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

### Relevant Questions 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. $$\displaystyle{G}{r}{a}{d}{e}\ {M}{o}{s}{t}{i}\mp{\quad\text{or}\quad}{\tan{{t}}}{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{l}{\left|{c}\right|}{c}{\left|{c}\right|}{c}\right\rbrace}&{4}{m}{a}{t}{h}{r}{m}{\left\lbrace{t}{h}\right\rbrace}&{5}{m}{a}{t}{h}{r}{m}{\left\lbrace{t}{h}\right\rbrace}&{6}{m}{a}{t}{h}{r}{m}{\left\lbrace{t}{h}\right\rbrace}&\ \text{ Total }\ \backslash{h}{l}\in{e}{G}{r}{a}{d}{e}{s}&{49}&{50}&{69}&{168}\backslash{h}{l}\in{e}{A}{t}{h}\le{t}{i}{c}&{24}&{36}&{38}&{98}\backslash{h}{l}\in{e}{P}{o}{p}\underline{{a}}{r}&{19}&{22}&{28}&{69}\backslash{h}{l}\in{e}{T}{o}{t}{a}{l}&{92}&{108}&{135}&{335}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ Suppose we select one of these students at random. Find P(athletic | 5th grade). 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.
$$\begin{array}{c|c}& 4th\ grade & 5th\ grade & 6th\ grade &Total \\ \hline Grades &49&50&69&168\\ \text{Athletic} &24&36&38&98\\ \text{Popular}\ &19&22&28&69\\ \hline \text{Total} & 92 & 108 & 135 &335 \end{array}$$
Suppose we select one of these students at random. What's the probability of each of the following? The student is a sixth-grader or rated good grades as Important. 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. Is there a relationship between gender and relative finger length? To find out, we randomly selected 452 U.S. high school students who completed a survey. The two-way table summarizes the relationship between gender and which finger was longer on the left hand (index finger or ring finger).
$$\begin{array} {lc} & \text{Gender} \ \text {Longer finger} & \begin{array}{l|c|r|r} & \text { Female } & \text { Male } & \text { Total } \\\hline \text { Index finger } & 78 & 45 & 123 \\\hline \text{ Ring finger } & 82 & 152 & 234 \\ \hline \text { Same length } & 52 & 43 & 95 \\ \hline \text { Total } & 212 & 240 & 452 \end{array}\ \end{array}$$
Suppose we randomly select one of the survey respondents. Define events R: ring finger longer and F: female. Given that the chosen student does not have a longer ring finger, what's the probability that this person is male? Write your answer as a probability statement using correct symbols for the events. A survey of 4826 randomly selected young adults (aged 19 to 25 ) asked, "What do you think are the chances you will have much more than a middle-class income at age 30? The two-way table summarizes the responses.
$$\begin{array} {c|cc|c} & \text { Female } & \text { Male } & \text { Total } \\ \hline \text { Almost no chance } & 96 & 98 & 194 \\ \hline \text { Some chance but probably not } & 426 & 286 & 712 \\ \hline \text { A 50-50 chance } & 696 & 720 & 1416 \\ \hline \text { A good chance } & 663 & 758 & 1421 \\ \hline \text { Almost certain } & 486 & 597 & 1083 \\ \hline \text { Total } & 2367 & 2459 & 4826 \end{array}$$
Choose a survey respondent at random. Define events G: a good chance, M: male, and N: almost no chance. Given that the chosen student didn't say "almost no chance," what's the probability that this person is female? Write your answer as a probability statement using correct symbols for the events. A survey of 4826 randomly selected young adults (aged 19 to 25 ) asked, "What do you think are the chances you will have much more than a middle-class income at age 30?" The two-way table summarizes the responses. PSK\begin{array} {lc} & \text{Gender} \ \text {Opinion} & \begin{array}{l|c|c|c} & Female & Male & Total \\ \hline Almost no chance & 96 & 98 & 194 \\ \hline \begin{array}{l} Some chance but \\ robably not \end{array} & 426 & 286 & 712 \\ \hline A 50-50 chance & 696 & 720 & 1416 \\ \hline A good chance & 663 & 758 & 1421 \\ \hline Almost certain & 486 & 597 & 1083 \\ \hline Total & 2367 & 2459 & 4826 \end{array}\ \end{array}ZSK Choose a survey respondent at random. Define events G: a good chance, M: male, and N: almost no chance. Find P(G | M). Interpret this value in context. Is there a relationship between gender and relative finger length? To find out, we randomly selected 452 U.S. high school students who completed a survey. The two-way table summarizes the relationship between gender and which finger was longer on the left hand (index finger or ring finger). $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{l}{c}\right\rbrace}&\text{Gender}\backslash\text{Longer finger}&{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{l}{\left|{c}\right|}{r}{\mid}{r}\right\rbrace}&\ \text{ Female }\ &\ \text{ Male }\ &\ \text{ Total }\ \backslash{h}{l}\in{e}\ \text{ Index finger }\ &{78}&{45}&{123}\backslash{h}{l}\in{e}\ \text{ Ring finger }\ &{82}&{152}&{234}\backslash{h}{l}\in{e}\ \text{ Same length }\ &{52}&{43}&{95}\backslash{h}{l}\in{e}\ \text{ Total }\ &{212}&{240}&{452}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\backslash{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ Suppose we randomly select one of the survey respondents. Define events R: ring finger longer and F: female. Find P(R|F). Interpret this value in context. The Pew Research Center asked a random sample of 2024 adult cellphone owners from the United States their age and which type of cell phone they own: iPhone, Android, or other (including non-smartphones). The two-way table summarizes the data.
$$\begin{array}{c|ccc|c} & 18-34 & 35-54 & 55+ & \text { Total } \\ \hline \text { iPhone } & 169 & 171 & 127 & 467 \\ \text { Androod } & 214 & 189 & 100 & 503 \\ \text { Other } & 134 & 277 & 643 & 1054 \\ \hline \text { Total } & 517 & 637 & 870 & 2024 \end{array}$$
Suppose we select one of the survey respondents at random. What's the probability that: The person is not age 18 to 34 and does not own an iPhone? A random sample of 88 U.S. 11th- and 12th-graders was selected. The two-way table summarizes the gender of the students and their response to the question "Do you have allergies?" Suppose we choose a student from this group at random.
$$\begin{array}{c|cc|c} & \text { Female } & \text { Male } & \text { Total } \\ \hline \text{ Yes } & 19 & 15 & 34 \\ \text{ No } & 24 & 30 & 54 \\ \hline \text{ Total } & 43 & 45 & 88\\ \end{array}\$$
What is the probability that the student is female or has allergies?
$$(a)\frac{19}{88}$$
(b)\frac{39}{88}\)
(c)\frac{58}{88}\)
(d)\frac{77}{88}\) The accompanying two-way table was constructed using data in the article “Television Viewing and Physical Fitness in Adults” (Research Quarterly for Exercise and Sport, 1990: 315–320). The author hoped to determine whether time spent watching television is associated with cardiovascular fitness. Subjects were asked about their television-viewing habits and were classified as physically fit if they scored in the excellent or very good category on a step test. We include MINITAB output from a chi-squared analysis. The four TV groups corresponded to different amounts of time per day spent watching TV (0, 1–2, 3–4, or 5 or more hours). The 168 individuals represented in the first column were those judged physically fit. Expected counts appear below observed counts, and MINITAB displays the contribution to $$\displaystyle{x}^{{{2}}}$$ from each cell.
State and test the appropriate hypotheses using $$\displaystyle\alpha={0.05}$$
$$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}&{a}\mp,\ {1}&{a}\mp,\ {2}&{a}\mp,\ {T}{o}{t}{a}{l}\backslash{h}{l}\in{e}{1}&{a}\mp,\ {35}&{a}\mp,\ {147}&{a}\mp,\ {182}\backslash{h}{l}\in{e}&{a}\mp,\ {25.48}&{a}\mp,\ {156.52}&{a}\mp,\backslash{h}{l}\in{e}{2}&{a}\mp,\ {101}&{a}\mp,\ {629}&{a}\mp,\ {730}\backslash{h}{l}\in{e}&{a}\mp,\ {102.20}&{a}\mp,\ {627.80}&{a}\mp,\backslash{h}{l}\in{e}{3}&{a}\mp,\ {28}&{a}\mp,\ {222}&{a}\mp,\ {250}\backslash{h}{l}\in{e}&{a}\mp,\ {35.00}&{a}\mp,\ {215.00}&{a}\mp,\backslash{h}{l}\in{e}{4}&{a}\mp,\ {4}&{a}\mp,\ {34}&{a}\mp,\ {38}\backslash{h}{l}\in{e}&{a}\mp,\ {5.32}&{a}\mp,\ {32.68}&{a}\mp,\backslash{h}{l}\in{e}{T}{o}{t}{a}{l}&{a}\mp,\ {168}&{a}\mp,\ {1032}&{a}\mp,\ {1200}\backslash{h}{l}\in{e}$$
$$\displaystyle{C}{h}{i}{s}{q}={a}\mp,\ {3.557}\ +\ {0.579}\ +\ {a}\mp,\ {0.014}\ +\ {0.002}\ +\ {a}\mp,\ {1.400}\ +\ {0.228}\ +\ {a}\mp,\ {0.328}\ +\ {0.053}={6.161}$$
$$\displaystyle{d}{f}={3}$$