Exercises: Contingency tables

Copy and complete the following contingency table in your notebook, showing 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 transport to work, the following absolute results were obtained:

State the joint absolute frequency of people who are "Under 30 years old" and travel by "Walking", and calculate the marginal absolute frequency of those who use a "Car".

From the following contingency table of absolute frequencies, draw up the corresponding table of overall relative frequencies (expressing the results as simplified fractions).

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

You can do it on paper and send a photo with your answer.

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

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

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

Write down the marginal distribution of the variable "Crust type", indicating its absolute frequencies.

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

A language academy 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 survey conducted at an Andalusian secondary school on sustainable mobility and physical activity, data from $80$ Year 10 students (4.º de ESO) were collected in the following contingency table:

The teacher asks the following question: “What is the proportion of students who do not do sports given that they have a transit pass?”. Lucas reasons as follows: “Since there are $16$ students who have a transit pass and do not do sports out of the $80$ students in total, the proportion is $\frac{16}{80} = 0,2$ ($20\%$).”

1. a) Explain Lucas's conceptual error precisely: what type of frequency did he calculate, and which one was actually requested?
2. b) Correct Lucas's calculation by clearly pointing out what the reference total (denominator) should be, and express the correct conditional frequency $h(\text{Does not do sports} \mid \text{Has transit pass})$ as a simplified fraction and as a percentage.

During a road safety campaign carried out by the Local Police of Seville regarding the mandatory use of helmets on electric scooters (personal mobility vehicles), the following data were recorded in a random checkpoint of $200$ users:

1. a) Before calculating any fraction or percentage, carefully observe the data in the two age columns and make a prior visual estimate: in which age group do you intuit that helmet use is proportionally higher? Justify your intuition by mentally comparing each cell with the marginal total of its column.
2. b) Check your estimate by calculating exactly the conditional relative frequencies $h(\text{Wears a helmet} \mid \text{Under 25})$ and $h(\text{Wears a helmet} \mid \text{25 or older})$ expressed as percentages.
3. c) Assess whether your initial estimate was correct and write a brief conclusion about whether helmet use seems to depend on the age group in this sample.

In a school with $300$ 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 students in the school do not do extracurricular activities and also do not like drama. Construct the contingency table of absolute frequencies and determine how many students do not do extracurricular activities but are signed up for drama.

You can do it on paper and send a photo with your answer.

A medical study analyses 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 success percentage (cured) for each of the treatments separately and reason, using these percentages, which of the two treatments appears to be more effective.

Complete the following contingency table, given that the overall relative frequencies of some cells are given as fractions 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 sets off the alarm for $35$ of the pieces containing prohibited items and also gives a false alarm for $24$ pieces containing no prohibited items. Construct the contingency table and calculate the percentage of luggage pieces that actually contain prohibited items among all those for which the alarm went off.

Reconstruct a $2 \times 2$ contingency table with variables $X$ and $Y$ using 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 satisfying $X_1$ also satisfy $Y_1$.

The following Venn diagram shows the distribution of $110$ students at 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 studying French given that they do not study English.

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

Calculate the marginal distribution of the variable "Satisfaction" as 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 / Moist). A total of $240$ trees were counted, of which $150$ are pines. It is known that $60\%$ of the oaks grow in moist soil, whereas only $40$ pines grow in moist soil. Construct the contingency table and determine what percentage of total trees grow in dry soil.

You can do it on paper and send a photo with your answer.

Based on plausible data from the Survey of Cultural Habits and Practices in Spain published by the Ministry of Culture and Sport, a study was carried out with a sample of $250$ young people aged between $15$ and $19$ to relate two variables: whether they regularly read books in their free time (at least one per month) and whether they are subscribed to audiovisual streaming platforms (Netflix, Prime Video, etc.).

• Out of the total of $250$ young people, $160$ state that they are subscribed to streaming platforms.

• Among those subscribed to streaming platforms, exactly $64$ state that they regularly read books.

• In the overall sample, $118$ young people state that they regularly read books in their free time.

1. a) Organise all the given information in a complete contingency table of absolute frequencies ($2 \times 2$), first calculating the marginal totals and the missing cells.
2. b) If a person is chosen at random from those who are NOT subscribed to streaming platforms, what is the proportion that they are a regular reader? Express the result as an irreducible fraction and as a percentage rounded to one decimal place.
3. c) Compare the conditional proportions of regular readers in both groups: $h(\text{Reads} \mid \text{Streaming})$ and $h(\text{Reads} \mid \text{No streaming})$. In light of the data, can it be stated that being subscribed to streaming platforms is associated with a lower proportion of book readers? Justify your answer.

In the canteen service of a school, the menu preferences of a single day with $200$ diners (teachers and students) are analysed:

Three students discuss how to check whether the choice of menu type is independent of the group (being a teacher or a student):

• Irene argues: “Since the difference in diners choosing the traditional menu between students and teachers is very large ($112 - 28 = 84$), the choice clearly depends on the group.”

• Marta proposes comparing the proportion of vegetarians among teachers with the proportion of vegetarians among students: $h(\text{Vegetarian} \mid \text{Teachers}) \stackrel{?}{=} h(\text{Vegetarian} \mid \text{Students})$.

• Sergio proposes comparing the proportion of vegetarians among teachers with the overall marginal proportion of vegetarians in the entire school: $h(\text{Vegetarian} \mid \text{Teachers}) \stackrel{?}{=} h(\text{Vegetarian})$.

1. a) Explain why the method proposed by Irene is flawed and demonstrate why absolute differences are not valid when the groups have different total sizes.
2. b) Perform the calculations proposed by Marta and Sergio. Do both methods reach the same conclusion regarding whether the variables are independent?
3. c) Based on contingency table theory (“the conditional distribution coincides with the marginal distribution”), justify which of the two valid methods (Marta's or Sergio's) represents the definition of statistical independence more directly and rigorously.

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

Given 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 across 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.

In two different hospitals, the success of a complex surgical procedure has been recorded, dividing cases into "Mild" and "Severe".

San Omar Hospital:

La Paz Hospital:

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

An electronics factory has two assembly machines. Machine I produces $60\%$ of the total components and Machine II produces the remaining $40\%$. Quality control reveals that $2\%$ of Machine I's components are defective, whereas the defect rate for Machine II is $5\%$.

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

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

Given that the ratio of $a$ to $c$ is $\frac{2}{3}$ and the ratio of $b$ to $d$ 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 $C$.

Convert the information in 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 study of $600$ consumers classifies their drink preferences (Water, Soft drink, Juice) and their age group (Young, Adult). The table is almost empty, but the following clues are known:

1. The total number of Young people is double that of Adults.

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

3. The number of Young people who prefer Soft drinks is $150$.

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

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

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

You can do it on paper and send a photo with your answer.

In a population of $400$ inhabitants, two variables are analysed: "Plays sport" (Yes/No) and "Smokes" (Yes/No). The marginal distributions are known: $240$ people play sport 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 smoking habits and playing sport in this population.

You can do it on paper and send a photo with your answer.

In a community health study conducted at a health centre in Valencia with a sample of $N = 120$ adult patients, two variables are analysed: “Uses dental floss daily” (Yes / No) and “Has active cavities” (Yes / No). The complete table is not available, but the following conditions are known:

1. Exactly $40\%$ of the patients in the sample use dental floss daily.

2. In the entire sample, there are a total of $48$ people with active cavities.

3. It is verified that using dental floss significantly reduces the incidence of cavities: the conditional frequency of having cavities among those who floss is exactly half the conditional frequency of having cavities among those who do not:

1. a) Assign an unknown $x$ to the joint frequency of patients who use dental floss and have cavities. Set up an equation relating the two conditional frequencies to their respective marginal totals and solve for the value of $x$.
2. b) Reconstruct the complete contingency table ($2 \times 2$) with all its joint and marginal absolute frequencies, verifying that all values are integers and consistent.
3. c) If the presence of cavities and daily flossing were statistically independent variables, how many patients who floss should have cavities? Compare this theoretical figure with the actual value obtained in your table and write a conclusion about the preventive impact of flossing.

Civil Protection and the State Meteorological Agency (AEMET) are evaluating the effectiveness of weather warnings for torrential rainfall in a Mediterranean coastal region during a historical series of $180$ autumn days. The recorded data are as follows:

• AEMET issued an orange- or red-level weather warning on $30$ of the $180$ days.

• Actual torrential rains were recorded on the ground on $24$ of the $180$ days.

• On $18$ of the days when the warning was active, torrential rains actually occurred.

1. a) Model the situation by constructing a complete contingency table ($2 \times 2$) of absolute frequencies, crossing the variables “Warning active (Yes / No)” and “Actual torrential rain (Yes / No)” with their corresponding marginal frequencies.
2. b) In a municipal evaluation meeting, two opposing opinions arise:

• The emergency management technician states: “The warning system is very effective: $75\%$ of the times it rained torrentially, the population was warned with an alert.”

• A neighbourhood association complains: “The system creates too much unjustified alarm: $40\%$ of the issued warnings are false alarms (it does not end up raining torrentially).”

Calculate the conditional frequencies $h(\text{Warning active} \mid \text{Torrential rain})$ and $h(\text{Torrential rain} \mid \text{Warning active})$, and mathematically interpret why both statements are correct but answer inverse conditional questions.

1. c) Formally prove whether the activation of the warning and the occurrence of torrential rain are dependent or independent variables by comparing the conditional frequency $h(\text{Torrential rain} \mid \text{Warning active})$ with the marginal frequency $h(\text{Torrential rain})$. Explain what practical consequences it would have for public safety if both variables turned out to be statistically independent.