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The accompanying frequency distribution summarizes data on the number of times smokers who had successfully quit smoking attempted to quit before their final successful attempt ("Demographic Variables, Smoking Variables, and Outcome Across Five Studies," Health Psychology [2007]: 278-287). \begin{tabular}{cc} Number of Attempts & Frequency \\ \hline 0 & 778 \\ 1 & 306 \\ 2 & 274 \\ \(3-4\) & 221 \\ 5 or more & 238 \\ \hline \end{tabular} Assume that no one had made more than 10 unsuccessful attempts, so that the last entry in the frequency distribution can be regarded as \(5-10\) attempts. Summarize this data set using a histogram. Be careful- the class intervals are not all the same width, so you will need to use a density scale for the histogram. Also remember that for a discrete variable, the bar for 1 will extend from 0.5 to 1.5. Think about what this will mean for the bars for the 3-4 group and the \(5-10\) group.

Step by step solution

01

Identify the groups and their corresponding frequency.

The groups are defined as follows: '0', '1', '2', '3-4', and '5-10'. Their corresponding frequencies are: 778, 306, 274, 221, and 238 respectively.

02

Determine the width and height (density) of each group.

Given this is a density scale, the width for groups '0', '1', '2' is 1 (e.g., 0.5 to 1.5), for '3-4' it's 2 (e.g., 2.5 to 4.5), and for '5-10' it's 6 (e.g., 4.5 to 10.5). To determine the density (height) for each category, divide the frequency by the width for each group.

03

Draw the histogram.

Create a horizontal axis representing the number of attempts and a vertical axis representing the density. Draw bars for each group, where the width represents the group's range and the height corresponds to the density calculated in the previous step.

04

Label the histogram and verify.

Label the histogram appropriately, including a title, a label for each axis, and each specific group. Ensure the total area of bars equals the sum of frequencies. This can be checked by adding up the areas of the bars (width times height for each), which should equal the total frequency of 1817.

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Key Concepts

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

Histogram

A histogram is a graphical representation used to visualize the distribution of a dataset. It consists of bars where each bar denotes the frequency (how many times data points occur) of items in consecutive numerical intervals.

When creating a histogram, like the one for summarizing the quit attempts by smokers, it involves a few steps. First, you need to determine the intervals or groups, and then, count the number of occurrences (frequency) within each interval. With unequal intervals, like in our exercise with the '3-4' and '5-10' attempt groups, you must use the density, rather than the frequency, to ensure the area of the bars accurately reflects the number of observations.

Essentially, a histogram offers a visual summary of numerous data points, which can be particularly useful when dealing with large datasets, making the trends and patterns more apparent through its visual nature.

Density Scale

The concept of a density scale in histograms is crucial when your class intervals are of unequal width. Typically, histograms show frequency by the height of the bars, but this approach changes when intervals vary because it would misrepresent the data.

In this case, the height of each bar is adjusted according to the interval width, ensuring the bar's area is proportional to the number of data points it includes. To calculate the density (height) for each bar, divide the frequency by the width of the interval. This ensures the visual representation is accurate, with the area of each bar corresponding to the actual data it represents, allowing for meaningful comparisons between bars.

Data Summarization

Data summarization involves reducing a large amount of raw data into a form that makes it easier to understand and interpret. Summaries can take the form of tables, charts, graphs, or statistical measures such as means and medians.

In our example concerning smokers attempting to quit, the frequency distribution table is a form of data summarization that shows how often each event occurs. The transition from this table to a histogram further simplifies the data, providing a visual snapshot that highlights the distribution and density of the data across differing numbers of attempts. This graphical form is often more approachable for many and helps with quick identification of patterns such as the most common attempt count at which smokers successfully quit.

Discrete Variable

A discrete variable is a type of variable that takes on a countable number of distinct values. Examples of discrete variables include things like the number of times smokers attempt to quit before succeeding, as discussed in our exercise.

These variables differ from continuous variables, which can take on any value within a range. When representing discrete variables in a histogram, it's important that the bars reflect the discrete nature of the data. Thus, a bar representing a single attempt should extend from 0.5 to 1.5 to ensure that the histogram accurately conveys the discrete intervals. This also avoids any misleading impressions that the data could be continuous.