6.2.1: Exercises
- Page ID
- 48851
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- In a Washington Post-University of Maryland poll5 of 1503 randomly selected US adults, 55% said they strongly support gender equity in sports. In the following exercises, you will construct a 99% confidence interval for the proportion of US adults who strongly support gender equity in sports.
- Identify the relevant information:
- What is the sample size?
- What is the sample proportion?
- Write a sentence to describe what the population parameter is in context.
- What is the sample size?
- Step 1: Is the sampling distribution of sample proportions approximately normal? Why or why not?
- Step 2: Compute the critical value from the standard normal distribution that corresponds to a confidence level of 99%. Write the function you use in desmos to find a critical value from the standard normal distribution.
- Step 3: Compute the margin of error \(E \approx Z_c \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\). Round to three decimal places.
- Step 4: Give the lower and upper limits of the 99% confidence interval for the population proportion (\(p\)). Then write the interval in interval notation.
- Step 5: Interpret the interval in context. Is it likely that a majority of US adults strongly support gender equity in sports? Use the interval to support your answer.
- Identify the relevant information:
- In a CNN poll6 of 1002 randomly selected US adults, 371 approve of the supreme court's decision to overturn Roe v. Wade. Construct a 95% confidence interval for the true proportion of US adults who approve of the supreme court’s decision to overturn Roe v. Wade.
- Write a sentence describing \(p\) in context.
- Step 1: Verify that the sampling distribution of sample proportions is approximately normal. Justify your answer.
- Step 2: Compute the critical value.
- Step 3: Compute the sample proportion and the margin of error rounded to three decimal places.
- Step 4: Compute the interval in interval notation.
- Step 5: Interpret the interval in context.
- Is it possible that a majority approves of the supreme court’s decision to overturn Roe v Wade? Use the interval to support your answer.
- Write a sentence describing \(p\) in context.
- According to a poll7 conducted by Gallup of 800 randomly surveyed US adults, 312 respondents were satisfied with the quality of the environment. Construct a 90% confidence interval using the five step process. Round the margin of error to three decimal places.
Reference
5 Liz Clarke, Scott Clement, and Emily Guskin, “Most Americans support gender equity in sports scholarships, poll finds,” Washingtonpost.com, June 22, 2022, accessed September 27, 2022, https://www.washingtonpost.com/sports/2022/06/22/title-ix-poll-americans-support-gender-equity/
6 Jennifer Agiesta, “About two-thirds of Americans disapprove of overturning Roe v. Wade, see negative effect for the nation ahead,” CNN.com, July 28, 2022, accessed September 27, 2022, https://www.cnn.com/2022/07/28/politics/cnn-poll-abortion-roe-v-wade/index.html
7 Jeffrey M. Jones, “Americans Offer Gloomy State of the Nation Report,” Gallup.com, February 2, 2022, accessed September 27, 2022, https://news.gallup.com/poll/389309/americans-offer-gloomy-state-nation-report.aspx