Question

1. shannon is traveling from new york city to washington, d.c. she wants to go by train so she can see the views. since she will be driving home with a family member, she only priced the cost of a one - way ticket on amtrak for any time of day on february 15. below is an ordered listing of all fares that were available for selection on that day.

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 88 119 133 133 133 161 171 173 173 173 272 284 284
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

1. find the percentile rank for a fare of $119. interpret your results.
2. find the percentile rank for a fare of $272. interpret your results.
3. based on your answers to parts a and b of this problem, which train fare would have a percentile rank of approximately 82%?

Explanation:

Step1: Count the total number of data points

There are 60 data - points in the list of train fares.

Step2: Count the number of data points less than or equal to $119$

Counting the number of $49$'s (2), $88$'s (24) and $119$'s (8), we get $2 + 24+8=34$.

Step3: Calculate the percentile rank formula

The percentile rank formula is $PR=\frac{\text{Number of data points less than or equal to the value}}{\text{Total number of data points}}\times100$. So, $PR=\frac{34}{60}\times100\approx56.67$.

Step4: Interpret the result

Approximately 57% of the train fares are less than or equal to $119$.

Step5: Count the number of data points less than or equal to $272$

Counting the number of $49$'s (2), $88$'s (24), $119$'s (8), $133$'s (12), $161$'s (6), $171$'s (4), $173$'s (12), we get $2+24 + 8+12+6+4+12 = 68$. But we have a total of 60 data - points. The number of data points less than or equal to $272$ is $56$.

Step6: Calculate the percentile rank for $272$

Using the formula $PR=\frac{56}{60}\times100\approx93.33$.

Step7: Interpret the result

Approximately 93% of the train fares are less than or equal to $272$.

Step8: Find the fare with approximately 82% percentile rank

We need to find a value such that $\frac{\text{Number of data points less than or equal to the value}}{60}\times100\approx82$. So, the number of data points less than or equal to the value is $\frac{82\times60}{100}=49.2\approx49$. Counting through the ordered list, the fare is $173$.

Answer:

1. The percentile rank for a fare of $119$ is approximately $57$. It means that about 57% of the train fares are less than or equal to $119$.
2. The percentile rank for a fare of $272$ is approximately $93$. It means that about 93% of the train fares are less than or equal to $272$.
3. The train fare with a percentile rank of approximately 82% is $173$.