Answer :
To find the 49th percentile of the given data set, we will follow these steps:
1. Determine the Position:
- With a given data set size of [tex]\( n = 117 \)[/tex], the percentile position can be calculated using the formula:
[tex]\[
\text{position} = \left(\frac{\text{percentile}}{100}\right) \times (n - 1)
\][/tex]
- Substituting the given values, we get:
[tex]\[
\text{position} = \left(\frac{49}{100}\right) \times (117 - 1) = 0.49 \times 116 = 56.84
\][/tex]
2. Locate the Necessary Indices:
- The decimal position, 56.84, indicates that the 49th percentile lies between the 56th and 57th values in the ordered data set.
- Note that indices start from zero, so these correspond to the 57th and 58th values in conventional counting.
3. Interpolation:
- Since the position is not a whole number, we need to interpolate between the values at the 56th (index 56) and 57th (index 57) positions.
- If the two calculated positions are 56 and 57, the values at these indices are 64.5 and 64.6, respectively.
4. Calculate the 49th Percentile Value:
- Determine the fractional part of the position, which is 0.84 (from 56.84).
- Use the formula for linear interpolation:
[tex]\[
\text{Value} = (1 - \text{fraction}) \times \text{Value}_{\text{lower}} + \text{fraction} \times \text{Value}_{\text{upper}}
\][/tex]
- Substituting the known values:
[tex]\[
\text{Value} = (1 - 0.84) \times 64.5 + 0.84 \times 64.6
\][/tex]
- Calculate:
[tex]\[
\text{Value} = 0.16 \times 64.5 + 0.84 \times 64.6 = 10.32 + 54.384 = 64.684
\][/tex]
Therefore, the 49th percentile of the data set is approximately 64.684.
1. Determine the Position:
- With a given data set size of [tex]\( n = 117 \)[/tex], the percentile position can be calculated using the formula:
[tex]\[
\text{position} = \left(\frac{\text{percentile}}{100}\right) \times (n - 1)
\][/tex]
- Substituting the given values, we get:
[tex]\[
\text{position} = \left(\frac{49}{100}\right) \times (117 - 1) = 0.49 \times 116 = 56.84
\][/tex]
2. Locate the Necessary Indices:
- The decimal position, 56.84, indicates that the 49th percentile lies between the 56th and 57th values in the ordered data set.
- Note that indices start from zero, so these correspond to the 57th and 58th values in conventional counting.
3. Interpolation:
- Since the position is not a whole number, we need to interpolate between the values at the 56th (index 56) and 57th (index 57) positions.
- If the two calculated positions are 56 and 57, the values at these indices are 64.5 and 64.6, respectively.
4. Calculate the 49th Percentile Value:
- Determine the fractional part of the position, which is 0.84 (from 56.84).
- Use the formula for linear interpolation:
[tex]\[
\text{Value} = (1 - \text{fraction}) \times \text{Value}_{\text{lower}} + \text{fraction} \times \text{Value}_{\text{upper}}
\][/tex]
- Substituting the known values:
[tex]\[
\text{Value} = (1 - 0.84) \times 64.5 + 0.84 \times 64.6
\][/tex]
- Calculate:
[tex]\[
\text{Value} = 0.16 \times 64.5 + 0.84 \times 64.6 = 10.32 + 54.384 = 64.684
\][/tex]
Therefore, the 49th percentile of the data set is approximately 64.684.
