Contingency tables
Does wearing glasses depend on whether you are a boy or a girl? Are smoking and playing sport related? To find out you need to study two variables at the same time, not separately: a contingency table.
Joint and marginal frequencies
| Wear glasses | Do not wear glasses | Total (marginal) | |
|---|---|---|---|
| Boys | $12$ | $18$ | $30$ |
| Girls | $10$ | $15$ | $25$ |
| Total (marginal) | $22$ | $33$ | $55$ |
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…
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.
| Wear glasses | Do not wear glasses | Total | |
|---|---|---|---|
| Boys | $12$ | $18$ | |
| Girls | $15$ | $25$ | |
| Total |
In a survey of $80$ people about their main mode of transport to work, the following absolute results were obtained: | | Car | Public Transport | Walking | Total | |…
In a survey of $80$ people about their main mode of transport to work, the following absolute results were obtained:
| Car | Public Transport | Walking | Total | |
|---|---|---|---|---|
| Under 30 years old | $10$ | $22$ | $8$ | $40$ |
| Over 30 years old | $25$ | $10$ | $5$ | $40$ |
| Total | $35$ | $32$ | $13$ | $80$ |
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".
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 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.
Conditional distributions
Sometimes you want to ask a more precise question: of those who satisfy one condition, what proportion satisfy the other? This is the conditional frequency.
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…
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:
| Spanish | English | Total | |
|---|---|---|---|
| European | $40$ | $60$ | $100$ |
| American | $30$ | $20$ | $50$ |
| Total | $70$ | $80$ | $150$ |
Calculate the conditional relative frequency that an attendee prefers the documentation in English, given that they are European.
In a company of $500$ employees, the relationship between their department and their level of job satisfaction is studied. | | Satisfied | Neutral | Dissatisfied | Total…
In a company of $500$ employees, the relationship between their department and their level of job satisfaction is studied.
| Satisfied | Neutral | Dissatisfied | Total | |
|---|---|---|---|---|
| Sales | $120$ | $60$ | $20$ | $200$ |
| Production | $90$ | $110$ | $100$ | $300$ |
| Total | $210$ | $170$ | $120$ | $500$ |
Calculate the marginal distribution of the variable "Satisfaction" as percentages and the conditional distribution of the level of satisfaction within the "Production" department.
Statistical independence
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…
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.
Two statistical variables are said to be independent if the conditional relative frequencies are equal to the corresponding marginal relative frequencies; that is, the…
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:
| Type I | Type II | Total | |
|---|---|---|---|
| Category Alpha | $12$ | $18$ | $30$ |
| Category Beta | $8$ | $y$ | $8+y$ |
| Total | $20$ | $18+y$ | $38+y$ |
Calculate the exact value of $y$ so that the two variables ("Category" and "Type") are perfectly independent.
Reconstructing incomplete tables
Many problems give only part of the table and ask you to complete the rest, using the fact that each row and each column adds up to its marginal total, sometimes by setting up an equation with one unknown.
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…
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.
Consider the following contingency table where some absolute frequencies depend on a parameter $x$: | | Event A | Event B | Total | | :--- | :---: | :---: | :---: | |…
Consider the following contingency table where some absolute frequencies depend on a parameter $x$:
| Event A | Event B | Total | |
|---|---|---|---|
| Condition 1 | $x$ | $2x$ | |
| Condition 2 | $10$ | $x+5$ | |
| Total |
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.
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…
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?
Back to the start: In the first table, the proportion of boys who wear glasses is $12/30 = 0{,}40$ ($40\,\%$) and that of girls is $10/25 = 0{,}40$ ($40\,\%$); since both match the overall marginal frequency ($22/55 = 0{,}40$), wearing glasses and sex are statistically independent.