12.10: Practice and Exploration
- Page ID
- 65588
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- In a research study of the economic conditions of individuals living in a large city, a researcher states the hypothesis, “The mean income of African American residents is less than that of white residents.” Identify the population and the parameter for this hypothesis. How would this hypothesis be written mathematically?
- A research team is considering the accessibility of voting in different voting districts in the United States. One hundred districts will be randomly sampled and demographic information about the residents, the number of polling places, and the total number of voting machines will be observed from the last election. The authors of the study would like to prove that voting districts with large populations of people of color have fewer polling places and voting machines. If this were to be tested using a statistical hypothesis test, what would the null and alternative hypotheses be? How could you write them mathematically?
- A researcher intends to do a study of voter turnout in an upcoming election. The researcher strongly feels that voter turnout will be less that 40%. If this were to be tested using a statistical hypothesis test, what would the null and alternative hypotheses be? How could you write them mathematically?
- In a study of environmental justice, a standard set of air pollution measurements were taken at 1,000 randomly selected houses in a large city. Demographic and socioeconomic data were also collected for each of the houses. The researchers running the study want to prove that air pollution is more of a problem in poorer neighborhoods. If this were to be tested using a statistical hypothesis test, what would the null and alternative hypotheses be? How could you write them mathematically?
- Suppose that a researcher is testing a null hypothesis against an alternative hypothesis and the test statistic is computed to be 2.7. The test statistic was computed by taking the difference between the estimate and the nearest value in the null hypothesis and dividing by the standard error. Interpret this value for the test statistic.
- Suppose that a researcher is testing the null hypothesis that a population mean is greater than 100. The observed sample mean is 97 and the estimated standard error is 5.7. How much evidence do you think there is against the null hypothesis?
- In a study of health care access and poverty, the researchers would like to prove that those who live in census blocks with lower median incomes have less access to health care. Describe in general terms what the null and alternative hypotheses for this study should be.
- Suppose that a research paper reports that the p-value for a statistical test is between less than 0.005. Would the null hypothesis be rejected when α=0.05? Would the null hypothesis be rejected when \(\alpha=0.01\)?
- Suppose that a research paper reports that the p-value for a statistical test is between 0.03 and 0.02. Would the null hypothesis be rejected when α=0.05? Would the null hypothesis be rejected when \(\alpha=0.01\)?
- Suppose that a research paper reports that the p-value for a statistical test is less than 0.025. Would the null hypothesis be rejected when α=0.05? Would the null hypothesis be rejected when \(\alpha=0.01\)?
- A research paper reports that the observed median income for African American individuals in a sample is greater than $30,000. Suppose the observed median income is $32,789. The paper also reports that the estimated standard error for the estimated median is $5,732. Can the authors really conclude that the actual median income for African American individuals in the population is greater than $30,000? Why or why not?
- Statistical theory can be used to show that selecting α to be smaller makes it more difficult to reject the null hypothesis. Explain in your own words why you think this happens.
- A researcher has computed a confidence interval for the mean number of doctor visits per year for a particular ethnic group. The interval was computed with \(\alpha=0.01\). The interval is \([3,9]\). What is the interpretation of this interval?
- Why would a researcher not want to use a 50% confidence interval?
- A researcher has computed a 99.9999% confidence interval for the mean age of individuals that regularly use a local community center. The confidence interval is \([8.8,39.2]\). The researcher is disappointed because the interval is so wide that it does not convey any real meaning. What can the researcher do to get a shorter interval?
- Using an online news service, find a news article about a social justice issue that is important to you that is based on an empirical analysis that relies on either statistical testing, confidence intervals, estimation, standard errors, or \(p\)-values. Identify, as clearly as you can, the hypotheses that the researcher is interested in and how they use the methods listed above to support their hypothesis. What levels of risk are specified? How are the conclusions interpreted?
- Using an online source or your library databases, find an academic research article about a social justice issue that is important to you that is based on an empirical analysis that relies on either statistical testing, confidence intervals, estimation, standard errors, or \(p\)-values. Identify, as clearly as you can, the hypotheses that the researcher is interested in and how they use the methods listed above to support their hypothesis. What levels of risk are specified? How are the conclusions interpreted?