Answer :
Sure, let's solve this problem step-by-step!
### Step-by-Step Solution:
#### a. Find the percentile rank for a fare of [tex]\( \$119 \)[/tex].
First, we list out all the fares:
[tex]\[
49, 88, 88, 88, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 284,
49, 88, 88, 88, 119, 133, 133, 161, 161, 173, 173, 173, 173, 272, 284,
88, 88, 88, 119, 119, 133, 133, 161, 161, 173, 173, 173, 272, 272, 284,
88, 88, 88, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 272, 284,
88, 88, 88, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 284, 284,
88, 88, 88, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 284, 284
\][/tex]
In total, there are 90 fares.
To calculate the percentile rank for a fare of [tex]\( \$119 \)[/tex]:
1. Count how many fares are less than [tex]\( 119 \)[/tex].
2. Count how many fares are equal to [tex]\( 119 \)[/tex].
3. Use the percentile rank formula.
Total number of fares below [tex]\( \$119 \)[/tex] is [tex]\( 12 \)[/tex].
Total number of fares equal to [tex]\( \$119 \)[/tex] is [tex]\( 6 \)[/tex].
The formula to find percentile rank is:
[tex]\[ \text{Percentile Rank} = \left( \frac{\text{Number of values below}}{\text{Total number of values}} \right) \times 100 + \left( \frac{\text{Number of values equal}}{\text{Total number of values}} \right) \times 50 \][/tex]
Let's plug in the values:
[tex]\[ \text{Percentile Rank} = \left( \frac{12}{90} \right) \times 100 + \left( \frac{6}{90} \right) \times 50 \][/tex]
[tex]\[ \text{Percentile Rank} = 13.33 + 3.33 \][/tex]
[tex]\[ \text{Percentile Rank} \approx 26 \][/tex]
The percentile rank for a fare of [tex]\( \$119 \)[/tex] is approximately 26%.
#### b. Find the percentile rank for a fare of [tex]\( \$272 \)[/tex].
To calculate the percentile rank for a fare of [tex]\( \$272 \)[/tex]:
1. Count how many fares are less than [tex]\( 272 \)[/tex].
2. Count how many fares are equal to [tex]\( 272 \)[/tex].
3. Use the percentile rank formula.
Total number of fares below [tex]\( \$272 \)[/tex] is [tex]\( 65 \)[/tex].
Total number of fares equal to [tex]\( \$272 \)[/tex] is [tex]\( 7 \)[/tex].
Let's plug in the values:
[tex]\[ \text{Percentile Rank} = \left( \frac{65}{90} \right) \times 100 + \left( \frac{7}{90} \right) \times 50 \][/tex]
[tex]\[ \text{Percentile Rank} = 72.22 + 3.89 \][/tex]
[tex]\[ \text{Percentile Rank} \approx 87 \][/tex]
The percentile rank for a fare of [tex]\( \$272 \)[/tex] is approximately 87%.
#### c. Based on your first two answers, which train fare would have a percentile rank of approximately [tex]\( 82\% \)[/tex] ?
To find the fare with a percentile rank of approximately 82%:
1. Sort the fares.
2. Find the position in the sorted list that corresponds to the 82nd percentile.
Since 90 fares exist, the index to look at is:
[tex]\[ \text{Index} = \left( \frac{82}{100} \times 90 \right) - 1 \][/tex]
[tex]\[ \text{Index} \approx 73 \][/tex]
So we look at the fare in the 74th position (since we start from 0). The fare at this position corresponds to [tex]\( \$173 \)[/tex].
The fare with a percentile rank of approximately [tex]\( 82\% \)[/tex] is [tex]\($173\)[/tex].
### Final Answers:
a. The percentile rank for a fare of [tex]\( \$119 \)[/tex] is [tex]\( 26\% \)[/tex].
b. The percentile rank for a fare of [tex]\( \$272 \)[/tex] is [tex]\( 87\% \)[/tex].
c. The fare with a percentile rank of approximately [tex]\( 82\% \)[/tex] is [tex]\( \$173 \)[/tex].
### Step-by-Step Solution:
#### a. Find the percentile rank for a fare of [tex]\( \$119 \)[/tex].
First, we list out all the fares:
[tex]\[
49, 88, 88, 88, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 284,
49, 88, 88, 88, 119, 133, 133, 161, 161, 173, 173, 173, 173, 272, 284,
88, 88, 88, 119, 119, 133, 133, 161, 161, 173, 173, 173, 272, 272, 284,
88, 88, 88, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 272, 284,
88, 88, 88, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 284, 284,
88, 88, 88, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 284, 284
\][/tex]
In total, there are 90 fares.
To calculate the percentile rank for a fare of [tex]\( \$119 \)[/tex]:
1. Count how many fares are less than [tex]\( 119 \)[/tex].
2. Count how many fares are equal to [tex]\( 119 \)[/tex].
3. Use the percentile rank formula.
Total number of fares below [tex]\( \$119 \)[/tex] is [tex]\( 12 \)[/tex].
Total number of fares equal to [tex]\( \$119 \)[/tex] is [tex]\( 6 \)[/tex].
The formula to find percentile rank is:
[tex]\[ \text{Percentile Rank} = \left( \frac{\text{Number of values below}}{\text{Total number of values}} \right) \times 100 + \left( \frac{\text{Number of values equal}}{\text{Total number of values}} \right) \times 50 \][/tex]
Let's plug in the values:
[tex]\[ \text{Percentile Rank} = \left( \frac{12}{90} \right) \times 100 + \left( \frac{6}{90} \right) \times 50 \][/tex]
[tex]\[ \text{Percentile Rank} = 13.33 + 3.33 \][/tex]
[tex]\[ \text{Percentile Rank} \approx 26 \][/tex]
The percentile rank for a fare of [tex]\( \$119 \)[/tex] is approximately 26%.
#### b. Find the percentile rank for a fare of [tex]\( \$272 \)[/tex].
To calculate the percentile rank for a fare of [tex]\( \$272 \)[/tex]:
1. Count how many fares are less than [tex]\( 272 \)[/tex].
2. Count how many fares are equal to [tex]\( 272 \)[/tex].
3. Use the percentile rank formula.
Total number of fares below [tex]\( \$272 \)[/tex] is [tex]\( 65 \)[/tex].
Total number of fares equal to [tex]\( \$272 \)[/tex] is [tex]\( 7 \)[/tex].
Let's plug in the values:
[tex]\[ \text{Percentile Rank} = \left( \frac{65}{90} \right) \times 100 + \left( \frac{7}{90} \right) \times 50 \][/tex]
[tex]\[ \text{Percentile Rank} = 72.22 + 3.89 \][/tex]
[tex]\[ \text{Percentile Rank} \approx 87 \][/tex]
The percentile rank for a fare of [tex]\( \$272 \)[/tex] is approximately 87%.
#### c. Based on your first two answers, which train fare would have a percentile rank of approximately [tex]\( 82\% \)[/tex] ?
To find the fare with a percentile rank of approximately 82%:
1. Sort the fares.
2. Find the position in the sorted list that corresponds to the 82nd percentile.
Since 90 fares exist, the index to look at is:
[tex]\[ \text{Index} = \left( \frac{82}{100} \times 90 \right) - 1 \][/tex]
[tex]\[ \text{Index} \approx 73 \][/tex]
So we look at the fare in the 74th position (since we start from 0). The fare at this position corresponds to [tex]\( \$173 \)[/tex].
The fare with a percentile rank of approximately [tex]\( 82\% \)[/tex] is [tex]\($173\)[/tex].
### Final Answers:
a. The percentile rank for a fare of [tex]\( \$119 \)[/tex] is [tex]\( 26\% \)[/tex].
b. The percentile rank for a fare of [tex]\( \$272 \)[/tex] is [tex]\( 87\% \)[/tex].
c. The fare with a percentile rank of approximately [tex]\( 82\% \)[/tex] is [tex]\( \$173 \)[/tex].
