4.3.1: Exercises
- Page ID
- 48788
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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 CDC survey, 37% of US high school students reported regular mental health struggles during the pandemic. An educator is concerned about the mental health of her students. She randomly surveys 8 students. Don tells Isabel that the probability that exactly 6 of the 8 students reported struggling with their mental health during the pandemic can be found using the following formula: \(P(6)=(0.37)^6(0.63)^2\). What should Isabel say to Don to help him understand binomial probability better?
- In a CDC survey, 37% of US high school students reported regular mental health struggles during the pandemic. An educator is concerned about the mental health of her students. She randomly surveys 8 students. Fill in the following table with the appropriate values. Round calculated values to four decimal places.
|
\(X=\) number of students with mental health struggles |
\({ }_n C_x\) |
\(P(X)\) |
|
0 |
\({ }_8 C_0=\underline{\ \ \ \ \ \ \ \ \ \ }\) | \(\begin{aligned} P(0) & =(0.63)\overline{\ \ \ \ \ }\approx 0.0248 \end{aligned}\) |
|
1 |
\({ }_8 C_{\underline{\ \ \ \ \ }}=8\) | \(P(1)=8 \cdot(0.37)\overline{\ \ \ \ \ }\cdot(0.63)^7 \approx 0.1794\) |
|
2 |
\({}_8C_{\underline{\ \ \ \ \ }}=\underline{\ \ \ \ \ \ \ \ \ \ }\) | \(P(2)=\underline{\ \ \ \ \ } \cdot(\underline{\ \ \ \ \ })^2 \cdot(0.63)^6 \approx 0.2397\) |
|
3 |
\({}_{\underline{\ \ \ \ \ }}C_{\underline{\ \ \ \ \ }}=\underline{\ \ \ \ \ \ \ \ \ \ }\) | \(P(3)=\underline{\ \ \ \ \ } \cdot(\underline{\ \ \ \ \ })^3 \cdot(0.63)^{\overline{\ \ \ \ \ }}\approx 0.2815\) |
|
4 |
\({}_{\underline{\ \ \ \ \ }}C_{\underline{\ \ \ \ \ }}=\underline{\ \ \ \ \ \ \ \ \ \ }\) | \(P(4)=\underline{\ \ \ \ \ } \cdot(\quad)^{\underline{\ \ \ \ \ }}\cdot(\underline{\ \ \ \ \ })^{\underline{\ \ \ \ \ }}\approx 0.2067\) |
|
5 |
\({}_{\underline{\ \ \ \ \ }}C_{\underline{\ \ \ \ \ }}=\underline{\ \ \ \ \ \ \ \ \ \ }\) | \(P(\underline{\ \ \ \ \ })=\underline{\ \ \ \ \ } \cdot(\underline{\ \ \ \ \ })^{\underline{\ \ \ \ \ }}\cdot(\underline{\ \ \ \ \ })^{\underline{\ \ \ \ \ }}\approx 0.0971\) |
|
6 |
\({}_{\underline{\ \ \ \ \ }}C_{\underline{\ \ \ \ \ }}=\underline{\ \ \ \ \ \ \ \ \ \ }\) | \(P(\underline{\ \ \ \ \ })=\underline{\ \ \ \ \ } \cdot(\underline{\ \ \ \ \ })^{\underline{\ \ \ \ \ }}\cdot(\underline{\ \ \ \ \ })^{\underline{\ \ \ \ \ }}\approx \underline{\ \ \ \ \ }\) |
|
7 |
\({}_{\underline{\ \ \ \ \ }}C_{\underline{\ \ \ \ \ }}=\underline{\ \ \ \ \ \ \ \ \ \ }\) | \(P(\underline{\ \ \ \ \ })=\underline{\ \ \ \ \ } \cdot(\underline{\ \ \ \ \ })^{\underline{\ \ \ \ \ }}\cdot(\underline{\ \ \ \ \ })^{\underline{\ \ \ \ \ }}\approx \underline{\ \ \ \ \ }\) |
|
8 |
\({}_{\underline{\ \ \ \ \ }}C_{\underline{\ \ \ \ \ }}=\underline{\ \ \ \ \ \ \ \ \ \ }\) | \(P(\underline{\ \ \ \ \ })=\underline{\ \ \ \ \ } \cdot(\underline{\ \ \ \ \ })^{\underline{\ \ \ \ \ }}\cdot(\underline{\ \ \ \ \ })^{\underline{\ \ \ \ \ }}\approx \underline{\ \ \ \ \ }\) |
- Use the discrete probability distribution from 1. to find the probability that at most five of the randomly selected students have reported regular mental health struggles during the pandemic.
- Use the discrete probability distribution from 1. to find the probability that at least six of the randomly selected students have reported regular mental health struggles during the pandemic.