To find the 30th percentile ([tex]\(P_{30}\)[/tex]) of the given radiation levels in [tex]\(\frac{W}{kg}\)[/tex], follow these steps:
1. Calculate the Position:
The first step is to find the position of the 30th percentile in the sorted data. For a dataset with [tex]\( N \)[/tex] values, the formula for the position [tex]\( k \)[/tex] is given by:
[tex]\[
k = P \times (N + 1)
\][/tex]
where [tex]\( P \)[/tex] is the percentile in decimal form. Here, [tex]\( P = 0.30 \)[/tex] and [tex]\( N = 50 \)[/tex]:
[tex]\[
k = 0.30 \times (50 + 1) = 0.30 \times 51 = 15.3
\][/tex]
2. Identify the Indices and Determine Interpolation:
Since the position [tex]\( k = 15.3 \)[/tex] is not an integer, we will need to interpolate between the 15th and 16th values in the sorted dataset:
- The 15th value corresponds to the integer part of the position.
- The 16th value corresponds to the next integer position.
3. Sort the Data:
Given that the data is already sorted, the 15th and 16th values can be identified directly from the list:
[tex]\[
\text{15th value (data_sorted[14])} = 0.97
\][/tex]
[tex]\[
\text{16th value (data_sorted[15])} = 0.99
\][/tex]
4. Calculate the Weight:
Since [tex]\( k = 15.3 \)[/tex], the fractional part [tex]\( 0.3 \)[/tex] indicates how close the 30th percentile is to the 16th value. The weight for interpolation can be determined as:
[tex]\[
\text{weight} = k - \lfloor k \rfloor = 15.3 - 15 = 0.3
\][/tex]
5. Interpolate:
Now we interpolate between the 15th and 16th values using the weight:
[tex]\[
P_{30} = \text{15th value} + \text{weight} \times (\text{16th value} - \text{15th value})
\][/tex]
[tex]\[
P_{30} = 0.97 + 0.3 \times (0.99 - 0.97)
\][/tex]
[tex]\[
P_{30} = 0.97 + 0.3 \times 0.02
\][/tex]
[tex]\[
P_{30} = 0.97 + 0.006 = 0.976
\][/tex]
Hence, the 30th percentile [tex]\( P_{30} \)[/tex] is:
[tex]\[
P_{30} = 0.98 \, \frac{W}{kg} \quad (\text{rounded to two decimal places})
\][/tex]
Therefore, the 30th percentile of the radiation levels for these cell phones is [tex]\( 0.98 \, \frac{W}{kg} \)[/tex].
