Question

Researchers carried out a survey of fourth-, fifth-, and sixth-grade students in Michigan.

Two-way tables
ANSWERED
asked 2020-10-19

Researchers carried out a survey of fourth-, fifth-, and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. This two-way table summarizes the survey data. 

\(\begin{array} {|c|} & Grade \ Most important & \begin{array}{l|c|c|c|c} & \begin{array}{c} 4 \mathrm{th} \ grade \end{array} & \begin{array}{c} 5 \mathrm{th} \ \text { grade } \end{array} & \begin{array}{c} 6 \mathrm{th} \ grade \end{array} & Total \\ \hline Grades & 49 & 50 & 69 & 168 \\ \hline Athletic & 24 & 36 & 38 & 98 \\ \hline Popular & 19 & 22 & 28 & 69 \\ \hline Total & 92 & 108 & 135 & 335 \end{array} \ \end{array}\)

 Suppose we select one of these students at random. What's the probability that: The student is a sixth grader or a student who rated good grades as important?

Answers (1)

2020-10-20

Definitions: Completed rule \(\displaystyle{P}{\left({A}^{{c}}\right)}={P}{\left(\neg{A}\right)}={1}-{P}{\left({A}\right)}\) General addition rule for any two events:\(P(A or B) = P(A) + P(B) - P(A and B)\)

Solution

\(Grade\ Most important\begin{array}{l|c|c|c|c} & 4 \mathrm{th} & 5 \mathrm{th} & 6 \mathrm{th} & \text { Total } \\ \hline Grades & 49 & 50 & 69 & 168 \\ Athletic & 24 & 36 & 38 & 98 \\ Popular & 19 & 22 & 28 & 69 \\ \hline Total & 92 & 108 & 135 & 335 \end{array}\)

S = Sixth grader G = Grades We note that 135 of the 335 people in the table are 6th grades, because 135 is mentioned in the row ” Total” and in the column ”6th grade” of the given table. The probability is the number of favorable outcomes divided by the number of possible outcomes: \(P(5th\ grade) = \frac{\# of\ favorable\ outcomes}{\# of\ possible\ outcomes} = \frac{135}{335}\) We note that 168 of the 335 people in the table rated good grades as important, because 168 is mentioned in the row ” Grades” and in the column ”Total” of the given table. \(P(Athletic\ and\ 5th\ grade) = \frac{\# of\ favorable\ outcomes}{\# of\ possible\ outcomes} = \frac{168}{335}\) We note that 69 of the 335 people in the table are 6th graders who rated good grades as important, because 69 is mentioned in the row ” Grades” and in the column ”6th gradel” of the given table. \(P(G)=\frac{\# \text{of favor about comes}}{\# \text{of possible out comes}}=\frac{69}{335}\) Use the general addition rule: \(\displaystyle{P}{\left({S}{\quad\text{or}\quad}{G}\right)}={P}{\left({S}\right)}+{P}{\left({G}\right)}-{P}{\left({S}{\quad\text{and}\quad}{G}\right)}\)
\(\displaystyle={\frac{{{135}}}{{{335}}}}+{\frac{{{168}}}{{{335}}}}-{\frac{{{69}}}{{{335}}}}\)
\(\displaystyle={\frac{{{135}+{168}-{69}}}{{{335}}}}\)
\(\displaystyle={\frac{{{234}}}{{{335}}}}\)
\(\displaystyle\approx{0.6985}\)
\(\displaystyle={69.85}\%\)

0
 
Best answer

expert advice

Need a better answer?

Relevant Questions

asked 2021-05-18

Researchers carried out a survey of fourth-, fifth-, and sixth-grade students in Michigan. Students were asked if good grades, athletic ability, or being popular was most important to them. The two-way table summarizes the survey data. \(\begin{array} {lc} & \text{Grade} \ \text {Whatis mostimportant? What is most important? ​ } & \begin{array}{l|c|c|c|c} & \text { 4th grade } & \text { 5th grade } & \text { 6th grade } & \text { Total } \\ \hline \text { Grades } & 49 & 50 & 69 & 168 \\ \hline \text { Athletic } & 24 & 36 & 38 & 98 \\ \hline \text { Popular } & 19 & 22 & 28 & 69 \\ \hline \text { Total } & 92 & 108 & 135 & 335 \end{array} \ \end{array}\) Identify the explanatory and response variables in this context.

asked 2021-02-09
Researchers carried out a survey of fourth-, fifth- and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. The two-way table summarizes the survey data.
\(\begin{array}{c|c} & 4th\ grade & 5th\ grade & 6th\ grade &Total \\ \hline Grades &49&50&69&168\\ Athletic &24&36&38&98\\ Popular\ &19&22&28&69\\ \hline Total & 92 & 108 & 135 &335 \end{array}\)
Suppose we select one of these students at random. What's the probability of each of the following? The student is not a sixth-grader and did not rate good grades as important.
asked 2021-05-05
The two-way table summarizes data on whether students at a certain high school eat regularly in the school cafeteria by grade level. \text{Grade}\ \text{Eat in cafeteria} \begin{array}{l|r|r|r|r|r} & 9 \mathrm{th} & 10 \mathrm{th} & 11 \mathrm{th} & 12 \mathrm{th} & \text { Total } \ \hline \text { Yes } & 130 & 175 & 122 & 68 & 495 \ \hline \text { No } & 18 & 34 & 88 & 170 & 310 \ \hline \text { Total } & 148 & 209 & 210 & 238 & 805 \end{array} If you choose a student at random who eats regularly in the cafeteria, what is the probability that the student is a 10th-grader?
asked 2021-05-27

You randomly survey students in your school about whether they liked a recent school play. The two-way table shows the results. Find and interpret the marginal frequencies.

\(\begin{array}{|c|c|}\hline & & \text{Student} \\ \hline & & \text{Liked} & \text{Did Not like} \\ \hline \text{Gender} & \text{Male} & 48 & 12 \\ \hline & \text{Female} & 56 & 14 \\ \hline \end{array}\)

...