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

Two-way tables
ANSWERED 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? 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: $$\displaystyle{P}{\left({S}{\quad\text{or}\quad}{G}\right)}={P}{\left({S}\right)}+{P}{\left({G}\right)}-{P}{\left({S}{\quad\text{and}\quad}{G}\right)}$$
$$\displaystyle={\frac{{{135}}}{{{335}}}}+{\frac{{{168}}}{{{335}}}}-{\frac{{{69}}}{{{335}}}}$$
$$\displaystyle={\frac{{{135}+{168}-{69}}}{{{335}}}}$$
$$\displaystyle={\frac{{{234}}}{{{335}}}}$$
$$\displaystyle\approx{0.6985}$$
$$\displaystyle={69.85}\%$$