91Ó°ÊÓ

Suppose that three book publishers were interested in the number of fiction paperbacks adult consumers purchase per month. Each publisher conducted a survey. In the survey, adult consumers were asked the number of fiction paperbacks they had purchased the previous month. The results are as follows: $$\begin{array}{|l|l|}\hline \\# \text { of books } & {\text { Freq. }} & {\text { Rel. Freq. }} \\ \hline 0 & {10} \\ \hline 1 & {12} \\ \hline 2 & {12} \\ \hline 2 & {16} \\ \hline 3 & {12} \\ \hline 4 & {8} \\ \hline 5 & {6} \\ \hline 6 & {2} \\ \hline 8 & {2} \\ \hline\end{array}$$ Table 2.62 Publisher A $$\begin{array}{|l|l|l|}\hline \\# \text { of books } & {\text { Freq. }} & {\text { Rel. Freq. }} \\ \hline 0 & {18} \\ \hline 1 & {24} \\ \hline 2 & {24} \\ \hline 3 & {22} \\ \hline 3 & {22} \\ \hline 5 & {10} \\ \hline 7 & {5} \\ \hline 9 & {1} \\ \hline\end{array}$$ Table 2.63 Publisher B $$\begin{array}{|l|l|l|}\hline \text { of books } & {\text { Freq. }} & {\text { Rel. Freq. }} \\ \hline 0-1 & {20} \\ \hline 2-3 & {35} \\ \hline 4-5 & {12} \\ \hline 6-7 & {2} \\ \hline 8-9 & {1} \\ \hline\end{array}$$ Table 2.64 Publisher C a. Find the relative frequencies for each survey. Write them in the charts. b. Using either a graphing calculator, computer, or by hand, use the frequency column to construct a histogram for each publisher's survey. For Publishers A and B, make bar widths of one. For Publisher C, make bar widths of two. c. In complete sentences, give two reasons why the graphs for Publishers A and B are not identical. d. Would you have expected the graph for Publisher C to look like the other two graphs? Why or why not? e. Make new histograms for Publisher A and Publisher B. This time, make bar widths of two. f. Now, compare the graph for Publisher C to the new graphs for Publishers A and B. Are the graphs more similar or more different? Explain your answer.

Step by step solution

01

Calculate Total Frequency (Publisher A)

Sum the frequencies given in the table for Publisher A: \(10 + 12 + 12 + 16 + 12 + 8 + 6 + 2 + 2 = 80\). This is the total frequency for Publisher A.

02

Calculate Relative Frequencies (Publisher A)

Divide each frequency by the total frequency calculated in Step 1. For Publisher A, for instance, the relative frequency for 0 books is \(\frac{10}{80} = 0.125\). Repeat for all other frequencies to fill out the Relative Frequency column: \(\frac{10}{80}, \frac{12}{80}, \frac{12}{80}, \frac{16}{80}, \frac{12}{80}, \frac{8}{80}, \frac{6}{80}, \frac{2}{80}, \frac{2}{80}\).

03

Calculate Total Frequency (Publisher B)

Sum the frequencies for Publisher B: \(18 + 24 + 24 + 22 + 22 + 10 + 5 + 1 = 126\). This is the total frequency for Publisher B.

04

Calculate Relative Frequencies (Publisher B)

Divide each frequency by the total frequency calculated in Step 3. For Publisher B, for example, the relative frequency for 0 books is \(\frac{18}{126} \approx 0.143\). Repeat this process for all frequencies to complete the Relative Frequency column: \(\frac{18}{126}, \frac{24}{126}, \frac{24}{126}, \frac{22}{126}, \frac{22}{126}, \frac{10}{126}, \frac{5}{126}, \frac{1}{126}\).

05

Calculate Total Frequency (Publisher C)

Sum the frequencies for Publisher C: \(20 + 35 + 12 + 2 + 1 = 70\). This is the total frequency for Publisher C.

06

Calculate Relative Frequencies (Publisher C)

Divide each frequency by the total frequency calculated in Step 5. For Publisher C, for the first category \(0-1\): \(\frac{20}{70} \approx 0.286\). Continue this for all other frequency intervals: \(\frac{20}{70}, \frac{35}{70}, \frac{12}{70}, \frac{2}{70}, \frac{1}{70}\).

07

Construct Histograms for Publishers A, B, and C

For Publisher A and B, create histograms with each bar representing a single book purchase (bar width = 1). For Publisher C, since intervals are broader, create bars with widths of two book purchases.

08

Compare Graphs of Publishers A and B

Graphs for Publishers A and B are not identical due to: 1) Different sample sizes leading to different heights of bars for equivalent categories, 2) Differences in the distribution or shape of purchase frequencies.

09

Compare C's Graph with A and B

The graph for Publisher C differs from Publishers A and B due to broader categories (bar widths of two), which smoothens out some details present in A and B's single book categories.

10

New Histograms for Publishers A and B (Bar Widths of Two)

Re-group the data for Publishers A and B to have bar widths of two books (e.g., 0-1, 2-3, etc.). Each new category will sum the frequencies of two adjacent original categories.

11

Compare the New Graphs to Publisher C

After grouping and redrawing, the new histograms for A and B might appear more similar to C due to similar category widths, reducing detailed variability in single-book data.

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Key Concepts

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

Relative Frequency

Relative frequency is a way to express how often something happens compared to the total occurrences. It's like saying out of 100 times, how many times a particular event happened.

To find the relative frequency:

• First, calculate the total frequency by adding all the individual frequencies together.
• For example, in Publisher A's survey, the total frequency is calculated as \(10 + 12 + 12 + 16 + 12 + 8 + 6 + 2 + 2 = 80\).
• Then, take each individual frequency and divide it by this total frequency.

Why Use Relative Frequency?

Relative frequency gives a proportion or a percentage which is easier to understand at a glance, especially when comparing different datasets. For instance, if you know that 0.125 of the respondents bought no books last month, it means that 12.5% of people surveyed did not purchase any books.

This approach also standardizes data, making it easier for comparison even across differing sample sizes, like comparing Publisher A's survey results to those of Publisher B and C, each with different total respondent counts.

Frequency Distribution

A frequency distribution is essentially a table that displays the number of occurrences of different outcomes in a dataset. It's a neat way to organize data to understand the most common and rare occurrences.

In the surveys conducted by the publishers, frequency distributions help to outline how many people bought each number of books last month:

• For Publisher A, the distribution shows, for instance, that 12 people bought 1 book.
• It shows a pattern of buying behavior among the respondents.

Interpreting Frequency Distributions

The key insights emerge when you interpret these frequencies. You can spot trends, such as which number of books had the highest frequency, indicating the most common purchase behavior. In our provided data, checking across all publishers helps identify variations in reading habits per survey. With frequency distributions, you're not only looking at who purchased but also understanding how purchasing tendencies differ across sampled groups.

Data Visualization

Data visualization turns complex datasets into easily digestible images like histograms, which were used to display the survey results from our publishers.

Histograms are clever tools for visualizing frequency distributions. They free data from rows and columns, turning them into a bar chart format where:

• Each bar represents the count for a range or specific value.
• Bar height shows the frequency, allowing you to see at a glance which values are most common.

Creating Histograms

When constructing a histogram:

• Decide on bar widths (like 1 book for Publishers A and B).
• Plot each data range on the x-axis and frequency on the y-axis to understand distributions visually.

This approach helps to easily see differences and similarities across datasets, identify peaks (where the most frequent occurrences are), and understand purchasing patterns.

Effectively visualized data enables faster conclusions and easier comparisons, as seen when interpreting purchase results through the lens of Publisher C compared to Publishers A and B.