5.7: Practice and Exploration
- Page ID
- 63733
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)-
The following question is used in a survey on the financial resources of individuals in a large city:
Which of the following types of financial accounts do you have (mark all that apply)?
- Savings Account
- Checking Account
- Retirement Account
- Investment Account
- Other
The survey software is programmed to count the number of boxes that have been checked as the response to this question. What is the measurement scale for this data?
- In a survey that tracks the online habits of residents in a city, a question asks for the age of the individual. Instead of asking for a numerical age, the question asks the individual to select one of the categories: Youth (17 years or younger), Young Adult (18 to 20 years), Adult (21 to 64 years), and Senior (65 years or older). What is the measurement scale for this data?
- A research article on barriers to graduation for college students from lower-income backgrounds includes data on whether a student has unpaid fees, and tracks what type of unpaid fee the student has. The data are coded on using the table below:
|
Code |
Meaning |
|
0 |
No unpaid fee |
|
1 |
Unpaid tuition fee |
|
2 |
Unpaid laboratory fee |
|
3 |
Unpaid non-academic fee |
|
4 |
Unpaid parking fee |
In the research article the authors average the coded values over all the students. The resulting average is 2.3. The authors then state that on average students are stuck between unpaid laboratory and non-academic fees. What is the measurement scale of the data? Do you feel that the conclusions are valid?
- One of the variables in a survey of undergraduate students asks what year they intend to graduate. For example, a student will enter the number 2027 if they intend to graduate in the year 2027. What is the measurement scale for this data? Would the measurement scale change if the question was re-worded to ask, “How many years until you graduate?” Why or why not?
- A study of automobile affordability has data with variables corresponding to gas mileage and the retail price of each automobile. The data also contains a variable that identifies whether the vehicle is a hybrid or not. What are the measurement scales for each of these variables?
- An employment study asks individuals in which sector of the economy they are employed. The question uses the categories “education”,, “business”,, “service”,, “skilled labor”,, and “other.”. When studying this data, it is it valid to compute the percentage of observations in each sector? Knowing that this is nominal data, explain why it is valid to compute these percentages.
- A study of the levels of education attained by individuals codes the data as follows:
|
Education Level |
Code |
|
Attained grade level 𝑘 where 𝑘 can be 1 to 12 |
𝑘 |
|
Completed 𝑘 years of undergraduate education |
12+𝑘 |
|
Completed 𝑘 years of graduate education |
16+𝑘 |
Hence, for example, if an individual attained an 8th grade education, then the response would be coded as 8, and an individual who has completed 3 years of undergraduate education would be coded as in 15. What is the measurement scale of this data? When deciding on the measurement scale think carefully about how college education is counted.
- In a large health database one of the variables is the weight of the individual in kilograms. What is the measurement scale for this data?
- The following question is used in a survey on the home life of college students:
Which of the following types of pets do you have at your place of residence (mark all that apply)?
- Cat
- Dog
- Rabbit
- Fish
- Reptile
- Other
The survey software is programmed to record the list of which boxes have been marked. What is the measurement scale for this data?
- In a survey to measure support for restarting the push for the Equal Rights Amendment to the United State Constitution, individuals are asked how much they support the amendment based on the measurement scale –1 (oppose), 0 (no feeling), and 1 (support). The individuals are then asked to read an essay by a famous scientist that supports the amendment. The individuals are then asked how much they support the amendment based on the measurement scale –2 (strongly opposed), –1 (oppose), 0 (no feeling), 1 (support), and 2 (strongly support). The data manager then computes how much the support of the individual has changed by taking the result from the second question and subtracting the result of the first question. That is, if an individual stated that they had no feeling about the amendment prior to reading the essay, and the stated that they strongly supported the amendment after reading the essay, then the final score would be 2−0=2. Do you feel that this calculation is valid for the data from these two questions? Why or why not? Can you suggest some improvements?
- In a study of family wealth, individuals are asked about their family structure. Specifically, the study asks everyone to identify their marital status using the choices in the first column of the table shown below.
|
Status |
Data Code |
|
Single |
0 |
|
Partner |
1 |
|
Married |
2 |
|
Divorced |
3 |
|
Separated |
4 |
The status partner refers to cohabitation with another individual to whom they are not legally married. What is the measurement scale for this data? Given this measurement scale, do you think it would be permissible to compute the average response for the individuals in the study? Why or why not?
- A historical researcher is interested in the racial views of working-class individuals who lived in a racially integrated city at the turn of the 20th century. To study these views the researcher has obtained eyewitness statements contained in historical county and city judicial records from the period 1895 to 1905. The researcher intends to summarize the views of these records by quoting representative passages that show either racial bias or racial tolerance. Are the data in this study qualitative or quantitative? If the data are quantitative, what is the measurement scale?
- Two researchers are developing a project that will study the association between attitudes about civil rights, race, and age. The researchers have decided to use a simplified form of the questions used on census forms in the United States that asks individuals to identify their race and ethnicity. This will allow the researchers to directly connect their data with census data available from the United States Census Bureau. The questions have the following form:
Are you of Hispanic, Latino, or Spanish origin?
- Yes
- No
What race do you consider yourself?
- African American
- American Indian or Alaska Native
- Asian
- Native Hawaiian or Pacific Islander
- White
- None of these categories describe my race
During the discussions about developing the survey, the question of how the race and ethnicity data were to be stored electronically became important. One researcher wanted to store the data using characters. For the first question each response would be stored as with Y or N, and for the second question the codes AF, AI, AS, NH, WH, and NO would be used. The second researcher objected to this scheme and suggested numbering the categories as 0 and 1 for the first question and 0–5 for the second question. Does the representation of the data as either characters or numbers affect the measurement scales for the data? Why? Suppose that the second researcher states that the second method is better because they could then compute the average race of the respondents. How would you respond to that statement?
- A researcher interested in economic justice is studying programs that provide tax benefits for individuals who donate items to a charitable organization. The researcher is interested in the fact that for many the tax benefits will never be realized because their income is insufficient for such tax benefits to have any effect. In these cases, the researcher would like to argue that other types of benefits should be offered to those individuals. As part of the study, the researcher obtains data from several such programs. The data have been blinded so the researcher cannot identify the individuals involved, but data on income, demographics, and value of donation are available. Based on the income and demographics, the researcher can determine if the individual would likely be able to benefit from the tax benefit. The researcher creates a variable with the following coded values:
|
Code |
Meaning |
|
0 |
Not likely to get tax benefit |
|
1 |
Even chance of getting tax benefit |
|
3 |
Likely to get tax benefit |
Identify the measurement scale for this variable. Would it make sense for the researcher to compute the percentage of individuals in each category? Would it make sense for the researcher to compute the average of this variable?
- A historical researcher is interested in the racial views of working-class individuals who lived in a racially integrated city at the turn of the 20th century. To study these views, the researcher has obtained eyewitness statements contained in historical county and city judicial records from the period 1895 to 1905. For each statement the researcher will use specially designed computer software that will compute how many words are contained in the statement, and how many of those words are on the list of derogatory racial terms. The researcher will then compute the derogatory term rate by dividing the number of derogatory racial terms in the statement by the total number of words in the statement. Are the data in this study qualitative or quantitative? If the data are quantitative, what is the measurement scale?
- Using an online news service, find a news article about an issue that is important to you personally that is based on an empirical analysis. Identify, as clearly as you can, the population that the authors of the news article have used to observe their data and gather information. What are the variables in the data they obtain? What are the measurement scales of each of the variables? Do the authors do any calculations that seem questionable to you based on the measurement scales of the variables? Why or why not?
- Using an online source or your library find an academic research article about an issue that is important to you personally that is based on an empirical analysis. Identify, as clearly as you can, the population that the authors of the news article have used to observe their data and gather information. What are the variables in the data they obtain? What are the measurement scales of each of the variables? Do the authors do any calculations that seem questionable to you based on the measurement scales of the variables? Why or why not?