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The paper "Modeling and Measurements of Bus Service Reliability" (Transportation Research [1978]: \(253-256\) ) studied various aspects of bus service and presented data on travel times (in minutes) from several different routes. The accompanying frequency distribution is for bus travel times from origin to destination on one particular route in Chicago during peak morning traffic periods: $$ \begin{array}{crc} \begin{array}{c} \text { Travel } \\ \text { Time } \end{array} & \text { Frequency } & \begin{array}{c} \text { Relative } \\ \text { Frequency } \end{array} \\ \hline 15 \text { to }<16 & 4 & .02 \\ 16 \text { to }<17 & 0 & .00 \\ 17 \text { to }<18 & 26 & .13 \\ 18 \text { to }<19 & 99 & .49 \\ 19 \text { to }<20 & 36 & .18 \\ 20 \text { to }<21 & 8 & .04 \\ 21 \text { to }<22 & 12 & .06 \\ 22 \text { to }<23 & 0 & .00 \\ 23 \text { to }<24 & 0 & .00 \\ 24 \text { to }<25 & 0 & .00 \\ 25 \text { to }<26 & 16 & .08 \\ \hline \end{array} $$ a. Construct the corresponding histogram. b. Compute (approximately) the following percentiles: i. 8 6th iv. 95 th ii. \(15 \mathrm{th}\) v. 10 th iii. 90 th

Step by step solution

01

Construct the histogram

Each travel time interval represents a bar in the histogram and the relative frequency values determine the height of each bar. The X-axis represents the travel time while the Y-axis represents the relative frequency.

02

Compute the percentiles

To compute the percentiles, we use the cumulative frequency distribution and the formula for percentiles which is: Percentile = \( \frac{n}{100} × (N + 1) \), where n is the percentile we want (e.g. 10th, 15th) and N is the total number of observations (201).

03

Calculate the 10th percentile

First, calculate the position of the 10th percentile using the formula: \( \frac{10}{100} × 202 \) = round off to 20. The 20th observation lies in the travel time interval 18 to < 19. So, the 10th percentile is approximately 18 minutes.

04

Calculate the 15th percentile

Similarly, the 15th percentile position is: \( \frac{15}{100} × 202 \) = round off to 30. The 30th observation lies in the travel time interval 18 to < 19, hence the 15th percentile is approximately 18 minutes.

05

Calculate the 86th percentile

The 86th percentile position is: \( \frac{86}{100} × 202 \) = round off to 173. The 173rd observation lies within the interval 21 to < 22. Hence, the 86th percentile is approximately 21 minutes.

06

Calculate the 90th percentile

The 90th percentile position is: \( \frac{90}{100} × 202 \) = round off to 182. The 182nd observation lies within the interval 25 to < 26. Hence, the 90th percentile is approximately 25 minutes.

07

Calculate the 95th percentile

Finally, the 95th percentile position is: \( \frac{95}{100} × 202 \) = round off to 192. The 192nd observation lies within 25 to < 26. Hence, the 95th percentile is approximately 25 minutes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Histogram

A histogram is a graphical representation of the distribution of numerical data. It is a type of bar chart where the data is grouped into ranges, known as bins, and the frequency of data within each bin is represented by the height of the bar. To create a histogram based on the given exercise, which involves bus travel times, each bar would represent an interval of travel times, and the length of the bar indicates how many observations fall within that range.

For instance, if we have a bar for the '18 to <19' minute interval with a relative frequency of .49, this means nearly half of the travel times were within this period. This visual information allows us to quickly perceive the distribution of bus travel time and identify patterns such as the most common travel time intervals during the peak morning traffic in Chicago. Histograms make it easier to see where the bulk of observations lies and can be used to estimate measures of central tendency and variability.

Percentiles

Percentiles are measures that divide a set of observations into 100 equal parts, each representing 1% of the ordered data. They are useful for providing a relative standing of a particular value within a data set. For example, a student scoring in the 90th percentile on a test did better than 90% of the other students.

In the bus travel time data set, computing percentiles can help transportation analysts understand the variability and reliability of travel times. For instance, the 15th percentile calculated in the solution (approximately 18 minutes) indicates that 15% of the bus trips took less than 18 minutes. Likewise, the 90th percentile being approximately 25 minutes means that 90% of the trips are completed within 25 minutes or less. Percentiles can assist in pinpointing performance targets for improving bus service efficiency.

Cumulative Frequency

Cumulative frequency is the running total of frequencies through the data set in ascending order. It allows us to determine how many observations fall below a particular value. When analyzing data such as bus travel times, the cumulative frequency gives a comprehensive picture of the data's distribution.

In practice, when calculating percentiles, as demonstrated in the solution, cumulative frequency helps in locating the position of a specific observation representative of the percentile. For the 90th percentile, by accumulating the frequencies up to the 90th percent position, we can ascertain that most of the observations fall within a specific travel time interval. This information is critical for transportation planners when setting realistic expectations and optimizing route schedules to minimize delays.

Statistics

Statistics is the field that involves collecting, analyzing, interpreting, presenting, and organizing data. In our exercise, statistics allows us to summarize complex data like bus travel times into meaningful information that can guide decision-making. It involves the use of various methods and concepts, including histograms, percentiles, and cumulative frequencies, to describe and understand the data.

Statistics helps to provide insights into trends, which can then be used to predict future patterns or to assess the reliability of a service. By utilizing tools such as percentile calculations and histograms, stakeholders can develop strategies to improve the service quality, like adjusting bus departure times to reduce travel variability and ensure punctuality.