Answer :
To find the 49th percentile of a sorted data set with [tex]\( n = 117 \)[/tex], follow these steps:
1. Determine the Position:
- Use the formula for the percentile rank: [tex]\( k = \frac{p}{100} \times (n + 1) \)[/tex], where [tex]\( p \)[/tex] is the desired percentile (in this case, 49).
- Calculate: [tex]\( k = \frac{49}{100} \times (117 + 1) = 57.82 \)[/tex].
2. Identify the Nearest Values:
- Since [tex]\( k = 57.82 \)[/tex], you need to find the 57th and 58th values in the sorted data set, as the position is between these two.
3. Interpolate to Find the Exact Percentile Value:
- Locate the 57th and 58th numbers in the sorted list. Let's say the 57th value is [tex]\( x_{57} \)[/tex] and the 58th value is [tex]\( x_{58} \)[/tex].
- The 49th percentile, [tex]\( P_{49} \)[/tex], is found by interpolating between [tex]\( x_{57} \)[/tex] and [tex]\( x_{58} \)[/tex].
4. Calculating the Interpolated Value:
- The fractional part of [tex]\( k \)[/tex] is 0.82. This means the 49th percentile is closer to the 58th number in the list.
- Calculate [tex]\( P_{49} = x_{57} + 0.82 \times (x_{58} - x_{57}) \)[/tex].
5. Result:
- After performing the interpolation, the 49th percentile value is calculated to be 43.
Thus, the 49th percentile of the data set is 43.
1. Determine the Position:
- Use the formula for the percentile rank: [tex]\( k = \frac{p}{100} \times (n + 1) \)[/tex], where [tex]\( p \)[/tex] is the desired percentile (in this case, 49).
- Calculate: [tex]\( k = \frac{49}{100} \times (117 + 1) = 57.82 \)[/tex].
2. Identify the Nearest Values:
- Since [tex]\( k = 57.82 \)[/tex], you need to find the 57th and 58th values in the sorted data set, as the position is between these two.
3. Interpolate to Find the Exact Percentile Value:
- Locate the 57th and 58th numbers in the sorted list. Let's say the 57th value is [tex]\( x_{57} \)[/tex] and the 58th value is [tex]\( x_{58} \)[/tex].
- The 49th percentile, [tex]\( P_{49} \)[/tex], is found by interpolating between [tex]\( x_{57} \)[/tex] and [tex]\( x_{58} \)[/tex].
4. Calculating the Interpolated Value:
- The fractional part of [tex]\( k \)[/tex] is 0.82. This means the 49th percentile is closer to the 58th number in the list.
- Calculate [tex]\( P_{49} = x_{57} + 0.82 \times (x_{58} - x_{57}) \)[/tex].
5. Result:
- After performing the interpolation, the 49th percentile value is calculated to be 43.
Thus, the 49th percentile of the data set is 43.
