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 (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

Organize the Data

The problem provides a frequency distribution table with bus route lengths (in km) and the frequencies for these lengths across defined intervals. The intervals and frequencies provide the complete data necessary for the exercise.

02

Draw a Histogram for Part (a)

To draw a histogram, plot the length intervals on the x-axis and the frequencies on the y-axis. Each interval corresponds to a bar whose height reflects the corresponding frequency. Use the given intervals: - Ranging from 6 to less than 8 km up to 40 to less than 45 km. - Frequencies: 6, 23, 30, 35, 32, 48, 42, 40, 28, 27, 26, 14, 27, 11, 2. Label axes appropriately to create a clear representation of the data distribution.

03

Calculate Proportions for Part (b)

First, calculate the proportion of route lengths less than 20 km by adding the frequencies of the relevant intervals: 6-<8, 8-<10, 10-<12, 12-<14, 14-<16, 16-<18. Total these frequencies (6 + 23 + 30 + 35 + 32 + 48 = 174).Next, calculate the total frequency: the sum of all frequencies in the table (361).Proportion of routes < 20 km = \( \frac{174}{361} \).For routes of at least 30 km, sum the frequencies for intervals 30-<35, 35-<40, 40-<45: (27 + 11 + 2 = 40).Proportion of routes \( \geq 30 \) km = \( \frac{40}{361} \).

04

Determine 90th Percentile for Part (c)

To find the 90th percentile, identify the cumulative frequency and the position of the 90th percentile, which is \( 0.9 \times 361 \approx 325 \).Examine cumulative frequencies:- The 325th route falls within the 22-<24 km interval where cumulative frequency exceeds 325.Thus, the 90th percentile for the bus route length is approximately 22 to 24 km.

05

Determine the Median for Part (d)

The median is at the 50th percentile, thus at \( 0.5 \times 361 \equiv 180.5 \).Locate the interval containing 180.5 by considering cumulative frequencies:- Cumulative frequency reaches 180.5 in the 14-<16 km interval.Consequently, the median route length is approximately in the 14 to 16 km range.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Histogram

A histogram is a type of bar graph that offers a visual representation of data distribution, making it easier to see how data clusters. In our example, the histogram displays the frequency distribution of bus route lengths in different intervals.

To create a histogram, follow these steps:

• Identify the data intervals, in this case, the lengths of bus routes that range from 6 to less than 8 kilometers, and so on up to 40 to less than 45 kilometers.
• For each interval, create a bar where the height represents the frequency of data within that interval. This means if, for example, the frequency is 23 in the interval 8-<10 km, the bar will reach up to 23 on the y-axis.
• Label the x-axis with the different length intervals and the y-axis with frequency counts.
• Each bar in the histogram should touch the one next to it, showing continuous data intervals. This touch-point signifies that each length could be part of neighboring intervals due to measurement continuity.

By examining such a histogram, patterns and trends in the data are easily observable, like which length of routes occur most frequently.

Percentile Calculation

Percentiles are a way to understand the distribution of the data by showing the relative standing of a value compared to the rest. In our bus route exercise, you needed to calculate the 90th percentile.

To calculate this percentile:

• First, multiply the total number of observations by the desired percentile. Hence, for the 90th percentile, calculate \(0.9 \times 361\).
• The result, 325, indicates the position within your ordered data that corresponds to the 90th percentile.
• With cumulative frequencies, find the first interval where this position falls in. Our solution shows that this is the interval 22-<24 km, meaning that 90% of bus routes are 24 km or less.

This process helps you understand the long tail of data and identify thresholds where a majority of data points fall below.

Median Calculation

The median of a dataset represents the middle point, indicating that half of the observations lie below and half lie above this value. In frequency distribution analyses, such as with bus routes, calculating the median involves a clear understanding of cumulative frequencies.

To find the median:

• Determine the position of the median by multiplying the total number of observations by \(0.5\). Here, you find \(0.5 \times 361 = 180.5\).
• Identify in which frequency interval this cumulative frequency position falls. For our data, this position is in the interval 14-<16 km.
• Hence, the median bus route length is between 14 and 16 km, where half of the routes are shorter, and half are longer.

The median is a strong measure of central tendency that is not distorted by extreme values or outliers and is very useful for understanding skewed distributions.