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The report "Findings from the 2008 Administration of the College Senior Sunvey" (Higher Education Research Institute, 2009 ) asked a large number of college seniors how they would rate themselves compared to the average person of their age with respect to physical health. The accompanying relative frequency table summarizes the responses for men and women. \begin{tabular}{lcc} \multicolumn{1}{c} { Rating of Physical Health } & \(\frac{\text { Relative Frequency }}{\text { Men }}\) & Women \\ \hline Highest \(10 \%\) & .220 & .101 \\ Above average & .399 & .359 \\ Average & .309 & .449 \\ Below average & .066 & .086 \\ Lowest \(10 \%\) & .005 & .005 \\ \hline \end{tabular} a. Construct a comparative bar graph of the responses that allows you to compare the responses of men and women. b. There were 8110 men and 15,260 women who responded to the survey. Explain why it is important that the comparative bar graph be constructed using the relative frequencies rather than the actual numbers of people (the frequencies) responding in each category. c. Write a few sentences commenting on how college seniors perceive themselves with respect to physical health and how men and women differ in their perceptions.

Short Answer

Expert verified
To make a valid comparison between the responses of men and women, relative frequencies are used to construct the comparative bar graph. This is because the number of men and women respondents are different. They provide a fair comparison by adjusting for the different group sizes. Observations show differences in how men and women perceive their physical health, which can be discussed in detail based on the populated graph.

Step by step solution

01

Understanding the Data

Initially, familiarize yourself with the data given in the frequency table. Here, the responses for health, presented as 'highest 10%', 'above average', 'average', 'below average', and 'lowest 10%', are given for men and women as relative frequencies.
02

Constructing a Comparative Bar Graph

Using the relative frequencies for each category of physical health ratings, construct a comparative bar graph for both men and women. The y-axis should represent the relative frequency and the x-axis should represent the categories of health ratings. Different colors or patterns could be used to distinguish between the bars that represent men and women.
03

Importance of Using Relative Frequencies

It's crucial to use relative frequencies in this context because they provide a meaningful comparison between groups of different sizes. Since there were different numbers of men and women respondents (8110 men and 15260 women), simply comparing the numbers of people in each category would present a skewed picture. Relative frequencies adjust for different group sizes and allow for a more fair comparison.
04

Observations and Conclusions

The differences between men and women's self-perceived health can be observed from the graph and few conclusions can be drawn. Write few sentences about how men and women's perception about their physical health differs based on their relative frequencies in the graph. Check which gender is most likely to rate their health 'highest 10%', 'above average', 'average', 'below average', or the 'lowest 10%'.

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Key Concepts

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

Relative Frequency
Understanding the concept of relative frequency is vital in analyzing survey data. Relative frequency refers to how often something happens compared to the total number of outcomes. For example, if we were to look at a group of students and count how many have brown hair, we might say 30 out of 100 students do, which is the frequency. But to discuss relative frequency, we'd say 30%, conveying the proportion of the whole.

  • It offers a way to compare data across different groups or categories.
  • In our survey example, the relatively higher percentage of men ranking their health as 'highest 10%' can be compared with that of women, despite differing total numbers of respondents.
  • This normalization allows us to make comparisons that are fair and not skewed by sample size.

When translating these relative frequencies into a bar graph, the visual representation will accurately reflect these proportional differences without bias towards the sheer volume of respondents.
Data Visualization
Data visualization is the graphical representation of information and data. By using visual elements like charts, graphs, and maps, data visualization tools provide an accessible way to see and understand trends, outliers, and patterns in data.

  • In the case of our comparative bar graph, it brings to life the abstract numbers from the survey.
  • It simplifies complex data making it easier to compare the self-perception of physical health between men and women.
  • Colors or patterns are essential in distinguishing the different data sets and making the information clear and digestible at a glance.

Data visualization is not just about making pretty pictures; it's a crucial step in data analysis that ensures information is conveyed accurately and efficiently.
Perception of Physical Health
Perception of physical health is a subjective evaluation of one's own well-being and fitness. It's influenced by various personal, social, and cultural factors and can significantly affect an individual's behavior and self-esteem.

From the survey results, we can draw insights into how different groups view their health:
  • We see that men are more likely to rate their health as 'highest 10%' or 'above average' compared to women.
  • This could be indicative of differing standards or societal expectations placed on men and women concerning physical health.
  • Understanding these perceptions is essential as they can influence individual health choices and policy-making.

Analyzing self-perception through survey outcomes informs not only academic understanding but can also highlight areas for public health improvement.
Survey Data Analysis
Survey data analysis involves examining the responses provided by survey participants to draw conclusions. This process can expose a wealth of insights about populations and subgroups, provided the right methods are applied.

Key aspects include:
  • Considering the sample size and demographics to ensure results are representative.
  • Using relative frequency to compare between different respondent groups, as seen in the physical health survey.
  • Interpreting the visualized data correctly to glean accurate perceptions and trends.

In our exercise, by using both relative frequency and comparative bar graphs, we can analyze the varying self-perception of physical health between genders, despite asymmetric sample sizes—a sophisticated and accurate approach in survey data analysis.

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Most popular questions from this chapter

An exam is given to students in an introductory statistics course. What is likely to be true of the shape of the histogram of scores if: a. the exam is quite easy? b. the exam is quite difficult? c. half the students in the class have had calculus, the other half have had no prior college math courses, and the exam emphasizes mathematical manipulation? Explain your reasoning in each case.

The article "Housework around the World" (USA Today. September 15,2009 ) included the percentage of women who say their spouses never help with household chores for five different countries. \begin{tabular}{lc} Country & Percentage \\ \hline Japan & \(74 \%\) \\ France & \(44 \%\) \\ United Kingdom & \(40 \%\) \\ United States & \(34 \%\) \\ Canada & \(31 \%\) \\ \hline \end{tabular} a. Display the information in the accompanying table in a bar chart. b. The article did not state how the author arrived at the given percentages. What are two questions that you would want to ask the author about how the data used to compute the percentages were collected? c. Assuming that the data that were used to compute these percentages were collected in a reasonable way, write a few sentences describing how the five countries differ in terms of spouses helping their wives with housework.

The article "Frost Belt Feels Labor Drain" (USA Today. May 1, 2008 ) points out that even though total population is increasing, the pool of young workers is shrinking in many states. This observation was prompted by the data in the accompanying table. Entries in the table are the percent change in the population of 25 - to 44 -year-olds over the period from 2000 to 2007 . A negative percent change corresponds to a state that had fewer 25 - to 44 -year-olds in 2007 than in 2000 (a decrease in the pool of young workers). a. The smallest value in the data set is -11.9 and the largest value is \(22.0 .\) One possible choice of stems for a stem-and-leaf display would be to use the tens digit, resulting in stems of \(-1,-0,0,1,\) and 2 . Notice that because there are both negative and positive values in the data set, we would want to use two 0 stems-one where we can enter leaves for the negative percent changes that are between 0 and -9.9 , and one where we could enter leaves for the positive percent changes that are between 0 and 9.9 . Construct a stem-and-leaf plot using these five stems. (Hint: Think of each data value as having two digits before the decimal place, so 4.1 would be regarded as 04.1.) b. Using two-digit stems would result in more than 30 stems, which is more than we would usually want for a stem-and-leaf display. Describe a strategy for using repeated stems that would result in a stemand-leaf display with about 10 stems. c. The article described "the frost belt" as the cold part of the country-the Northeast and Midwestnoting that states in the frost belt generally showed a decline in the number of people in the \(25-\) to 44 -year-old age group. How would you describe the group of states that saw a marked increase in the number of 25 - to 44 -year-olds?

3.36 Construct a histogram corresponding to each of the five frequency distributions, \(\mathrm{I}-\mathrm{V},\) given in the following table, and state whether each histogram is symmetric bimodal, positively skewed, or negatively skewed: \begin{tabular}{cccccc} & \multicolumn{5}{c} { Frequency } \\ \cline { 2 - 6 } Class Interval & I & II & III & IV & V \\ 0 to \(<10\) & 5 & 40 & 30 & 15 & 6 \\ 10 to \(<20\) & 10 & 25 & 10 & 25 & 5 \\ 20 to \(<30\) & 20 & 10 & 8 & 8 & 6 \\ 30 to \(<40\) & 30 & 8 & 7 & 7 & 9 \\ 40 to \(<50\) & 20 & 7 & 7 & 20 & 9 \\ 50 to \(<60\) & 10 & 5 & 8 & 25 & 23 \\ 60 to \(<70\) & 5 & 5 & 30 & 10 & 42 \\ \hline \end{tabular}

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?

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