91Ó°ÊÓ

Lengths of bus routes for any particular transit system will typically vary from one route to another. The article "Planning of City Bus Routes" \((J\). of the Institution of Engineers, 1995: 211-215) gives the following information on lengths \((\mathrm{km})\) for one particular system: \(\begin{array}{lccccc}\text { Length } & 6-<8 & 8-<10 & 10-<12 & 12-<14 & 14-<16 \\ \text { Frequency } & 6 & 23 & 30 & 35 & 32 \\ \text { Length } & 16-<18 & 18-<20 & 20-<22 & 22-<24 & 24-<26 \\ \text { Frequency } & 48 & 42 & 40 & 28 & 27 \\ \text { Length } & 26-<28 & 28-<30 & 30-<35 & 35-<40 & 40-<45 \\\ \text { Frequency } & 26 & 14 & 27 & 11 & 2\end{array}\) a. Draw a histogram corresponding to these frequencies. b. What proportion of these route lengths are less than 20 ? What proportion of these routes have lengths of at least 30 ? c. Roughly what is the value of the \(90^{\text {th }}\) percentile of the route length distribution? d. Roughly what is the median route length?

Step by step solution

01

Draw the Histogram

To draw the histogram, mark the lengths on the horizontal axis in intervals like 6-<8, 8-<10, etc., and frequencies on the vertical axis. For each interval, draw a bar up to the frequency value. This visualizes the distribution of the bus route lengths.

02

Calculate Proportion for Routes Less Than 20

Add the frequencies for length intervals less than 20: 6, 23, 30, 35, 32, 48, and 42. This totals to 216. Then, calculate the proportion by dividing 216 by the total number of routes (351), giving a proportion of approximately 0.615.

03

Calculate Proportion for Routes at Least 30

Add the frequencies for length intervals from 30 onwards: 27, 11, and 2. This totals to 40. Calculate the proportion by dividing 40 by the total number of routes (351), giving a proportion of approximately 0.114.

04

Calculate the 90th Percentile

To find the 90th percentile, determine which interval contains the 316th route (0.9 × 351 = 315.9). Cumulative frequency shows it falls in the 24-<26 interval. Hence, the 90th percentile is roughly around 25 km.

05

Determine the Median Route Length

The median is the 176th route value (since 351/2 = 175.5). From cumulative frequency, the 176th value falls in the 16-<18 interval. Thus, the median route length is around 17 km.

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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 that helps you visualize the frequency distribution of data, in this case, the lengths of bus routes. To create a histogram:

1. Place the intervals (or bins) on the horizontal axis, such as 6-<8 km, 8-<10 km, etc.
2. Plot the frequency of each interval on the vertical axis.
3. Draw bars for each interval where the height represents the frequency. The bars touch each other, indicating that the data is continuous.

This gives a clear picture of how the bus route lengths are distributed. You can quickly see the most common lengths and any variations in the frequency. Histograms are particularly useful for identifying the shape of the data distribution, whether it's skewed or symmetrical, and any outliers that might exist.

Percentile

The concept of a percentile is useful in understanding a specific value's position within a data set. It indicates the value below which a percentage of data falls. For example, the 90th percentile means 90% of the data points are less than this value.

To calculate the 90th percentile in this context:

1. Multiply the total number of data points (351) by 0.9 to find the 90th percentile's position. This yields 315.9, indicating the 316th data point.
2. By using cumulative frequency, you can locate the interval where this data point lies. In this example, it falls in the 24-<26 km interval, suggesting the 90th percentile is approximately 25 km.

Percentiles are crucial in statistics for comparing different data sets or individuals within a dataset to determine where they stand relative to the rest.

Median

The median is the middle value in a data set when it is ordered from smallest to largest. It divides the data into two equal halves. If the data set has an odd number of entries, the median is the middle number. For an even number of entries, it's the average of the two central numbers.

Here's how to determine the median with our bus route data:

1. Order all bus route lengths from shortest to longest using cumulative frequency.
2. Since the total number of routes is 351, the median corresponds to the 176th value (since 351/2 = 175.5).
3. Using the cumulative frequencies, identify that the 176th route length resides in the 16-<18 km interval.

Thus, the median route length is approximately 17 km. The median provides insight into the central tendency of the data, especially useful when you have skewed data distributions where the mean might be misleading.

Cumulative Frequency

Cumulative frequency is a running total of frequencies through different data classes. It illustrates how many data points fall below a certain level, which aids in understanding the distribution over intervals.

To calculate cumulative frequency, start at the beginning of the dataset:

• Add the frequency of a class to the sum of frequencies of all previous classes.
• Continue this until all classes are accounted for.

This method is critical when determining percentiles or the median in a data set.

In this exercise, you use cumulative frequency to locate specific percentiles and median values application. By summing frequencies progressively, the cumulative frequency can tell us which interval contains the data point of interest, helping you uncover more profound insights into route lengths and distribution patterns.