Get help with two way tables questions

The General Social Survey (GSS) asked a random sample of adults their opinion about whether astrology is very scientific, sort of scientific, or not at all scientific. Here is a two-way table of counts for people in the sample who had three levels of higher education:
\(\begin{array}{l|c|c|c|c} & \text { Associate's } & \text { Bachelor's } & \text { Master's } & \text { Total } \\ \hline \begin{array}{l} \text { Not al all } \\ \text { scientific } \end{array} & 169 & 256 & 114 & 539 \\ \hline \begin{array}{l} \text { Very or sort } \\ \text { of scientific } \end{array} & 65 & 65 & 18 & 148 \\ \hline \text { Total } & 234 & 321 & 132 & 687 \end{array}\)
State appropriate hypotheses for performing a chi-square test for independence in this setting.

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. What is most important?
\(\begin{array}{l|c|c|c|c} & \text { 4th grade } & \text { 5th grade } & \text { 6th grade } & \text { Total } \\ \hline \text { Grades } & 49 & 50 & 69 & 168 \\ \hline \text { Athletic } & 24 & 36 & 38 & 98 \\ \hline \text { Popular } & 19 & 22 & 28 & 69 \\ \hline \text { Total } & 92 & 108 & 135 & 335 \end{array}\)
Identify the explanatory and response variables in this context.

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.
\(\begin{array}{|c|c|}\hline & \text{4th grade} & \text{5th grade} & \text{6th grade} & \text{Total} \\ \hline \text{Grades} & 49 & 50 & 69 & 168 \\ \text{Athletic} & 24 & 36 & 38 & 98 \\ \hline \text{Popular} & 19 & 22 & 28 & 69 \\ \hline \text{Total} & 92 & 108 & 135 & 335 \\ \hline \end{array}\)
Suppose we select one of these students at random. What's the probability that: The student is not a sixth grader and did not rate good grades as important?

Suppose that each student in a sample had been categorized with respect to political views, marijuana usage, and religious preference, with the categories of this latter factor being Protestant, Catholic, and other. The data could be displayed in three different two-way tables, one corresponding to each category of the third factor. With pijk = P(political category i, marijuana category j, and religious category k), the null hypothesis of independence of all three factors states that \(p_{ijk}=p_i..p_j..p_k\)
Let nijk denote the observed frequency in cell (i, j, k). Show how to estimate the expected cell counts assuming that H0 is true (\(\widehat{e}_{ijk}=n\widehat{p}_{ijk}\), so the \(\widehat{p}_{ijk}\)’s must be determined). Then use the general rule of thumb to determine the number of degrees of freedom for the chi-squared statistic.

Is there a relationship between gender and relative finger length? To investigate, a random sample of 452 U.S. high school students was selected. The two-way table shows the gender of each student and which finger was longer on his or her left hand (index finger or ring finger). State the hypotheses for a test of the relationship between gender and relative finger length for U.S. high school students.
\(\begin{array}{|l|c|r|c|} \hline & \text { Female } & \text { Male } & \text { Total } \\ \hline \text { Index finger } & 78 & 45 & 123 \\ \hline \text { Ring finger } & 82 & 152 & 234 \\ \hline \text { Same length } & 52 & 43 & 95 \\ \hline \text { Total } & 212 & 240 & 452 \\ \hline \end{array}\)

A study in Sweden looked at former elite soccer players, people who had played soccer but not at the elite level, and people of the same age who did not play soccer. Here is a two-way table that classifies these individuals by whether or not they had arthritis of the hip or knee by their mid-fifties:
\(\text{Soccer level} \\ \begin{array}{ll|c|c|c} & & \text { Ellte } & \text { Non-elite } & \text { Did not play } \\ \hline \text { Whether person } & \text { Yes } & 10 & 9 & 24 \\ \hline \text { developed arthritis } & \text { No } & 61 & 206 & 548 \end{array}\)
Researchers suspected that the more serious soccer players were more likely to develop arthritis later in life. Do the data confirm this suspicion? Calculate appropriate percentages to support your answer.

Using data from the 2000 census, a random sample of 348 U.S. residents aged 18 and older was selected. The two-way table summarizes the relationship between marital status and housing status for these residents.

In 1912 the Titanic struck an iceberg and sank on its first voyage. Some passengers got off the ship in lifeboats, but many died. The following two-way table gives information about adult passengers who survived and who died, by class of travel.
\(\begin{array} {lc} & \text{Class} \\ \text {Survived} & \begin{array}{c|c|c|c} & \text { First } & \text { Second } & \text { Third } \\ \hline \text { Yes } & 197 & 94 & 151 \\ \hline \text { No } & 122 & 167 & 476 \\ \end{array}\\ \end{array}\)
Suppose we randomly select one of the adult passengers who rode on the Titanic. Define event D as getting a person who died and event F as getting a passenger in first class. Find P (not a passenger in first class and survived).

Do male and female students have different favorite seasons? The two-way table shows the favorite season and gender for a simple random sample of 89 high school juniors and seniors in the United States from the Census At School database. Is there convincing evidence of an association between gender and favorite season for students like those who participated in the Census At School survey?

A study in Sweden looked at former elite soccer players, people who had played soccer but not at the elite level, and people of the same age who did not play soccer. Here is a two-way table that classifies these individuals by whether or not they had arthritis of the hip or knee by their mid-50s.
\(\begin{array} {lc} & \text{Soccer experience} \ \text {Arthritis} &\begin{array}{l|c|c|c|c} & & & \text { Did not } & \\ & \text { Elite } & \text { Non-elite } & \text { play } & \text { Total } \\ \hline \text { Yes } & 10 & 9 & 24 & 43 \\ \hline \text { No } & 61 & 206 & 548 & 815 \\ \hline \text { Total } & 71 & 215 & 572 & 858 \end{array} \end{array}\)
Suppose we choose one of these players at random. What is the probability that the player has arthritis?

A study in Sweden looked at former elite soccer players, people who had played soccer but not at the elite level, and people of the same age who did not play soccer. Here is a two-way table that classifies these individuals by whether or not they had arthritis of the hip or knee by their mid-fifties:

\(\text{Soccer level} \\ \begin{array}{ll|c|c|c} & & \text { Ellte } & \text { Non-elite } & \text { Did not play } \\ \hline \text { Whether person } & \text { Yes } & 10 & 9 & 24 \\ \hline \text { developed arthritis } & \text { No } & 61 & 206 & 548 \end{array}\)

What percent of the elite soccer players developed arthritis? What percent of those who got arthritis were elite soccer players?

Mutually exclusive versus independent. The two-way table summarizes data on the gender and eye color of students in a college statistics class. Imagine choosing a student from the class at random. Define event A: student is male, and event B: student has blue eyes.

\(\text{Gender}\ \text{Eye color}\begin{array}{l|c|c|c} & \text { Male } & \text { Female } & \text { Total } \\ \hline \text { Blue } & & & 10 \\ \hline \text { Brown } & & & 40 \\ \hline \text { Total } & 20 & 30 & 50 \end{array}\)

Copy and complete the two-way table so that events A and B are mutually exclusive.

Is there a relationship between gender and relative finger length? To investigate, a random sample of 452 U.S. high school students was selected. The two-way table shows the gender of each student and which finger was longer on his or her left hand (index finger or ring finger). State the hypotheses for a test of the relationship between gender and relative finger length for U.S. high school students.
\(\begin{array} {lc} & \text{Gender} \ \text {Longerfinger Longer finger} & \begin{array}{l|c|r|r} & \text { Female } & \text { Male } & \text { Total } \\ \hline \text { Index finger } & 78 & 45 & 123 \\ \hline \text { Ring finger } & 82 & 152 & 234 \\ \hline \text { Same length } & 52 & 43 & 95 \\ \hline \text { Total } & 212 & 240 & 452 \end{array} \\ \end{array}\)