Exercises: Contingency tables

In a secondary school, a survey was conducted among the $100$ Year 12 students about their preferred extracurricular activity. The data have started to be organised in the following contingency table, but some values are missing.

	Sport	Music	Total
Girls	$35$		$55$
Boys		$20$
Total			$100$

Copy the table into your exercise book and complete it by finding the missing absolute frequencies.

A video game shop registered its sales from this morning according to console type and game format. $12$ Nintendo Switch games were sold in physical format and $8$ in digital format. For the PlayStation 5, $15$ were sold in physical format and $10$ in digital format.

Organise this information in a two-way (contingency) table including the marginal frequencies (the total for each row and column).

A group of $50$ people were asked whether they regularly play sport and whether they are smokers. The results are shown in the following table:

	Plays sport	Does not play sport	Total
Smoker	$5$	$15$	$20$
Non-smoker	$25$	$5$	$30$
Total	$30$	$20$	$50$

Calculate the table of joint relative frequencies. (Express each value as the result of dividing the absolute frequency by the sample total $N=50$).

Based on the following table on athletes and smokers, answer the following questions:

	Smokers	Non-smokers
Athletes	10	30
Non-athletes	25	35

1. a) What is the marginal frequency of "Athletes"? What percentage of the total do they represent?
2. b) What is the marginal frequency of "Non-smokers"? What percentage of the total do they represent?

A courier company analyses the mode of transport used by its workers according to their place of residence:

	Lives in city	Lives outside	Total
Public Transport	$40$	$10$	$50$
Own Vehicle	$20$	$30$	$50$
Total	$60$	$40$	$100$

If we randomly choose a worker who lives in the city, what is the relative frequency that they use public transport? (Calculate the conditional frequency).

A study has been carried out on the use of glasses in a class of $30$ students. We know that there are $16$ girls, of whom $4$ wear glasses. We also know that there are $10$ students in total (boys and girls) who wear glasses.

Complete the contingency table and determine how many boys do not wear glasses.

A movie theater analyzes popcorn preference (salted or sweet) according to movie type (animation or horror).

	Animation	Horror	Total
Salted	$24$	$36$	$60$
Sweet	$56$	$16$	$72$
Total	$80$	$52$	$132$

Look at the table and answer:

1. a) What is the majority preference (mode) among viewers of animation movies?
2. b) What percentage of viewers who prefer sweet popcorn went to see a horror movie?

Given the following relative frequency table about reading habits and library use:

	Reads a lot	Reads little	Total
Uses the library	$0.25$	$0.05$	$0.30$
Doesn't use the library	$0.15$	$0.55$	$0.70$
Total	$0.40$	$0.60$	$1.00$

If the total sample is $200$ people, calculate how many people belong to the category of those who "Read a lot" and "Don't use the library". Write the complete operation and the numerical result.

In a study on digital content consumption among 200 teenagers, the preferred platform for watching series and movies was analyzed according to the main access device. The data collected are as follows: 65 mainly use their cell phone (of which 40 prefer TikTok and the rest Netflix), 85 use a computer (50 prefer Netflix and the rest YouTube), and 50 use a Smart TV (35 prefer Netflix and the rest YouTube).

a) Organize the information into a contingency table of absolute frequencies relating the variables "Device" and "Platform".

b) Calculate the marginal frequencies of each variable.

c) What percentage of the total number of teenagers prefers Netflix?

A survey was conducted among students at a school about their favorite extracurricular activity (Sports or Music), differentiating by year (Year $1^\circ$ or Year $2^\circ$ of high school). The results are shown in the following incomplete table:

Year	Sports	Music	Total
Year $1^\circ$	$42$		$70$
Year $2^\circ$		$24$
Total	$78$		$130$

a) Complete the missing values in the contingency table.

b) If we choose a student at random from Year $2^\circ$ of high school, what is the probability that their favorite activity is Sports?

c) Among the students who prefer Music, what percentage belongs to Year $1^\circ$ of high school?

To investigate the relationship between using Artificial Intelligence (AI) to do homework and academic performance, $150$ students were surveyed. The data show that $60$ students use AI frequently, of which $15$ have excellent marks. On the other hand, of the students who do not use AI, $45$ have excellent marks.

a) Construct the table of relative frequencies with respect to the total.

b) Calculate the distribution of the variable "AI use" conditional on having "Excellent marks".

c) Can we state that the variables are independent? Justify your answer by comparing a joint relative frequency with the product of the corresponding marginal frequencies.

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

A video game company wants to know whether the type of console users own is related to their subscription to premium services. $300$ users are analyzed:

Subscription	Console A	Console B	Console C
Premium	$45$	$60$	$15$
Basic	$75$	$90$	$15$

a) Obtain the table of relative frequencies conditioned by columns (type of console).

b) Based on the results of the previous part, on which console is it most common to find a user with a Premium subscription?

c) Represent the distribution of the subscription by type of console using a stacked bar chart.

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

In a sports club, $60\%$ of the members are men and $40\%$ are women. It is known that $30\%$ of the men swim, while $55\%$ of the women swim. The remaining members play tennis.

a) If the club has a total of $500$ members, construct the contingency table of absolute frequencies.

b) Calculate the marginal relative frequency of members who play tennis.

c) What is the majority profile in the club (combination of gender and sport)?

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

The reading habits of a group of people are analyzed according to their educational level. The data are presented in this table of relative frequencies with respect to the total sample:

Education	Reads daily	Reads occasionally	Does not read
Secondary	$0.12$	$0.18$	$0.10$
University	$0.35$	$0.20$	$0.05$

a) Calculate the marginal relative frequencies of the variable "Education".

b) If it is known that a person never reads, which is more likely: that they have secondary or university education? Justify your answer by comparing the corresponding conditional frequencies.

c) If the total sample were $400$ people, how many of them would be university-educated individuals who read occasionally?

A study was conducted on the preference for type of diet (Omnivorous or Vegetarian) depending on the area of residence (Urban or Rural). The following data were obtained:

Residence	Omnivorous	Vegetarian	Total
Urban	$120$	$80$	$200$
Rural	$90$	$10$	$100$
Total	$210$	$90$	$300$

a) Calculate the row relative frequency table.

b) Is there a dependency between the area of residence and the type of diet? Compare the conditional distributions of diet for urban and rural residents.

c) What value should the frequency of "Rural-Vegetarian" have for the variables to be independent, maintaining the marginal totals for residence?

A high school wants to analyze whether the use of public transportation to travel to school depends on the distance from the school. $120$ students are classified as "Near" ($< 2$ km) and "Far" ($\geq 2$ km).

a) Complete the table knowing that $25\%$ of the students live near, and that of the students who live far, $80\%$ use public transportation. Furthermore, no students living near use public transportation.

b) Obtain the relative frequency table with respect to the total.

c) Calculate the relative frequency of "living far" given "not using public transportation".

A study on the impact of social media on the emotional well-being of 1st-year Bachillerato students classified a sample of 200 students according to daily screen time and self-reported life satisfaction level. The incomplete data obtained are as follows:

• 40% of the total sample reports "Low" life satisfaction.

• There are 50 students who use social media "Less than 1 hour", of whom only 10% have "Low" satisfaction.

• The number of students with "High" satisfaction who use social media "Between 1 and 3 hours" is twice the number of those with "High" satisfaction who use it "More than 3 hours".

• In total, 70 students use social media "Between 1 and 3 hours".

Based on this information:

1. a) Reconstruct the complete contingency table with absolute frequencies.
2. b) Calculate the conditional distribution of the variable "Life Satisfaction" for each of the levels of "Screen time".
3. c) Can we claim that the variables are independent? Justify your answer by comparing the conditional relative frequencies.

A technology consulting firm analyzes the relationship between the use of Artificial Intelligence (AI) for studying and the average grade in a sample of students. It is known that:

1. $30\%$ of the students never use AI.

2. $25\%$ of the students have an average grade of "Outstanding".

3. The total sample size is 400 students.

4. It has been observed that the events "Having an Outstanding grade" and "Never using AI" are independent.

Calculate:

1. a) The number of students who never use AI and have an Outstanding grade.
2. b) The probability that a student regularly uses AI given that their grade is not Outstanding.
3. c) Complete the contingency table by dividing the sample into two usage categories (AI / No AI) and two grade categories (Outstanding / Other).

In a health center, the relationship between competitive sports participation and the type of injuries sustained in young people is investigated. We have the following table of relative frequencies conditional on columns (type of sport):

Injury / Sport	Basketball ($D_1$)	Soccer ($D_2$)	Athletics ($D_3$)
Muscular ($L_1$)	$0.30$	$0.55$	$0.40$
Bone ($L_2$)	$0.70$	$0.45$	$0.60$

If we know that $40\%$ of the young people in the sample play soccer, $35\%$ play basketball and the rest do athletics:

1. a) Obtain the joint relative frequency table.
2. b) Calculate the probability that a young person chosen at random plays basketball given that they have suffered a bone injury.
3. c) In which sport is it most likely to suffer a muscular injury? Does the answer change if we analyze absolute frequencies with respect to the total population? Justify your answer.

A municipality wants to promote the use of sustainable transportation. After surveying 500 citizens about their main mode of transportation ($X$) and their age group ($Y$), the following marginal frequency data were obtained:

• $n_{Bus} = 180$; $n_{Bici} = 120$; $n_{Coche} = 200$.

• $n_{Joven} = 150$; $n_{Adulto} = 250$; $n_{Senior} = 100$.

In addition, the following information is known:

1. $60\%$ of those who use a bicycle are young people.

2. Seniors do not use bicycles at all.

3. $50\%$ of car users are adults.

4. The event "Being a bus user" and "Being an adult" has a joint absolute frequency of 90 people.

1. a) Determine the complete absolute frequency table.
2. b) Study the dependence between age and car use by comparing the marginal and conditional distributions.
3. c) What is the most frequent transportation profile for a senior citizen?

The labor market for recent graduates in a region is analysed. The variables are "Type of Qualification" ($X$: University or Vocational) and "Employment Status" ($Y$: Employed, Unemployed, or Internship). Part of the records from a sample of 1 000 individuals have been lost, but the conditional distributions by rows (Type of Qualification) are preserved:

$X \setminus Y$	Employed	Unemployed	Internship
University	$0.65$	$0.10$	$0.25$
Vocational	$0.75$	$0.15$	$0.10$

Knowing that the marginal frequency of "Employed" is 690 people:

1. a) Calculate how many individuals in the sample have university qualifications and how many have vocational qualifications.
2. b) Determine the distribution of the variable "Type of Qualification" conditional on being "Unemployed".
3. c) If a company is looking for candidates for an internship contract, in which qualification group will it find a larger absolute number of candidates according to this study?

A sociological study on reading habits analyzes the relationship between the preferred format (Paper, E-book, Audio) and the number of books read per year (Fewer than 5, Between 5 and 12, More than 12).

It is known that in the sample:

1. The "Paper" format is preferred by $50\%$ of readers.

2. The "Audio" format is the least frequent, with only $10\%$.

3. Among "E-book" readers, the three categories of book quantity have the same relative frequency ($1/3$ each).

4. The event "Reading more than 12 books" is independent of the "Paper" format, and it is known that $40\%$ of all readers read more than 12 books.

5. $10\%$ of the total sample are "Paper" readers who read "Fewer than 5" books.

6. None of the "Audio" users read fewer than 5 books per year.

Find:

1. a) The joint relative frequency table.
2. b) The distribution of the number of books read conditioned on "Audio" users, knowing that none of them read fewer than 5 books per year.
3. c) What percentage of readers who read "Between 5 and 12" books prefer the Paper format?

In a study on music streaming consumption among teenagers, two platforms ($A$ and $B$) and account types (Free or Premium) are compared.

The percentage data indicate:

• $60\%$ of users use platform $A$.

• $70\%$ of users of platform $B$ have a Free account.

• In the total sample, the percentage of Premium accounts is $40\%$.

1. a) Calculate the percentage of platform $A$ users who have a Premium account.
2. b) If we choose a user with a Premium account, what is the probability that they belong to platform $B$?
3. c) Determine whether there is independence between the chosen platform and the account type. Justify numerically.

An e-commerce platform wants to analyse customer loyalty. It records whether the customer made a purchase in the last month ($C$) and whether they are subscribed to the newsletter ($S$). The data from a sample of 1 000 customers are:

• $35\%$ of customers have made a purchase in the last month.

• Of those who have not made a purchase, $80\%$ are not subscribed to the newsletter.

• The probability of having made a purchase in the last month given that they received the newsletter is $0.6$.

1. a) Draw up the contingency table of absolute frequencies.
2. b) If a customer is not subscribed to the newsletter, what is the probability that they made a purchase in the last month?
3. c) An analyst states: "The newsletter triples the probability of a purchase". Based on the conditional relative frequencies, confirm or refute this statement with precise calculations.