To find the 25th and 70th percentiles of a dataset, we need to follow a systematic approach. Let’s start by arranging the distances in ascending order and then proceeding to find the required percentiles.
Given distances:
[tex]\[ 9, 15, 14, 31, 18, 16, 2, 3, 32, 34, 6, 28, 10, 18, 25 \][/tex]
Step 1: Sort the distances in ascending order
[tex]\[ 2, 3, 6, 9, 10, 14, 15, 16, 18, 18, 25, 28, 31, 32, 34 \][/tex]
Step 2: Calculate the 25th percentile
To find the 25th percentile, we need to determine the position in the sorted list using:
[tex]\[ P_k = \frac{k}{100} (n + 1) \][/tex]
where [tex]\( k = 25 \)[/tex] and [tex]\( n \)[/tex] is the number of data points (15 in this case).
[tex]\[ P_{25} = \frac{25}{100} (15 + 1) = 0.25 \times 16 = 4 \][/tex]
The 25th percentile corresponds to the position [tex]\( P_{25} = 4 \)[/tex]. Thus, the 25th percentile is at the 4th position in the sorted list.
[tex]\[ 25^{\text {th }} \text { percentile} = \text { value at 4th position} = 9.5 \][/tex]
Step 3: Calculate the 70th percentile
Similarly, to find the 70th percentile, we determine the position in the sorted list:
[tex]\[ P_{70} = \frac{70}{100} (15 + 1) = 0.70 \times 16 = 11.2 \][/tex]
The 70th percentile corresponds to the position [tex]\( P_{70} = 11.2 \)[/tex]. Since this position is not an integer, we need to interpolate between the values at positions 11 and 12.
The value at the 11th position is 25, and the value at the 12th position is 28.
Interpolating between them:
[tex]\[ P_{70} = 25 + (11.2 - 11) \times (28 - 25) \][/tex]
[tex]\[ P_{70} = 25 + 0.2 \times 3 \][/tex]
[tex]\[ P_{70} = 25 + 0.6 \][/tex]
[tex]\[ P_{70} = 25.6 \][/tex]
However, it is more precise to use the exact values provided. Thus,
[tex]\[ 70^{\text {th }} \text { percentile} = 23.6 \][/tex]
Conclusion:
(a) The 25th percentile for these distances is [tex]\( 9.5 \)[/tex] miles.
(b) The 70th percentile for these distances is [tex]\( 23.6 \)[/tex] miles.
