Note: Please make sure to properly format your answers. All dollar figures in the answers need to include the dollar sign and any amount over 1,000 should include the comma ($2,354.67). All percentage values in the answers need to include a percentage sign (%). For all items without specific rounding instructions, round your answers to two decimal places, show both decimal places (5.06).

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.

a. Find the percentile rank for a fare of $119.
b. Find the percentile rank for a fare of $272.
c. Based on your first two answers, which train fare would have a percentile rank of approximately 82%?

To solve this problem, we first need to understand how to calculate percentile rank. The percentile rank of a score (or fare in this case) is the percentage of scores in a distribution that are equal to or less than the given score.

The formula to find the percentile rank is:

\[
\text{Percentile Rank} = \left(\frac{\text{Number of values below the score} + 0.5 \times \text{Number of values equal to the score}}{\text{Total number of values}}\right) \times 100
\]

Let's assume we have a list of all fares available on that day. Since the fares are ordered, we can directly count the number of fares below and equal to a given fare. However, as the list of fares isn’t provided, we'll assume a hypothetical list for calculation purposes.

### a. Finding the Percentile Rank for a Fare of $119
Let's assume the list of fares in ascending order is as follows (based on typical pricing for such trips):
- $89
- $99
- $109
- **$119** (we assume this is the 4th value in a list of 10)
- $129
- $139
- $149
- $159
- $199
- $272

Total number of fares: 10

1. **Number of values below $119**: 3
2. **Number of values equal to $119**: 1

Using the formula:

\[
\text{Percentile Rank for } \$119 = \left(\frac{3 + 0.5 \times 1}{10}\right) \times 100 = \left(\frac{3.5}{10}\right) \times 100 = 35\%
\]

### b. Finding the Percentile Rank for a Fare of $272
Using the same hypothetical list:

1. **Number of values below $272**: 9
2. **Number of values equal to $272**: 1

Using the formula:

\[
\text{Percentile Rank for } \$272 = \left(\frac{9 + 0.5 \times 1}{10}\right) \times 100 = \left(\frac{9.5}{10}\right) \times 100 = 95\%
\]

### c. Finding the Train Fare with a Percentile Rank of Approximately 82%
Given our hypothetical list and percentile calculations, we would be looking for the fare that has 8 or more values below it but less than 9. 

If we interpolate between the $199 (rank 75%)$ and $272 (rank 95%)$, the fare that corresponds to approximately the 82nd percentile would be slightly above $199.

Since $199$ gives us a 75% rank, and $272$ gives us 95%, the value between them could be approximately $230 or $240. But typically, such fare would be around $199 to $240.

**Conclusion**: 
1. **Percentile Rank for $119**: **35%**
2. **Percentile Rank for $272**: **95%**
3. **Approximate Fare at 82nd Percentile**: Likely **$240**

---

Do you want more details on how interpolation was done, or have any questions? Here are some related questions:

1. How would the percentile rank change if there were more than 10 fare options?
2. What if two fares had the same value, how would that affect the percentile calculation?
3. Can you explain why $272 ranks so high on the percentile?
4. How could you use this percentile rank to choose a fare?
5. What does a 35% percentile rank indicate about the fare?

**Tip:** When calculating percentiles, always remember to include the term that accounts for scores equal to the value being ranked, as this ensures accuracy in ranking, especially in smaller data sets.

Learn how to calculate percentile ranks for train fares on February 15, using statistical methods. Includes step-by-step solutions for finding percentile ranks of specific fares ($119 and $272), and identifying approximate fare at the 82nd percentile.

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