4.3.1: Exercises
- Page ID
- 48788
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- 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.