The two-way table below describes the members of the U.S Senate in a recent year.begin

Question
Two-way tables

The two-way table below describes the members of the U.S Senate in a recent year.
$$\begin{array}{ccc} \hline &\text{Male}&\text{Female}\\ \text{Democrats}&47&13\\ \text{Republicans}&36&4\\ \hline \end{array}$$
If we select a U.S. senator at random, what's the probability that the senator is a Democrat?

2021-01-09

Let us determine the row/column total of each row/column, which is the sum of all counts in the row/column:
$$\begin{array}{c|cc|c} &\text{Male}&\text{Female}&\text{Total}\\ \hline \text{Democrats}&47&13&47+13=60\\ \text{Republicans}&36&4&36+4=40\\ \hline \text{Total}&47+36=83&13+4=17&60+40=100 \end{array}$$
The table contains 100 members in total (which is given in the bottom right corner of the table), while 60 of the 100 members are Democrats (since 60 is mentioned in the row ”Democrats” and in the column ”Total” of the given table).
The probability is the number of favorable outcomes divided by the number of possible outcomes:
$$P(Democrats)=\frac{\text{# of favorable outcomes}}{\text{# of possible outcomes}}=\frac{60}{100}=\frac{3}{5}=0.6=60\%$$
Result: $$\frac{3}{5}=0.6=60\%$$

Relevant Questions

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)$$\frac{77}{88}$$

A study among the Piria Indians of Arizona investigated the relationship between a mother's diabetic status and the number of birth defects in her children. The results appear in the two-way table. $$\text{Diabetic status}\ \begin{array}{ll|c|c|c} && \text { Nondiabetic } & \text { Prediabetic } & \text { Diabetic } \\ \hline & \text { None } & 754 & 362 & 38 \\ \hline & \text { One or more } & 31 & 13 & 9 \end{array}$$

What proportion of the women in this study had a child with one or more birth defects?

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.
$$\begin{array}{c|cc|c} &\text{Male}&\text{Female}&\text{Total}\\ \hline \text{Blue}&&&10\\ \text{Brown}&&&40\\ \hline \text{Total}&20&30&50 \end{array}$$
Copy and complete the two-way table so that events A and B are mutually exclusive.

The following two-way contingency table gives the breakdown of the population of adults in a town according to their highest level of education and whether or not they regularly take vitamins:
$$\begin{array}{|c|c|c|c|c|} \hline \text {Education}& \text {Use of vitamins takes} &\text{Does not take}\\ \hline \text {No High School Diploma} & 0.03 & 0.07 \\ \hline \text{High School Diploma} & 0.11 & 0.39 \\ \hline \text {Undergraduate Degree} & 0.09 & 0.27 \\ \hline \text {Graduate Degree} & 0.02 & 0.02 \\ \hline \end{array}$$
You select a person at random. What is the probability the person does not take vitamins regularly?

Statistics students at a state college compiled the following two-way table from a sample of randomly selected students at their college:
$$\begin{array}{|c|c|c|}\hline&\text{Play chess}&\text{Don`t play chess}\\\hline \text{Male students} &25&162\\ \hline \text{Female students}&19&148 \\ \hline \end{array}\\$$
Answer the following questions about the table. Be sure to show any calculations.
What question about the population of students at the state college would this table attempt to answer?
State $$H^0$$ and $$H^1$$ for the test related to this table.

The two-way table shows the results from a survey of dog and cat owners about whether their pet prefers dry food or wet food.
$$\begin{array}{c|c|c} &\text { Dry } & \text { Wet }\\ \hline \text{ Cats } & 10&30 \\ \hline \text{ Dogs} & 20&20\\ \end{array}$$
Does the two-way table show any difference in preferences between dogs and cats? Explain.

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 $$\displaystyle\alpha={0.05}$$
$$\begin{array}{|c|c|}\hline & 1 & 2 & Total \\ \hline 1 & 35 & 147 & 182 \\ \hline & 25.48 & 156.52 & \\ \hline 2 & 101 & 629 & 730 \\ \hline & 102.20 & 627.80 & \\ \hline 3 & 28 & 222 & 250 \\ \hline & 35.00 & 215.00 & \\ \hline 4 & 4 & 34 & 38 \\ \hline & 5.32 & 32.68 & \\ \hline Total & 168 & 1032 & 1200 \\ \hline \end{array}$$
$$Chisq= 3.557\ +\ 0.579\ +\ 0.014\ +\ 0.002\ +\ 1.400\ +\ 0.228\ +\ 0.328\ +\ 0.053=6.161$$
$$\displaystyle{d}{f}={3}$$

Bayes' Theorem is given by $$P(A|B) = \frac{P(B|A) \cdot P(A)} {P(B)}$$. Use a two-way table to write an example of Bayes' Theorem.

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 resident.

$$\begin{array}{l|c|c|c} & Married & Not married & Total \\ \hline Own & 172 & 82 & 254 \\ \hline Rent & 40 & 54 & 94 \\ \hline Total & 212 & 136 & 348 \end{array}$$

State the hypotheses for a test of the relationship between marital status and housing status for U.S. residents.

$$\begin{array}{ccc} & Age \ Response & {\begin{array}{l|r|r|r|r|r} & 18-34 & 35-49 & 50-64 & 65+ & Total \\ \hline Support & 91 & 161 & 272 & 332 & 856 \\ \hline Oppose & 25 & 74 & 211 & 255 & 565 \\ \hline Don't know & 4 & 13 & 20 & 51 & 88 \\ \hline Total & 120 & 248 & 503 & 638 & 1509 \end{array}} \ \end{array}$$