Exercises: Contingency tables

Copy and complete the following contingency table in your notebook, which shows the distribution of a group of students according to their gender and whether they wear glasses or not.

In a survey of $80$ people about their main mode of transportation to go to work, the following absolute results were obtained:

State the joint absolute frequency of the people who are "Under 30" and go "Walking", and calculate the marginal absolute frequency of those who use the "Car".

From the following contingency table of absolute frequencies, create the corresponding table of global relative frequencies (expressing the results as a simplified fraction).

In a sports club there are $120$ members. We know that $70$ play tennis, $50$ play pickleball, and of those who play tennis, $20$ are women. If there is a total of $45$ women in the club, construct the contingency table of absolute frequencies relating gender (Male / Female) to the sport played (Tennis / Pickleball).

Observe the following table which classifies the attendees of a conference according to their nationality and the language in which they preferred to receive the documentation:

Calculate the conditional relative frequency of an attendee preferring the documentation in English, given that they are European.

$200$ customers of a pizzeria were asked about their favorite type of crust. The results are shown in the following table:

Write the marginal distribution of the variable "Type of crust", indicating its absolute frequencies.

Given the following table relating the cell phone brand and the age group of its owners, identify the mode of the joint distribution and the mode of the marginal distribution of the variable "Brand".

A language school offers French and German classes in the morning and afternoon. The table shows the number of enrolled students:

Calculate what percentage of the total number of students is represented by those studying French in the afternoon.

In a high school of $300$ students, $60\%$ participate in extracurricular activities. Of the students who participate in these activities, $40\%$ are signed up for drama. Furthermore, it is known that $15\%$ of the total number of students in the school do not do extracurricular activities and do not like drama either. Construct the contingency table of absolute frequencies and determine how many students do not do extracurricular activities but are signed up for drama.

A medical study analyzes the effectiveness of two treatments (Treatment A and Treatment B) to cure a skin condition. The results after one month are as follows:

Calculate the percentage of success (cured) for each of the treatments separately and explain, based on these percentages, which of the two treatments seems to be more effective.

Complete the following contingency table knowing that the global relative frequencies of some cells are given as a fraction of the total $N = 150$.

In a quality control test, a metal detector inspects $1000$ pieces of luggage. It is known that $40$ pieces contain prohibited items. The detector triggers the alarm in $35$ of the pieces that have prohibited items and also gives a false alarm in $24$ pieces that do not contain anything prohibited. Construct the contingency table and calculate the percentage of pieces that actually contain prohibited items among all those in which the alarm was triggered.

Reconstruct a $2 \times 2$ contingency table with the variables $X$ and $Y$ from the following information:

• The total number of individuals is $N = 200$.

• The marginal frequency of $X_1$ is $120$.

• The marginal frequency of $Y_2$ is $90$.

• $30\%$ of the individuals who satisfy $X_1$ also satisfy $Y_1$.

The following Venn diagram shows the distribution of $110$ students from a language school according to whether they study English ($A$) or French ($B$).

Translate the information from the diagram into a contingency table where the variables are "Studies English" (Yes/No) and "Studies French" (Yes/No). Then, calculate the proportion of students who study French given that they do not study English.

In a company of $500$ employees, the relationship between the department they belong to and their level of job satisfaction is studied.

Calculate the marginal distribution of the variable "Satisfaction" in percentages and the conditional distribution of the level of satisfaction within the "Production" department.

A nature reserve classifies the trees in a plot according to their species (Pine / Oak) and the type of soil in which they grow (Dry / Damp). A total of $240$ trees have been counted, of which $150$ are pines. It is known that $60\%$ of the oaks grow in damp soil, while only $40$ pines grow in damp soil. Construct the contingency table and determine what percentage of the total number of trees grows in dry soil.

Consider the following contingency table where some absolute frequencies depend on a parameter $x$:

If it is known that the marginal relative frequency of "Condition 1" is equal to $0.6$, set up an equation, find the value of $x$ and reconstruct the complete table with numerical values.

Two statistical variables are said to be independent if the conditional relative frequencies are equal to the corresponding marginal relative frequencies; that is, the proportion remains constant in all rows and columns.

Given the following incomplete table:

Calculate the exact value of $y$ so that the two variables ("Category" and "Type") are perfectly independent.

The success of a complex surgical operation has been recorded in two different hospitals, separating the cases into "Mild" and "Severe".

St. Jude Hospital:

Mercy Hospital:

Construct a single global contingency table for each hospital (adding mild and severe cases) relating the "Hospital" to the "Result" (Success / Failure). Calculate the global percentage of success for each hospital and explain why there is a change in the perception of the best hospital when aggregating the data.

An electronic components factory has two assembly machines. Machine I produces $60\%$ of the total components and Machine II the remaining $40\%$. Quality control reveals that $2\%$ of the components from Machine I are defective, while the defect rate in Machine II is $5\%$.

For a batch of $2000$ components, create the contingency table relating the "Machine" (I / II) to the "Condition" (Defective / Correct). If a defective component is selected from this batch, what proportion of the defective ones comes from Machine II?

In a $2 \times 2$ contingency table with the variables $X$ and $Y$, the following proportional relationships between the absolute frequencies $a, b, c, d$ are known:

Knowing that the ratio between $a$ and $c$ is $\frac{2}{3}$ and the ratio between $b$ and $d$ is $\frac{4}{1}$, set up a system of equations to find the values of $a, b, c$ and $d$.

The following tree diagram represents the proportions of a population of $500$ individuals classified according to two dichotomous variables $A$ and $C$.

Transform the information from this diagram into a contingency table of absolute frequencies where the rows are $A$ and $B$, and the columns are $C$ and $D$. Calculate the marginal relative frequency of $C$.

A market research study on $600$ consumers classifies their drink preferences (Water, Soda, Juice) and their age range (Youth, Adult). The table is almost empty, but the following clues are known:

1. The total number of Youth is twice that of Adults.

2. The number of Adults who prefer Water is equal to the number of Adults who prefer Juice.

3. The Youth who prefer Soda are $150$.

4. The total number of people who prefer Juice is $140$.

5. The marginal frequency of those who prefer Water is $200$.

Deduce all the joint and marginal frequencies and construct the complete contingency table.

In a population of $400$ inhabitants, two variables are analyzed: "Plays sports" (Yes/No) and "Smokes" (Yes/No). The marginal distributions are known: $240$ people play sports and $160$ do not; on the other hand, $100$ people smoke and $300$ do not smoke.

Construct the contingency table of absolute frequencies assuming that there is perfect statistical independence between the habit of smoking and playing sports in this population.