Find expert answers and community-driven knowledge on IDNLearn.com. Ask your questions and get detailed, reliable answers from our community of experienced experts.
Sagot :
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].
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].
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for choosing IDNLearn.com. We’re here to provide reliable answers, so please visit us again for more solutions.