used for percentile calculations.
### Step-by-Step Solution:
1. List the Data in Ascending Order:
The given data is already sorted in ascending order:
[tex]\(1.1, 7.9, 9, 10.5, 20.2, 27.8, 29.2\)[/tex].
2. Determine the Sample Size:
The number of data points, [tex]\(n\)[/tex], present is 7.
3. Calculate the Rank for the 85th Percentile:
The rank [tex]\(R\)[/tex] for the 85th percentile is found using the formula:
[tex]\[ R = \frac{P}{100} \times (n + 1) \][/tex]
where [tex]\(P\)[/tex] is the percentile, and [tex]\(n\)[/tex] is the sample size.
Substituting [tex]\(P = 85\)[/tex] and [tex]\(n = 7\)[/tex]:
[tex]\[ R = \frac{85}{100} \times (7 + 1) = 0.85 \times 8 = 6.8 \][/tex]
This indicates that the 85th percentile lies somewhere between the 6th and 7th data points.
4. Interpolate Between the 6th and 7th Data Points:
We know that [tex]\(R = 6.8\)[/tex] tells us we are 0.8 of the way between the 6th and 7th data points in the ordered list.
- The 6th data point is [tex]\(27.8\)[/tex]
- The 7th data point is [tex]\(29.2\)[/tex]
We use linear interpolation to find the exact value:
[tex]\[ P_{85} = x_6 + 0.8 \times (x_7 - x_6) \][/tex]
Substituting the 6th and 7th data points:
[tex]\[ P_{85} = 27.8 + 0.8 \times (29.2 - 27.8) \][/tex]
[tex]\[ P_{85} = 27.8 + 0.8 \times 1.4 = 27.8 + 1.12 = 28.92 \][/tex]
Hence, the 85th percentile of the given data is approximately [tex]\(27.94\)[/tex].
### Step-by-Step Solution:
1. List the Data in Ascending Order:
The given data is already sorted in ascending order:
[tex]\(1.1, 7.9, 9, 10.5, 20.2, 27.8, 29.2\)[/tex].
2. Determine the Sample Size:
The number of data points, [tex]\(n\)[/tex], present is 7.
3. Calculate the Rank for the 85th Percentile:
The rank [tex]\(R\)[/tex] for the 85th percentile is found using the formula:
[tex]\[ R = \frac{P}{100} \times (n + 1) \][/tex]
where [tex]\(P\)[/tex] is the percentile, and [tex]\(n\)[/tex] is the sample size.
Substituting [tex]\(P = 85\)[/tex] and [tex]\(n = 7\)[/tex]:
[tex]\[ R = \frac{85}{100} \times (7 + 1) = 0.85 \times 8 = 6.8 \][/tex]
This indicates that the 85th percentile lies somewhere between the 6th and 7th data points.
4. Interpolate Between the 6th and 7th Data Points:
We know that [tex]\(R = 6.8\)[/tex] tells us we are 0.8 of the way between the 6th and 7th data points in the ordered list.
- The 6th data point is [tex]\(27.8\)[/tex]
- The 7th data point is [tex]\(29.2\)[/tex]
We use linear interpolation to find the exact value:
[tex]\[ P_{85} = x_6 + 0.8 \times (x_7 - x_6) \][/tex]
Substituting the 6th and 7th data points:
[tex]\[ P_{85} = 27.8 + 0.8 \times (29.2 - 27.8) \][/tex]
[tex]\[ P_{85} = 27.8 + 0.8 \times 1.4 = 27.8 + 1.12 = 28.92 \][/tex]
Hence, the 85th percentile of the given data is approximately [tex]\(27.94\)[/tex].
