91Ó°ÊÓ

The article cited in Exercise 20 also gave the following values of the variables \(y=\) number of culs-de-sac and \(z=\) number of intersections: $$ \begin{array}{llllllllllllllllllll} y & 1 & 0 & 1 & 0 & 0 & 2 & 0 & 1 & 1 & 1 & 2 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\\ z & 1 & 8 & 6 & 1 & 1 & 5 & 3 & 0 & 0 & 4 & 4 & 0 & 0 & 1 & 2 & 1 & 4 & 0 & 4 \end{array} $$ $$ \begin{array}{llllllllllllllllllll} y & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 2 & 0 & 1 & 2 & 2 & 1 & 1 & 0 & 2 & 1 & 1 & 0 \\\ z & 0 & 3 & 0 & 1 & 1 & 0 & 1 & 3 & 2 & 4 & 6 & 6 & 0 & 1 & 1 & 8 & 3 & 3 & 5 \\\ y & 1 & 5 & 0 & 3 & 0 & 1 & 1 & 0 & 0 & & & & & & & & & & \\ z & 0 & 5 & 2 & 3 & 1 & 0 & 0 & 0 & 3 & & & & & & & & & & \end{array} $$ a. Construct a histogram for the \(y\) data. What proportion of these subdivisions had no culs-de-sac? At least one cul-de-sac? b. Construct a histogram for the \(z\) data. What proportion of these subdivisions had at most five intersections? Fewer than five intersections?

Step by step solution

01

Analyze Unique Values of y

The dataset for the variable \( y \) (number of culs-de-sac) consists of the numbers: 0, 1, 2, 3, and 5. To build a histogram, we first need to count the frequency of each unique value. For instance, by counting, we find that \( y=0 \) occurs 11 times, \( y=1 \) occurs 19 times, \( y=2 \) occurs 7 times, \( y=3 \) occurs 1 time, and \( y=5 \) occurs 1 time.

02

Construct y Histogram and Proportions

Using the frequency data, construct a histogram where the x-axis represents the number of culs-de-sac and the y-axis represents their frequencies. Calculate the proportion of subdivisions with no culs-de-sac as the count of \( y=0 \) divided by the total number of subdivisions (\( n=39 \)): \( \frac{11}{39} \approx 0.282 \). For at least one cul-de-sac, subtract from 1: \( 1 - \frac{11}{39} = \frac{28}{39} \approx 0.718 \).

03

Analyze Unique Values of z

For variable \( z \) (number of intersections), analyze the frequency of each value. The observed numbers are: 0, 1, 2, 3, 4, 5, 6, and 8. By observation and counting, we find: \( z=0 \) counts 9, \( z=1 \) counts 10, \( z=2 \) counts 3, \( z=3 \) counts 7, \( z=4 \) counts 5, \( z=5 \) counts 3, \( z=6 \) counts 3, and \( z=8 \) counts 1.

04

Construct z Histogram and Proportions

Construct a histogram using \( z \) data with the x-axis depicting the number of intersections and the y-axis representing frequency. To find the proportion with at most five intersections, sum frequencies of \( z=0 \) to \( z=5 \) (totaling 37) and divide by 39: \( \frac{37}{39} \approx 0.949 \). For fewer than five intersections, sum \( z=0 \) to \( z=4 \) frequencies (totaling 34) and divide by 39: \( \frac{34}{39} \approx 0.872 \).

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Key Concepts

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

Culs-de-sac Frequency

When working with the culs-de-sac frequency, it is essential to understand the distribution of these features across a dataset. Culs-de-sac, which are dead-end streets, can significantly impact traffic flow and urban design. Analyzing their frequency involves counting how often various levels of culs-de-sac occur in a dataset. The dataset provided includes values for the variable \( y \) representing culs-de-sac, such as 0, 1, 2, 3, and 5.
To create an effective histogram, start by counting each occurrence. For example, in the given data, \( y=0 \) occurs 11 times. This means that 11 subdivisions have no culs-de-sac.
Understanding the frequency helps in visually interpreting data through a histogram. It facilitates the identification of predominant patterns, such as the most common number of culs-de-sac, which in this case is 1, appearing 19 times. This information can be decisive for city planners assessing the prevalence of culs-de-sac in different suburbs.

Intersections Frequency

Intersections, represented by the variable \( z \), are critical points where streets meet. By analyzing intersections frequency, we delve into urban layout efficiency and traffic congestion potential. In our dataset, intersections assume values like 0, 1, 2, 3, 4, 5, 6, and 8. Similar to the culs-de-sac analysis, we must tally up each occurrence.
The frequency count helps us determine statistics such as how many subdivisions have a certain number of intersections. For instance, \( z=0 \), reflecting no intersections, was noted 9 times in the dataset. This frequency count is crucial, as it lays the foundation for creating a histogram.
Histograms offer a visual representation, helping identify the density and distribution of intersections across urban areas. This analysis provides insights into areas that may need infrastructure improvement or traffic flow adjustments.

Proportion Calculation

Proportion calculation plays a pivotal role in understanding the extent of a particular feature within a dataset. When calculating the proportion of subdivisions with specific roadway features, we must consider the frequency of each feature compared to the total number of subdivisions.
For culs-de-sac, the proportion with no culs-de-sac is calculated by dividing the frequency of zero culs-de-sac (11) by the total subdivisions (39). This gives \( \frac{11}{39} \approx 0.282 \), indicating that about 28.2% of subdivisions have none.
Similarly, to find the proportion of subdivisions with at most five intersections, sum the frequencies for 0 through 5 intersections, which is 37, and divide by 39, yielding \( \frac{37}{39} \approx 0.949 \). This reveals that approximately 94.9% of subdivisions have up to five intersections.
These calculations help in understanding the broader structure of urban environments, providing necessary data for planning and development strategies.

Data Analysis Techniques

Data analysis techniques involve harnessing various methods to interpret and derive meaning from data. The construction of histograms and the calculation of proportions are fundamental techniques for data visualization and summary.
In our scenario, constructing a histogram involves plotting the frequency of occurrence for each unique number of culs-de-sac and intersections. This provides a clear visual framework for comparing and contrasting different data points.
Additionally, proportions offer succinct numerical insights, enabling a quick assessment of data characteristics. By applying these techniques, we can efficiently summarize large datasets, uncover trends, and make data-driven decisions.
Furthermore, these methods emphasize the significance of distinguishing between detailed frequency counts and broader proportion calculations. Together, they facilitate a holistic understanding of datasets, enhancing urban analysis and decision-making.