Answer :
Sure! Let's go through the details to solve the problems given.
### a. Find the percentile rank for a fare of \[tex]$119. Interpret your results.
To find the percentile rank for a fare of \$[/tex]119, we need to determine what percentage of fares are less than or equal to \[tex]$119.
Step-by-Step Solution:
1. List all the fares given and count how many individual fares are present.
2. Count how many fares are less than or equal to \$[/tex]119.
3. Divide the number of fares less than or equal to \[tex]$119 by the total number of fares, then multiply by 100 to get the percentile.
From the provided data, let's list the fares as:
\[ 49, 88, 88, 88, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 284, \]
\[ 49, 88, 88, 88, 119, 133, 133, 161, 161, 173, 173, 173, 173, 272, 284, \]
\[ 88, 88, 88, 119, 119, 133, 133, 161, 161, 173, 173, 173, 272, 272, 284, \]
\[ 88, 88, 88, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 272, 284, \]
\[ 88, 88, 88, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 284, 284, \]
\[ 88, 88, 88, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 284, 284 \]
In total, there are \( 6 \times 15 = 90 \) fares.
Count the number of fares that are \$[/tex]119 or less:
- We see 11 fares of [tex]$88 and 1 fare of $[/tex]49 in each list.
- We also see 1 instance of [tex]$119 in each list.
Total number across all lists = \( 6 \times 3 (fares \leq 119) = 18 \).
Now calculate the percentile rank for \$[/tex]119:
[tex]\[ \text{Percentile Rank} = \left(\frac{18}{90}\right) \times 100 = 20\% \][/tex]
So, the percentile rank for the fare \[tex]$119 is 20%. This means 30% of the fares are less than or equal to \$[/tex]119.
### b. Find the percentile rank for a fare of \[tex]$272. Interpret your results.
Step-by-Step Solution:
1. Count how many fares are less than or equal to \$[/tex]272.
2. Use the same method as in part (a) to calculate the percentile.
In the fares given:
- We need to count all fares that are less than or equal to \[tex]$272.
- We see that \$[/tex]272 appears 12 times across the lists.
Total cumulative by this fare tier is:
[tex]\( \text{Cumulative tally:} = \number of fares < \$272 + 12
\[
Count the fares less than or equal to \$272:
- \( \text{Fares < 272:} = 82
So in total, \( 82 + 12 = 92 \)[/tex].
[tex]\[ \text{Percentile Rank} = \left(\frac{92}{90}\right) \times 100 = 91.11\%
\][/tex]
So, the percentile rank for the fare \[tex]$272 is 91.11%. This means 91.11% of the fares are less than or equal to \$[/tex]272.
### c. Which train fare would have a percentile rank of approximately 82%?
To find the train fare corresponding to the 82nd percentile:
1. Sort the fares in ascending order.
2. Identify the fare that corresponds to the 82nd percentile.
From the ordered combined list:
In total number of fares = 90.
Position number in the sorted list for 82nd percentile is:
[tex]\[ 82 \% \times 90 = 73.8
\][/tex]
Approximately [tex]\(74th\)[/tex] position corrects to 173 fare tier. So, any of the 173 from the ascending position count,
Therefore, the train fare corresponding to the 82nd percentile is $173.
Hope this solution helps clarify the concept of percentile ranks! Let me know if you have any further questions.
### a. Find the percentile rank for a fare of \[tex]$119. Interpret your results.
To find the percentile rank for a fare of \$[/tex]119, we need to determine what percentage of fares are less than or equal to \[tex]$119.
Step-by-Step Solution:
1. List all the fares given and count how many individual fares are present.
2. Count how many fares are less than or equal to \$[/tex]119.
3. Divide the number of fares less than or equal to \[tex]$119 by the total number of fares, then multiply by 100 to get the percentile.
From the provided data, let's list the fares as:
\[ 49, 88, 88, 88, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 284, \]
\[ 49, 88, 88, 88, 119, 133, 133, 161, 161, 173, 173, 173, 173, 272, 284, \]
\[ 88, 88, 88, 119, 119, 133, 133, 161, 161, 173, 173, 173, 272, 272, 284, \]
\[ 88, 88, 88, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 272, 284, \]
\[ 88, 88, 88, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 284, 284, \]
\[ 88, 88, 88, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 284, 284 \]
In total, there are \( 6 \times 15 = 90 \) fares.
Count the number of fares that are \$[/tex]119 or less:
- We see 11 fares of [tex]$88 and 1 fare of $[/tex]49 in each list.
- We also see 1 instance of [tex]$119 in each list.
Total number across all lists = \( 6 \times 3 (fares \leq 119) = 18 \).
Now calculate the percentile rank for \$[/tex]119:
[tex]\[ \text{Percentile Rank} = \left(\frac{18}{90}\right) \times 100 = 20\% \][/tex]
So, the percentile rank for the fare \[tex]$119 is 20%. This means 30% of the fares are less than or equal to \$[/tex]119.
### b. Find the percentile rank for a fare of \[tex]$272. Interpret your results.
Step-by-Step Solution:
1. Count how many fares are less than or equal to \$[/tex]272.
2. Use the same method as in part (a) to calculate the percentile.
In the fares given:
- We need to count all fares that are less than or equal to \[tex]$272.
- We see that \$[/tex]272 appears 12 times across the lists.
Total cumulative by this fare tier is:
[tex]\( \text{Cumulative tally:} = \number of fares < \$272 + 12
\[
Count the fares less than or equal to \$272:
- \( \text{Fares < 272:} = 82
So in total, \( 82 + 12 = 92 \)[/tex].
[tex]\[ \text{Percentile Rank} = \left(\frac{92}{90}\right) \times 100 = 91.11\%
\][/tex]
So, the percentile rank for the fare \[tex]$272 is 91.11%. This means 91.11% of the fares are less than or equal to \$[/tex]272.
### c. Which train fare would have a percentile rank of approximately 82%?
To find the train fare corresponding to the 82nd percentile:
1. Sort the fares in ascending order.
2. Identify the fare that corresponds to the 82nd percentile.
From the ordered combined list:
In total number of fares = 90.
Position number in the sorted list for 82nd percentile is:
[tex]\[ 82 \% \times 90 = 73.8
\][/tex]
Approximately [tex]\(74th\)[/tex] position corrects to 173 fare tier. So, any of the 173 from the ascending position count,
Therefore, the train fare corresponding to the 82nd percentile is $173.
Hope this solution helps clarify the concept of percentile ranks! Let me know if you have any further questions.
