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Chapter 3: Problem 32

Example 3.19 used annual rainfall data for Albuquerque, New Mexico, to construct a relative frequency distribution and cumulative relative frequency plot. The National Climate Data Center also gave the accompanying annual rainfall (in inches) for Medford, Oregon, from 1950 to 2008 . \(\begin{array}{llllllllll}28.84 & 20.15 & 18.88 & 25.72 & 16.42 & 20.18 & 28.96 & 20.72 & 23.58 & 10.62 \\ 20.85 & 19.86 & 23.34 & 19.08 & 29.23 & 18.32 & 21.27 & 18.93 & 15.47 & 20.68 \\ 23.43 & 19.55 & 20.82 & 19.04 & 18.77 & 19.63 & 12.39 & 22.39 & 15.95 & 20.46 \\ 16.05 & 22.08 & 19.44 & 30.38 & 18.79 & 10.89 & 17.25 & 14.95 & 13.86 & 15.30 \\ 13.71 & 14.68 & 15.16 & 16.77 & 12.33 & 21.93 & 31.57 & 18.13 & 28.87 & 16.69 \\ 18.81 & 15.15 & 18.16 & 19.99 & 19.00 & 23.97 & 21.99 & 17.25 & 14.07 & \end{array}\) a. Construct a relative frequency distribution for the Medford rainfall data. b. Use the relative frequency distribution of Part (a) to construct a histogram. Describe the shape of the histogram. c. Construct a cumulative relative frequency plot for the Medford rainfall data. d. Use the cumulative relative frequency plot of Part (c) to answer the following questions: i. Approximately what proportion of years had annual rainfall less than 15.5 inches? ii. Approximately what proportion of years had annual rainfall less than 25 inches? iii. Approximately what proportion of years had annual rainfall between 17.5 and 25 inches?

Short Answer

Expert verified
You will need to create the plots and then interpret them to find the proportions of years with specific rainfall amounts. The exact proportions will depend on the results from your plots and how you interpret them.

Step by step solution

01

Construct a Relative Frequency Distribution

A relative frequency distribution is created by dividing each frequency by the total count of observations. First, count the frequency of rainfall within certain intervals or 'bins'. Determine suitable bins based on the range of data. Summarize this data in a table with one column showing the 'bins' and the second column showing the frequency. The relative frequency is calculated by dividing the frequency in each bin by the total count of data. Add this as a column to your table.
02

Construct a Histogram

A histogram can be created using the frequency distribution. The bins or intervals of rainfall represent the x-axis and the relative frequencies represent the y-axis. The shape of the histogram will give an indication of the distribution of the rainfall. Normal, skewed or bimodal are possible states, detailed interpretation depends on the created histogram.
03

Construct a Cumulative Relative Frequency Plot

To create a cumulative relative frequency plot, add up the relative frequencies as you move through your bins in ascending order and create a plot with these cumulative frequencies against the y-axis and the bins on the x-axis.
04

Interpret the Cumulative Relative Frequency Plot

Estimate the proportions of years with a certain amount of annual rainfall by locating the rainfall amount on the x-axis and reading the corresponding cumulative frequency on the y-axis. For example, to find the proportion of years with less than 15.5 inches of rain, locate 15.5 on the x-axis and read off the y-value. This represents the proportion of years with less than 15.5 inches of rain. Repeat this step for other rainfall amounts.

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Key Concepts

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

Relative Frequency Distribution
Understanding a relative frequency distribution is simpler than it sounds. It involves taking data and summarizing how often each unique value occurs in a dataset relative to the total number of observations. Think of it as a part of the whole, showing us the proportion of each value within the total dataset.

To start, let's use annual rainfall data for Medford, Oregon. First, organize the data into bins or groups based on the range of rainfall amounts. For example, if the data ranges between 10 to 35 inches, you might create bins like 10-15 inches, 15-20 inches, and so on. Next, count how many data points fall into each bin. Divide this number by the total number of observations. This fraction represents the relative frequency of each bin.

This relative frequency helps us understand not just how many but how common each range of rainfall is compared to the entire dataset. It's a crucial first step in constructing more visual forms of data representation like histograms.
Histogram
A histogram is a visual representation of data that uses bars to show the frequency of occurrences within defined intervals or bins. With the rainfall data, the histogram reveals patterns or trends in the distribution of the data at a glance. The height of each bar reflects the relative frequency calculated earlier.

To create the histogram, place the bins on the x-axis and the relative frequencies on the y-axis. Each bar corresponds to a bin, and the height reflects how common that range of rainfall is. For example, a taller bar at 10-15 inches indicates that this bin is more frequent than others.

The shape of the histogram describes the distribution. It can be normal (bell-shaped), skewed (more data on one side), or bimodal (two peaks). Observing the shape helps in understanding the overall data trends, pinpointing the average rainfall abundance, and detecting any outliers or exceptional values.
Frequency Distribution
Frequency distribution refers to the organization of data to show how frequently each value or range of values occurs. It is fundamental for gaining insights into the structure of any dataset, including the study of rainfall patterns.

When constructing a frequency distribution table, start by deciding on the bin intervals, which should cover the entire range of your data. Count and list the number of occurrences for each interval without considering their relative position within the dataset.

This table provides a foundational view of the dataset鈥檚 framework, aiding in understanding how typical or atypical certain values are, and it serves as a precursor to more detailed analysis through plotting a histogram. You鈥檙e practically categorizing rainfall amounts into easy-to-interpret sections, which is immensely helpful for more complex analyses.
Rainfall Data Analysis
Rainfall data analysis allows us to explore weather patterns over time. By analyzing the annual rainfall data for Medford, Oregon, we gain insights into variations and trends that have occurred over the decades from 1950 to 2008.

This type of analysis can inform future planning and policy decisions by highlighting possible patterns such as wet periods or droughts. To do this effectively, employ tools like relative frequency distributions, histograms, and cumulative plots.

Using these methods, you can predict likely future conditions by examining past trends closely. This analysis can also help in assessing the effectiveness of water conservation measures and contribute valuable data for environmental research. Overall, it's a powerful way to make historical data work for future preparedness and climate strategy.

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Most popular questions from this chapter

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.

The article "The Need to Be Plugged In" (Associated Press, December 22,2005 ) described the results of a survey of 1006 adults who were asked about various technologies, including personal computers, cell phones, and DVD players. The accompanying table summarizes the responses to questions about how essential these technologies were. \begin{tabular}{lccc} & \multicolumn{3}{c} { Relative Frequency } \\ \cline { 2 - 4 } Response & Personal Computer & Cell Phone & DVD Player \\ \hline Cannot imagine living without & .46 & .41 & .19 \\ Would miss but could do without & .28 & .25 & .35 \\ Could definitely live without \\ \hline \end{tabular} Construct a comparative bar chart that shows the distribution of responses for the three different technologies.

The National Confectioners Association asked 1006 adults the following question: "Do you set aside a personal stash of Halloween candy?" Fifty-five percent of those surveyed responded no, \(41 \%\) responded yes, and \(4 \%\) either did not answer the question or said they did not know (USA Today. October 22, 2009). Use the given information to construct a pie chart.

The report "Wireless Substitution: Early Release of Estimates from the National Health Interview Survey" (Center for Disease Control, 2009 ) gave the following estimates of the percentage of homes in the United States that had only wireless phone service at 6-month intervals from June 2005 to December 2008 . \begin{tabular}{lr} & Percent with Only \\ Date & Wireless Phone Service \\ \hline June 2005 & 7.3 \\ December 2005 & 8.4 \\ June 2006 & 10.5 \\ December 2006 & 12.8 \\ June 2007 & 13.6 \\ December 2007 & 15.8 \\ June 2008 & 17.5 \\ December 2008 & 20.2 \\ \hline \end{tabular} Construct a time-series plot for these data and describe the trend in the percent of homes with only wireless phone service over time. Has the percent increased at a fairly steady rate?

Student loans can add up, especially for those attending professional schools to study in such areas as medicine, law, or dentistry. Researchers at the University of Washington studied medical students and gave the following information on the educational debt of medical students on completion of their residencies (Annals of Internal Medicine [March 2002]: \(384-398\) ): \begin{tabular}{cc} Educational Debt (dollars) & Relative Frequency \\ \hline 0 to \(<5000\) & .427 \\ 5000 to \(<20,000\) & .046 \\ 20,000 to \(<50,000\) & .109 \\ 50,000 to \(<100,000\) & .232 \\ 100,000 or more & .186 \\ \hline \end{tabular} a. What are two reasons that you could not use the given information to construct a histogram with the educational debt intervals on the horizontal axis and relative frequency on the \(y\) -axis? b. Suppose that no student had an educational debt of \(\$ 150,000\) or more upon completion of his or her residency, so that the last class in the relative frequency distribution would be 100,000 to \(<150,000\). Summarize this distribution graphically by constructing a histogram of the educational debt data. (Don't forget to use the density scale for the heights of the bars in the histogram, because the interval widths aren't all the same.) c. Based on the histogram of Part (b), write a few sentences describing the educational debt of medical students completing their residencies.
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