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} {lc} & 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 goo

Question
Two-way tables
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. PSK\begin{array} {lc} & 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}ZSK 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: PSKP(A or B) = P(A) + P(B) - P(A and B) Solution \(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{l}{\left|{c}\right|}{c}{\left|{c}\right|}{c}\right\rbrace}&{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{4}{m}{a}{t}{h}{r}{m}{\left\lbrace{t}{h}\right\rbrace}\ \nabla{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}&{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{5}{m}{a}{t}{h}{r}{m}{\left\lbrace{t}{h}\right\rbrace}\ \nabla{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}&{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{6}{m}{a}{t}{h}{r}{m}{\left\lbrace{t}{h}\right\rbrace}\ \nabla{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}&{T}{o}{t}{a}{l}\backslash{h}{l}\in{e}{G}{r}{a}{d}{e}{s}&{49}&{50}&{69}&{168}\backslash{h}{l}\in{e}{A}{t}{h}\le{t}{i}{c}&{24}&{36}&{38}&{98}\backslash{h}{l}\in{e}{P}{o}{p}\underline{{a}}{r}&{19}&{22}&{28}&{69}\backslash{h}{l}\in{e}{T}{o}{t}{a}{l}&{92}&{108}&{135}&{335}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\) 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: \(\displaystyle{P}{\left({S}\right)}={\frac{{#{o}{f}{f}{a}{v}{\quad\text{or}\quad}{a}{b}\le{o}{u}{t}{c}{o}{m}{e}{s}}}{{#{o}{f}{p}{o}{s}{s}{i}{b}\le{o}{u}{t}{c}{o}{m}{e}{s}}}}={\frac{{{135}}}{{{335}}}}\) 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. \(\displaystyle{P}{\left({G}\right)}={\frac{{#{o}{f}{f}{a}{v}{\quad\text{or}\quad}{a}{b}\le{o}{u}{t}{c}{o}{m}{e}{s}}}{{#{o}{f}{p}{o}{s}{s}{i}{b}\le{o}{u}{t}{c}{o}{m}{e}{s}}}}={\frac{{{168}}}{{{335}}}}\) 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. \(\displaystyle{P}{\left({G}\right)}={\frac{{#{o}{f}{f}{a}{v}{\quad\text{or}\quad}{a}{b}\le{o}{u}{t}{c}{o}{m}{e}{s}}}{{#{o}{f}{p}{o}{s}{s}{i}{b}\le{o}{u}{t}{c}{o}{m}{e}{s}}}}={\frac{{{69}}}{{{335}}}}\) 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

Relevant Questions

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asked 2021-01-02
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asked 2020-12-09
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What is the probability that the student is female or has allergies?
\((a)\frac{19}{88}\)
(b)\frac{39}{88}\)
(c)\frac{58}{88}\)
(d)\frac{77}{88}\)
asked 2020-11-23
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State and test the appropriate hypotheses using \(\displaystyle\alpha={0.05}\)
\(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}&{a}\mp,\ {1}&{a}\mp,\ {2}&{a}\mp,\ {T}{o}{t}{a}{l}\backslash{h}{l}\in{e}{1}&{a}\mp,\ {35}&{a}\mp,\ {147}&{a}\mp,\ {182}\backslash{h}{l}\in{e}&{a}\mp,\ {25.48}&{a}\mp,\ {156.52}&{a}\mp,\backslash{h}{l}\in{e}{2}&{a}\mp,\ {101}&{a}\mp,\ {629}&{a}\mp,\ {730}\backslash{h}{l}\in{e}&{a}\mp,\ {102.20}&{a}\mp,\ {627.80}&{a}\mp,\backslash{h}{l}\in{e}{3}&{a}\mp,\ {28}&{a}\mp,\ {222}&{a}\mp,\ {250}\backslash{h}{l}\in{e}&{a}\mp,\ {35.00}&{a}\mp,\ {215.00}&{a}\mp,\backslash{h}{l}\in{e}{4}&{a}\mp,\ {4}&{a}\mp,\ {34}&{a}\mp,\ {38}\backslash{h}{l}\in{e}&{a}\mp,\ {5.32}&{a}\mp,\ {32.68}&{a}\mp,\backslash{h}{l}\in{e}{T}{o}{t}{a}{l}&{a}\mp,\ {168}&{a}\mp,\ {1032}&{a}\mp,\ {1200}\backslash{h}{l}\in{e}\)
\(\displaystyle{C}{h}{i}{s}{q}={a}\mp,\ {3.557}\ +\ {0.579}\ +\ {a}\mp,\ {0.014}\ +\ {0.002}\ +\ {a}\mp,\ {1.400}\ +\ {0.228}\ +\ {a}\mp,\ {0.328}\ +\ {0.053}={6.161}\)
\(\displaystyle{d}{f}={3}\)
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