91Ó°ÊÓ

The article cited in Exercise 18 also gave the following values of the variables \(y=\) number of culs-de-sac and \(z=\) number of intersections: \(\begin{array}{llllllllllllllllllll}y & 1 & 0 & 1 & 0 & 0 & 2 & 0 & 1 & 1 & 1 & 2 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\ z & 1 & 8 & 6 & 1 & 1 & 5 & 3 & 0 & 0 & 4 & 4 & 0 & 0 & 1 & 2 & 1 & 4 & 0 & 4 \\ y & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 2 & 0 & 1 & 2 & 2 & 1 & 1 & 0 & 2 & 1 & 1 & 0 \\ z & 0 & 3 & 0 & 1 & 1 & 0 & 1 & 3 & 2 & 4 & 6 & 6 & 0 & 1 & 1 & 8 & 3 & 3 & 5 \\ y & 1 & 5 & 0 & 3 & 0 & 1 & 1 & 0 & 0 & & & & & & & & & \\ z & 0 & 5 & 2 & 3 & 1 & 0 & 0 & 0 & 3 & & & & & & & & & \end{array}\) a. Construct a histogram for the y data. What proportion of these subdivisions had no culsde-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

Prepare the Data for y

List the values of the number of culs-de-sac \(y\) from the data: 1, 0, 1, 0, 0, 2, 0, 1, 1, 1, 2, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 2, 0, 1, 2, 2, 1, 1, 0, 2, 1, 1, 0, 1, 5, 0, 3, 0, 1, 1, 0, 0. Count the number of occurrences for each value in \(y\).

02

Construct y Histogram

Create a histogram to represent the distribution of \(y\) values. For each unique value of \(y\) (0, 1, 2, 3, 5), count how many times it appears and plot that frequency as the height of the corresponding bar.

03

Proportion of no Culs-de-Sac and at Least One Cul-de-Sac

Count the values in \(y = 0\) (those with no culs-de-sac) and calculate the proportion by dividing by the total number of subdivisions. There are 11 zeros out of 47 subdivisions. Hence, the proportion of no culs-de-sac is \(\frac{11}{47}\). For at least one cul-de-sac, subtract this value from 1. So, \(1 - \frac{11}{47} = \frac{36}{47}\).

04

Prepare the Data for z

List the values of the number of intersections \(z\) from the data: 1, 8, 6, 1, 1, 5, 3, 0, 0, 4, 4, 0, 0, 1, 2, 1, 4, 0, 4, 0, 3, 0, 1, 1, 0, 1, 3, 2, 4, 6, 6, 0, 1, 1, 8, 3, 3, 5, 0, 5, 2, 3, 1, 0, 0, 0, 3. Count the number of occurrences for each value in \(z\).

05

Construct z Histogram

Create a histogram to represent the distribution of \(z\) values. For each unique value of \(z\) (0 through 8), count how many times it appears and plot that frequency as the height of the corresponding bar.

06

Proportion of at Most and Fewer than Five Intersections

Count the values in \(z\) that are less than or equal to 5 (at most five) and divide by the total. The values \(z = 0, 1, 2, 3, 4, 5\) appear 42 times. Thus, the proportion is \(\frac{42}{47}\). For fewer than five intersections, count values \(z = 0, 1, 2, 3, 4\), which appear 36 times, giving a proportion of \(\frac{36}{47}\).

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Key Concepts

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

data distribution

When dealing with data, understanding its distribution is crucial. In the problem provided, we are asked to generate histograms based on the number of culs-de-sac and intersections. A histogram is a type of bar chart that represents the frequency of data across different categories. It provides a visual representation of how data is distributed across various values. For example, by counting how often each value of culs-de-sac occurs, we create a visual frequency distribution. The same is done for intersections. With this, you can easily see patterns or trends, such as which number of culs-de-sac or intersections is most common. Understanding these trends helps us draw valuable conclusions about the data set as a whole.

proportions

Proportions offer a way to express data relationships as fractions or percentages. In solving the exercise, we compute several proportions to describe part of the data relative to its whole. For instance, the proportion of subdivisions with no culs-de-sac can be calculated by dividing the number of subdivisions with zero culs-de-sac by the total number of subdivisions. This tells us how common it is for subdivisions to lack culs-de-sac. Similarly, proportions are used for describing subdivisions with particular counts of intersections. Proportions are powerful as they normalize values, making it easier to compare data sets of different sizes or to understand the prevalence of specific characteristics within a data set.

culs-de-sac

A cul-de-sac is a dead-end street that is typically used to refer to residential areas. In the dataset provided, the variable "y" represents the number of culs-de-sac in various subdivisions. Analyzing the distribution of culs-de-sac helps urban planners understand how residential areas are organized. For example, subdivisions with more culs-de-sac might indicate quieter, more isolated neighborhoods. In constructing the histogram for culs-de-sac data, understanding what each bar represents (the frequency of subdivisions having zero, one, or more culs-de-sac) enables better interpretation of how such street patterns are distributed across the city or area being studied. Recognizing these patterns can lead to more informed urban development decisions.

intersections

Intersections, represented by the variable "z" in the dataset, denote points where two or more roads meet. They are crucial in traffic flow and urban planning because intersections often dictate accessibility and connectivity within an area. By looking at intersections, we can gauge how interconnected or accessible a subdivision is. Constructing a histogram for intersections involves plotting the frequency of subdivisions with various numbers of intersections, from none to more complex configurations. By analyzing this data, we can discern trends, such as whether the area is densely or sparsely interconnected. Such insights are essential for planners to design streets that effectively balance traffic, accessibility, and residential tranquility.