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Chapter 11: Problem 2

A group of adults where asked how many cars they had in their household a. Complete the frequency table for the car number data b. Construct a histogram of the data c. Construct a pie chart of the data $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|l|l|} \hline 1 & 4 & 2 & 2 & 1 & 2 & 3 & 3 & 1 & 4 & 2 & 2 \\ \hline 1 & 2 & 1 & 3 & 2 & 2 & 1 & 2 & 1 & 1 & 1 & 2 \\ \hline \end{array} $$

Short Answer

Expert verified
Frequency Table: 1 car: 8, 2 cars: 9, 3 cars: 2, 4 cars: 2. Histogram and pie chart should represent these frequencies visually.

Step by step solution

01

Organize and Count the Data

To complete the frequency table, organize the data by counting the number of times each value occurs. From the data set, count the number of households for each number of cars. The counts are as follows: \(1\) car: \(8\) households, \(2\) cars: \(9\) households, \(3\) cars: \(2\) households, and \(4\) cars: \(2\) households.
02

Construct the Frequency Table

Using the counts from Step 1, complete the frequency table as follows: | Number of Cars | Frequency | |----------------|-----------| | 1 | 8 | | 2 | 9 | | 3 | 2 | | 4 | 2 |
03

Create a Histogram

To construct a histogram, plot the number of cars on the x-axis and the frequency on the y-axis. Create bars for each number of cars based on their frequency. The bar heights should match the frequencies from the table: \(1\) car height is \(8\), \(2\) cars height is \(9\), \(3\) cars height is \(2\), and \(4\) cars height is \(2\).
04

Calculate Total Households for Pie Chart

To create a pie chart, first calculate the total number of households, which is the sum of all frequencies: \(8 + 9 + 2 + 2 = 21\).
05

Calculate Angles for Pie Chart

Calculate the proportion of each car count relative to the total, and then convert these to angles for the pie chart. The angles are calculated by multiplying the proportion by \(360\) degrees:- \(1\) car: \((8/21) \times 360 \approx 137.14\) degrees- \(2\) cars: \((9/21) \times 360 \approx 154.29\) degrees- \(3\) cars: \((2/21) \times 360 \approx 34.29\) degrees- \(4\) cars: \((2/21) \times 360 \approx 34.29\) degrees.
06

Construct the Pie Chart

Draw a circle divided into sectors based on the angles calculated in Step 5. Each sector represents a different number of cars, and its size corresponds to the angle calculated: \(1\) car for \(137.14\) degrees, \(2\) cars for \(154.29\) degrees, \(3\) cars for \(34.29\) degrees, and \(4\) cars for \(34.29\) degrees.

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Key Concepts

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

Frequency Table
A frequency table is a very useful tool in data representation. It organizes data into categories or groups and shows how frequently each category occurs. In the case of the car number data, the frequency table helps us understand the distribution of cars across households.
To construct a frequency table, follow these steps:
  • First, list all the distinct values that your data can take. For our car data, these are 1, 2, 3, and 4 cars.
  • Count how often each value appears in the dataset. It's like tallying the occurrence of each category in the list.
  • Create two columns: one for the number of cars (the value) and one for frequency, which is the count of each value.
With the car data, you can see the number of households that have 1 car is 8, 2 cars is 9, and so on. This table not only summarizes the data but also makes comparisons easy.
Histogram
A histogram is a way of visualizing a frequency table. It resembles a bar chart but is specifically used to show frequency distributions. This makes it a great tool for quickly understanding where data is concentrated.
To construct a histogram from the frequency table:
  • First, draw two axes. Label the x-axis with the categories, here the number of cars (1, 2, 3, 4).
  • The y-axis will represent the frequency of each category.
  • Draw bars for each category. Each bar's height should match the corresponding frequency value.
For instance, the height of the bar for 2 cars will be taller because the frequency is higher compared to others. A histogram provides a clear graphical summary, which can be more intuitive to understand than raw numbers.
Pie Chart
Pie charts offer another method of data representation, ideal for showing proportions. Each "slice" of the pie corresponds to a category, and the size of each slice represents its proportion to the total.
To create a pie chart:
  • Calculate the total number of data points. For the car example, add up all households, which equals 21.
  • Determine the proportion of each category by dividing each frequency by the total.
  • Convert these proportions into angles by multiplying by 360 degrees.
In the given example, households with 2 cars take up the largest segment of the pie chart at approximately 154.29 degrees. This visual representation makes it easy to appreciate the distribution of data at a glance.
Data Analysis
Data analysis is the process of inspecting, cleaning, transforming, and modeling data to discover useful information for decision-making. It involves various techniques and tools to make sense of the collected data. In this exercise, we focus on organizing car data within households.
To analyze our data effectively:
  • Organize data using a frequency table. This shapes your understanding by highlighting the count of different categories.
  • Use histograms to visualize the frequency distribution. It emphasizes how data is spread across different categories.
  • Implement pie charts to visualize proportions, helping to quickly grasp the data's relative sizes.
By employing these methods, you'll gain insights into how many households prefer having 2 cars or why some might have no preference for more than 1 car. Each method presents unique insights, and combining them forms a comprehensive analysis, ultimately aiding in informed decision-making.

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Most popular questions from this chapter

A sample of 30 distance scores measured in yards has a mean of \(7,\) a variance of 16 , and a standard deviation of 4 . a. You want to convert all your distances from yards to feet, so you multiply each score in the sample by \(3 .\) What are the new mean, median, variance, and standard deviation? b. You then decide that you only want to look at the distance past a certain point. Thus, after multiplying the original scores by \(3,\) you decide to subtract 4 feet from each of the scores. Now what are the new mean, median, variance, and standard deviation?

Make up three data sets with 5 numbers each that have: a. the same mean but different standard deviations. b. the same mean but different medians. c. the same median but different means.

In your class, design a poll on a topic of interest to you and give it to the class. a. Summarize the data, computing the mean and five-number summary. b. Create a graphical representation of the data. c. Write several sentences about the topic, using your computed statistics as evidence in your writing.

The table below shows scores on a Math test. a. Complete the frequency table for the Math test scores b. Construct a histogram of the data construct a pie chart of the data $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|l|l|} \hline 80 & 50 & 50 & 90 & 70 & 70 & 100 & 60 & 70 & 80 & 70 & 50 \\ \hline 90 & 100 & 80 & 70 & 30 & 80 & 80 & 70 & 100 & 60 & 60 & 50 \\ \hline \end{array} $$

Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new AIDS antibody drug is currently under study. It is given to patients once the AIDS symptoms have revealed themselves. Of interest is the average length of time in months patients live once starting the treatment. Two researchers each follow a different set of 40 AIDS patients from the start of treatment until their deaths. The following data (in months) are collected. $$ \text { Researcher } 1: 3 ; 4 ; 11 ; 15 ; 16 ; 17 ; 22 ; 44 ; 37 ; 16 ; 14 ; 24 ; 25 ; 15 ; 26 ; 27 ; 33 ; 29 ; $$ \(35 ; 44 ; 13 ; 21 ; 22 ; 10 ; 12 ; 8 ; 40 ; 32 ; 26 ; 27 ; 31 ; 34 ; 29 ; 17 ; 8 ; 24 ; 18 ; 47 ; 33 ; 34\) Researcher \(2: 3 ; 14 ; 11 ; 5 ; 16 ; 17 ; 28 ; 41 ; 31 ; 18 ; 14 ; 14 ; 26 ; 25 ; 21 ; 22 ; 31 ; 2 ; 35 ;\) \(44 ; 23 ; 21 ; 21 ; 16 ; 12 ; 18 ; 41 ; 22 ; 16 ; 25 ; 33 ; 34 ; 29 ; 13 ; 18 ; 24 ; 23 ; 42 ; 33 ; 29\) a. Create comparative histograms of the data b. Create comparative boxplots of the data
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