Master Two Way Tables with Our Expert Answers

You roll a die. If it comes up a 6, you win 100. If not, you gettoroll again. If you get a 6 the second time, you win 50. If not, you lose. Find the expected amount you'll win.

In the article "Graphical Display of Two-Way Contingency Tables", R. Snee presented data on hair color and eye color among 592 students in an elementary statistics course at the University of Delaware. Raw data for that information are presented on the WeissStats site. Use the technology of your choice to do the following tasks, and interpret your results. a) Obtain both a frequency distribution and a relative-frequency distribution for the hair-color data. b) Get a pie chart of the hair-color data. c) Determine a bar chart of the hair-color data. d) Repeat parts (a)-(c) for the eye-color data.

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 gender and housing status for these residents.
$$\begin{array}{|cccc|}\hline & \text{Male}& \text{Female}& \text{Total}\\ \text{Own}& 132& 122& 254\\ \text{Rent}& 50& 44& 94\\ \text{Total}& 182& 166& 348\\ \hline\end{array}$$
Make a graph that displays the relationship between gender and housing status for the members of the sample. Briefly describe the association.

The ? relative frequencies are the sums of each row and column in a two-way table. (joint, marginal, or conditional)

Write a paragraph that describes the relationship between race (respondent's race) and cappun (opinion about capital punishment). Include a two-way table of counts and a relevant set of conditional percentages.

The two-way table gives the party affiliation of all members of the House of Representatives from eight different regions in the United States in July 2014 (there were 3 vacant seats at that time).

The two-way table summarizes information about eye color and gender in a random sample of 200 high school students.

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 (${\hat{e}}_{ijk}=n{\hat{p}}_{ijk}$, so the ${\hat{p}}_{ijk}$’s must be determined). Then use the general rule of thumb to determine the number of degrees of freedom for the chi-squared statistic.

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.
$\begin{array}{lc}& \text{Class}\\ \text{Survived}& \begin{array}{cccc}& \text{ First }& \text{ Second }& \text{ Third }\\ \text{ Yes }& 197& 94& 151\\ \text{ No }& 122& 167& 476\end{array}\end{array}$
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.
$\begin{array}{lcc}& \text{Soccer experience}\text{Arthritis}& \begin{array}{lcccc}& & & \text{ Did not }& \\ & \text{ Elite }& \text{ Non-elite }& \text{ play }& \text{ Total }\\ \text{ Yes }& 10& 9& 24& 43\\ \text{ No }& 61& 206& 548& 815\\ \text{ Total }& 71& 215& 572& 858\end{array}\end{array}$
Suppose we choose one of these players at random. What is the probability that the player has arthritis?

Describe how you can use a two -way table to organize data you collect from a 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-fifties:

$$\begin{gathered} \text{Soccer level} \\ \begin{array}{llccc}& & \text{ Ellte }& \text{ Non-elite }& \text{ Did not play }\\ \text{ Whether person }& \text{ Yes }& 10& 9& 24\\ \text{ developed arthritis }& \text{ No }& 61& 206& 548\end{array} \end{gathered}$$

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.

$\text{Gender}\text{Eye color}\begin{array}{lccc}& \text{ Male }& \text{ Female }& \text{ Total }\\ \text{ Blue }& & & 10\\ \text{ Brown }& & & 40\\ \text{ Total }& 20& 30& 50\end{array}$

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.
$\begin{array}{lcc}& \text{Gender}\text{Longerfinger Longer finger}& \begin{array}{lcrr}& \text{ Female }& \text{ Male }& \text{ Total }\\ \text{ Index finger }& 78& 45& 123\\ \text{ Ring finger }& 82& 152& 234\\ \text{ Same length }& 52& 43& 95\\ \text{ Total }& 212& 240& 452\end{array}\end{array}$

1. Who seems to have more variability in their shoe sizes, men or women?
a) Men
b) Women
c) Neither group show variability
d) Flag this Question
2. In general, why use the estimate of $n-1$ rather than n in the computation of the standard deviation and variance?
a) The estimate n-1 is better because it is used for calculating the population variance and standard deviation
b) The estimate n-1 is never used to calculate the sample variance and standard deviation
c) $n-1$ provides an unbiased estimate of the population and allows more variability when using a sample and gives a better mathematical estimate of the population
d) The estimate n-1 is better because it is use for calculation of both the population and sample variance as well as standard deviation.
$\begin{array}{|cc|}\hline \text{Shoe Size (in cm)}& \text{Gender (M of F)}\\ 25.7& M\\ 25.4& F\\ 23.8& F\\ 25.4& F\\ 26.7& M\\ 23.8& F\\ 25.4& F\\ 25.4& F\\ 25.7& M\\ 25.7& F\\ 23.5& F\\ 23.1& F\\ 26& M\\ 23.5& F\\ 26.7& F\\ 26& M\\ 23.1& F\\ 25.1& F\\ 27& M\\ 25.4& F\\ 23.5& F\\ 23.8& F\\ 27& M\\ 25.7& F\\ \hline\end{array}$
$\begin{array}{|cc|}\hline \text{Shoe Size (in cm)}& \text{Gender (M of F)}\\ 27.6& M\\ 26.9& F\\ 26& F\\ 28.4& M\\ 23.5& F\\ 27& F\\ 25.1& F\\ 28.4& M\\ 23.1& F\\ 23.8& F\\ 26& F\\ 25.4& M\\ 23.8& F\\ 24.8& M\\ 25.1& F\\ 24.8& F\\ 26& M\\ 25.4& F\\ 26& M\\ 27& M\\ 25.7& F\\ 27& M\\ 23.5& F\\ 29& F\\ \hline\end{array}$

Complete the two-way table.

 $\begin{array}{|cccc|}\hline \text{ Gender }& \text{ Response }& & \\ & \text{ Yes }& \text{ No }& \text{ Total }\\ \text{ Male }& & 52& \\ \text{ Female }& 48& & 110\\ \text{ Total }& 65& & \\ \hline\end{array}$

The two-way table below describes the members of the U.S Senate in a recent year.
Male Female
Democrats 47 13
Republicans 36 4
If we select a U.S. senator at random, what's the probability that the senator is a female and a Democrat?

A random sample of 1200 U.S. college students was asked, "What is your perception of your own body? Do you feel that you are overweight, underweight, or about right?" The two-way table summarizes the data on perceived body image by gender. $\begin{array}{cc}& \text{Gender}\\ \text{Body image}& \begin{array}{lccr}& \text{ Female }& \text{ Male }& \text{ Total }\\ \text{ About right }& 560& 295& 855\\ \text{ Overwerght }& 163& 72& 235\\ \text{ Underweight }& 37& 73& 110\\ \text{ Total }& 760& 440& 1200\end{array}\end{array}$

What percent of respondents feel that their body weight is about right?

A survey was designed to study how business operations vary according to their size. Companies were classified as small, medium, or large. Questionnaires were sent to 200 randomly selected businesses of each size. Because not all questionnaires were returned, researchers decided to investigate the relationship between the response rate and the size of the business. The data are given in the two-way table. $\begin{array}{lcc}& \text{Business size}\text{Response?}& \begin{array}{lcccc}& \text{ Small }& \text{ Medium }& \text{ Large }& \text{ Total }\\ \text{ Yes }& 125& 81& 40& 246\\ \text{ No }& 75& 119& 160& 354\\ \text{ Total }& 200& 200& 200& 600\end{array}\end{array}$

 Which variable should be used as the explanatory variable? Explain.