Exercises: Frequency table

In a survey of $20$ students, they were asked about the number of siblings ($X$) and the number of pets ($Y$). The results are shown in the following two-way table:

$X \setminus Y$	$0$	$1$	$2$
0	$2$	$4$	$1$
1	$3$	$5$	$2$
2	$1$	$1$	$1$

State how many students have exactly $1$ sibling and $2$ pets, and check that the sum of all joint frequencies equals the total number of students surveyed.

$10$ people were asked about the number of times they go to the cinema per month ($X$) and the number of books they read per month ($Y$). The responses obtained for each individual $(X,Y)$ were:

$(1, 0), (2, 1), (1, 0), (3, 2), (2, 1), (1, 1), (3, 2), (2, 0), (1, 0), (2, 1)$.

Construct the two-way table corresponding to these two variables, showing the joint absolute frequencies.

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

Given the following table of joint frequencies relating the Maths marks ($X$) and Physics marks ($Y$) of a group of students:

$X \setminus Y$	$5$	$6$	$7$
5	$4$	$2$	$0$
6	$1$	$5$	$3$
7	$0$	$2$	$6$

Write the table for the marginal distribution of variable $X$ (Maths marks), indicating each value of $X$ and its corresponding marginal frequency $f_{i\bullet}$.

From the same table as in the previous exercise, relating Maths marks ($X$) and Physics marks ($Y$):

$X \setminus Y$	$5$	$6$	$7$
5	$4$	$2$	$0$
6	$1$	$5$	$3$
7	$0$	$2$	$6$

Write the table for the marginal distribution of variable $Y$ (Physics marks), indicating each value of $Y$ and its marginal frequency $f_{\bullet j}$.

Complete the marginal totals (rows and columns) of the following contingency table relating daily study hours ($X$) and the number of failed subjects ($Y$):

$X \setminus Y$	$0$	$1$	$2$	Total $X$
1	$1$	$3$	$2$
2	$4$	$1$	$0$
3	$0$	$2$	$2$
Total $Y$

Copy the table into your exercise book and fill in the empty cells with the corresponding sums.

The following table shows the relationship between the age of several young people ($X$) and the number of hours per week they spend doing sport ($Y$):

$X \setminus Y$	$2$	$4$	$6$
14	$5$	$3$	$1$
15	$2$	$6$	$4$
16	$1$	$2$	$5$

Write the conditional distribution of the number of hours of sport ($Y$) for the young people who are exactly $15$ years old ($X=15$).

In a study on families, the number of cars ($X$) and the number of bicycles ($Y$) were recorded. We know that variable $X$ takes the values $0, 1, 2$ and variable $Y$ takes the values $0, 1$. Construct the two-way table knowing that:

• There are $2$ families with $(0,0)$ and $3$ families with $(0,1)$.

• There are $4$ families with $(1,0)$ and $5$ families with $(1,1)$.

• There is $1$ family with $(2,0)$ and $0$ families with $(2,1)$.

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

Given the following marginal distribution of variable $X$ with absolute frequencies $f_i$:

$X$	$1$	$2$	$3$
$f_i$	$4$	$5$	$1$

If we know that variable $Y$ takes only the value $Y=2$ for all individuals in the sample, construct the complete two-way table for the joint distribution of $(X,Y)$.

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

In a survey of $30$ 4th-year ESO students about the number of daily hours of leisure screen time ($X$: 1, 2, or 3 hours) and the days per week they practice sports outside school ($Y$: 0, 1, or 2 days), the following two-way frequency table was obtained:

$X \setminus Y$	$0$ days	$1$ day	$2$ days	Total $X$
1 hour	$2$	$5$	$4$	$11$
2 hours	$3$	$6$	$3$	$12$
3 hours	$4$	$2$	$1$	$7$
Total $Y$	$9$	$13$	$8$	$30$

Álvaro analyses these data and makes two common mistakes in his notebook:

1. He states: «To find out how many people participated in the study in total, I have to add the row totals ($11 + 12 + 7 = 30$) and the column totals ($9 + 13 + 8 = 30$), so a total of $60$ people were surveyed».

2. He states: «The cell containing the number $6$ indicates that there are $6$ students who do sport $2$ days a week and spend $1$ hour in front of screens».

Identify which concept Álvaro is confusing in each of his statements, explain why his reasoning is flawed, and write down the correct interpretation of both questions based on the table.

The Regional Transport Consortium of an autonomous community analyses youth mobility and interviews $40$ secondary school students about two variables: the travel pass/ticket they usually use ($X$: «Youth Pass», «Multi-trip card», or «Single ticket») and the number of days a week they use the bus or metro ($Y$: «1-2 days», «3-4 days», or «5-7 days»). From the interviews, the following data are obtained:

• $18$ young people have a Youth Pass; of these, none travel only 1-2 days and $12$ travel 5 to 7 days.

• $14$ young people use a Multi-trip card; of these, $6$ travel 1 to 2 days and $5$ travel 3 to 4 days.

• The remaining $8$ young people use a Single ticket; none travel 5 to 7 days, and those who travel 1 to 2 days are three times as many as those who travel 3 to 4 days.

1. a) Model this situation by constructing a complete two-way frequency table: place the travel passes/tickets ($X$) in the rows, the frequency of use ($Y$) in the columns, and calculate all joint frequencies and marginal row and column totals.
2. b) From the table, calculate what percentage of the total number of young people travel by public transport 5 to 7 days a week, and explain why it is consistent with reality that no Single ticket user belongs to this intensive use category.

A two-way table shows the relationship between the number of goals scored ($X$) and goals conceded ($Y$) by a football team in several matches. Some inner cells have been erased, but the marginal totals remain.

$X \setminus Y$	$0$	$1$	$2$	Total $X$
0	$2$	$a$	$0$	$5$
1	$b$	$3$	$1$	$8$
2	$1$	$1$	$c$	$4$
Total $Y$	$7$	$7$	$3$	$17$

Calculate the unknown values $a$, $b$ and $c$ so that the table is consistent.

The following table relates the amount of fertiliser in grams ($X$) and the height of the plant in centimetres ($Y$) after one month of growth:

$X \setminus Y$	$10$	$15$	$20$
5	$2$	$4$	$1$
10	$1$	$5$	$3$
15	$0$	$2$	$6$

Calculate the marginal distribution of plant height ($Y$) and find its marginal arithmetic mean $\bar{y}$.

Using the table from the previous exercise on fertiliser ($X$) and height ($Y$):

$X \setminus Y$	$10$	$15$	$20$
5	$2$	$4$	$1$
10	$1$	$5$	$3$
15	$0$	$2$	$6$

Calculate the conditional mean of plant height knowing that exactly $10\text{ g}$ of fertiliser was applied (that is, calculate $\bar{y}_{X=10}$).

In an economic study, the monthly income level in thousands of euros ($X$) is related to leisure expenditure in hundreds of euros ($Y$) for several families.

$X \setminus Y$	$1$	$2$	$3$
1	$12$	$6$	$2$
2	$5$	$10$	$5$
3	$1$	$4$	$5$

Answer the following questions by calculating the corresponding percentages:

1. a) What percentage of the total families surveyed has an income of $2$ thousand euros and spends $3$ hundred euros on leisure?
2. b) What percentage of families earning $3$ thousand euros spends $2$ hundred euros on leisure?

The heights in centimetres ($X$) and the number of medals won ($Y$) by a group of athletes have been collected in the following table, with variable $X$ grouped into intervals:

$X \setminus Y$	$0$	$1$	$2$
$[160, 170)$	$6$	$3$	$1$
$[170, 180)$	$4$	$8$	$2$
$[180, 190)$	$2$	$5$	$9$

Obtain the marginal distribution of heights ($X$) and calculate its marginal arithmetic mean $\bar{x}$, using the class midpoints of each interval.

Given the following two-way table on the number of rainy days per week ($X$) and the number of daily traffic accidents ($Y$) in a city:

$X \setminus Y$	$0$	$1$	$2$
1	$10$	$4$	$1$
2	$5$	$8$	$2$
3	$2$	$6$	$12$

Calculate the marginal variance of variable $Y$ (traffic accidents).

From the same data in the previous exercise on rainy days ($X$) and accidents ($Y$):

$X \setminus Y$	$0$	$1$	$2$
1	$10$	$4$	$1$
2	$5$	$8$	$2$
3	$2$	$6$	$12$

Calculate the conditional variance of the number of traffic accidents ($Y$) when the number of rainy days is exactly $X=3$.

The following scatter diagram shows the scores obtained in two aptitude tests, $X$ and $Y$, by a group of candidates. The numbers inside the circles indicate the joint absolute frequency (the number of candidates who obtained that pair of scores). Construct the two-way table associated with this graph and calculate the marginal mean for test $X$.

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

The State Meteorological Agency (AEMET) has recorded two variables over the $30$ days of November in a coastal region: the wind warning level ($X$: 0 = No warning, 1 = Yellow warning, 2 = Orange warning) and the accumulated precipitation level ($Y$: 0 = No rain, 1 = Moderate rain, 2 = Heavy rain). The records are summarised in the following table:

$X \setminus Y$	No rain ($0$)	Moderate ($1$)	Heavy ($2$)	Total $X$
No warning (0)	$11$	$4$	$0$	$15$
Yellow (1)	$2$	$5$	$3$	$10$
Orange (2)	$0$	$1$	$4$	$5$
Total $Y$	$13$	$10$	$7$	$30$

1. a) Estimate without calculating: Observe the position of the highest frequencies along the diagonal and in the corners of the table. Estimate whether a more severe wind warning tends to be associated with a higher rain level, or whether both variables seem to behave completely independently. Justify which cells you look at to make your estimate.
2. b) Check with data: Calculate the proportion of days with heavy rain ($Y=2$) within the group of days with no wind warning ($X=0$) and compare it numerically with the proportion of days with heavy rain within the group with an orange warning ($X=2$). Do these percentages confirm your initial estimate? Justify your answer.

In a study on healthy lifestyle habits based on indicators from the National Health Survey (INE), a primary care team collects data from $60$ adult patients. Two variables are analysed: daily consumption of fresh fruit ($X$: 0 = Less than 1 piece, 1 = 1 to 2 pieces, 2 = 3 or more pieces) and weekly physical activity level ($Y$: 0 = Sedentary, 1 = Moderate, 2 = Regular). When printing the report, an ink stain blurred four cells ($a$, $b$, $c$, and $d$):

$X \setminus Y$	Sedentary ($0$)	Moderate ($1$)	Regular ($2$)	Total $X$
$< 1$ piece (0)	$12$	$6$	$a$	$20$
1-2 pieces (1)	$8$	$b$	$10$	$25$
$\ge 3$ pieces (2)	$c$	$4$	$9$	$d$
Total $Y$	$22$	$17$	$21$	$60$

1. a) Reasonably deduce the value of the four unknown cells ($a$, $b$, $c$, and $d$) from the properties of marginal row and column totals, clearly indicating the order you followed to calculate them.
2. b) A doctor claims after seeing the results: «The percentage of sedentary people among those who barely eat fruit ($< 1$ piece) is more than three times the percentage of sedentary people among those who eat three or more pieces a day». Check with numerical calculations of relative frequencies whether the doctor's claim is true or false.

In a two-way table relating two variables $X$ and $Y$, the cell frequency for $X=2$ and $Y=3$ is an unknown value $k$.

$X \setminus Y$	$1$	$2$	$3$
1	$4$	$5$	$2$
2	$1$	$3$	$k$
3	$2$	$6$	$1$

If we know that the total sample size is $N=30$, determine the value of $k$ and then calculate the marginal standard deviation of variable $X$.

The following table relates the weekly training hours ($X$) and the race time in minutes ($Y$). Two joint absolute frequencies, $a$ and $b$, are unknown.

$X \setminus Y$	$15$	$20$	$25$
2	$4$	$2$	$1$
4	$a$	$5$	$3$
6	$1$	$b$	$4$

We know that in total there are $N=40$ athletes in the study and that the marginal mean of training hours is $\bar{x} = 4,25$. Set up a system of linear equations and find the exact values of $a$ and $b$.

Given the following two-dimensional contingency table for the discrete variables $X$ and $Y$:

$X \setminus Y$	$2$	$4$	$6$
10	$1$	$3$	$0$
20	$2$	$5$	$2$
30	$0$	$2$	$5$

Calculate the covariance $\sigma_{xy}$ of the joint distribution. Remember that the formula for covariance is $\sigma_{xy} = \frac{\sum x_i y_j f_{ij}}{N} - \bar{x}\bar{y}$.

An economic study groups both variable $X$ (advertising expenditure in thousands of euros) and variable $Y$ (sales in thousands of euros) into class intervals.

$X \setminus Y$	$[10, 20)$	$[20, 30)$	$[30, 40)$
$[0, 2)$	$2$	$5$	$1$
$[2, 4)$	$3$	$8$	$4$
$[4, 6)$	$0$	$2$	$5$

Using the class midpoints of each interval, calculate the marginal means $\bar{x}$ and $\bar{y}$, and the covariance of the bivariate distribution.

Demonstrate numerically a fundamental property of conditional means using the following table:

$X \setminus Y$	$1$	$3$	$5$
2	$3$	$1$	$0$
4	$1$	$4$	$1$
6	$0$	$2$	$8$

First, calculate the conditional means of $Y$ for each value of $X$: $\bar{y}_{X=2}$, $\bar{y}_{X=4}$ and $\bar{y}_{X=6}$. Next, calculate the marginal mean $\bar{y}$ and check that the weighted mean relationship holds: $\bar{y} = \frac{\sum \bar{y}_{X=x_i} \cdot f_{i\bullet}}{N}$.

Reconstruct a two-way table for the variables $X$ (number of breakdowns in a machine) and $Y$ (production line: $1$ or $2$). The following data are known:

• The marginal distribution of $X$ takes the values $0, 1, 2$ with marginal frequencies $f_{0\bullet} = 20$, $f_{1\bullet} = 15$ and $f_{2\bullet} = 5$.

• The conditional distribution of $Y$ given $X=0$ is equally likely (half of the breakdowns correspond to line $1$ and the other half to line $2$).

• For $X=1$, twice as many breakdowns occur on line $1$ as on line $2$.

• For $X=2$, all breakdowns occur on line $2$.

Draw the two-way table with all joint absolute frequencies $f_{ij}$.

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

In a class, the marks obtained in the first term ($X$) and in the second term ($Y$) are collected in the following two-way table:

$X \setminus Y$	$4$	$6$	$8$
4	$2$	$3$	$0$
6	$1$	$5$	$2$
8	$0$	$1$	$6$

A new single-variable $Z = X + Y$ is defined, representing the sum of the marks from both terms. From the two-way table, construct the absolute frequency table for variable $Z$ and calculate its arithmetic mean $\bar{z}$.

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

A packaging machine classifies products according to their weight in grams ($X$) and their length in millimetres ($Y$). Instead of absolute frequencies, the following table of joint relative frequencies $h_{ij} = \frac{f_{ij}}{N}$ is provided:

$X \setminus Y$	$10$	$20$	$30$
100	$0,10$	$0,20$	$0,15$
200	$0,05$	$0,30$	$0,20$

If we know that the marginal absolute frequency for products weighing $100\text{ g}$ is $f_{100\bullet} = 45$, determine the total number of evaluated products $N$ and reconstruct the joint absolute frequency table $f_{ij}$.

In a study on digital well-being at a school, $N = 50$ 4th-year ESO students participate. Two variables are analysed: the weekly time spent reading books for leisure ($X$: 1 = Less than 2 hours, 2 = 2 hours or more) and the daily recreational use of mobile phones ($Y$: 1 = Less than 3 hours, 2 = 3 hours or more). The individual data have been lost, but the final project report definitely includes the following three conditions:

1. Exactly $30$ of the $50$ surveyed students claim to use their mobile phone for $3$ hours or more per day ($Y = 2$).

2. Exactly half of the sample spends $2$ hours or more per week reading books ($X = 2$).

3. Among the students who read less than 2 hours per week ($X = 1$), the number of those who use their mobile phone for 3 hours or more per day is four times the number of those who use it for less than 3 hours.

1. a) Reconstruct the complete two-way frequency table (the four joint cells and all marginal row and column totals). State and justify each mathematical deduction or equation you use to find each cell.
2. b) Calculate the proportion of students with high mobile phone use ($Y = 2$) within the group of regular readers ($X = 2$) and compare it with the same proportion within the group of non-readers ($X = 1$). In light of both values, explain what kind of relationship or trend these two variables show in the analysed sample.

The Directorate-General for Traffic (DGT) records in a stretch of interurban road a sample of $100$ speeding offences, classified by road type ($X$: Conventional or Dual carriageway) and offence severity ($Y$: Minor, Serious, or Very serious). Due to a computer glitch, three internal cells ($x$, $y$, $z$) have been erased, although all marginal totals are preserved:

$X \setminus Y$	Minor ($L$)	Serious ($G$)	Very serious ($M$)	Total $X$
Conventional	$x$	$25$	$y$	$60$
Dual carriageway	$z$	$15$	$5$	$40$
Total $Y$	$50$	$40$	$10$	$100$

To calculate the three unknown frequencies, two members of the technical team propose different methods:

• Method A (Direct marginal deduction): First calculate $y$ using the «Very serious» column ($y + 5 = 10$), then $z$ with the «Dual carriageway» row ($z + 15 + 5 = 40$), and finally $x$ by subtracting in the «Minor» column or in the «Conventional» row.

• Method B (Global algebraic system): Set up a linear system of three equations with three unknowns by adding the rows and columns involving the unknowns:

$$\begin{cases} x + y + 25 = 60 \\ z + 15 + 5 = 40 \\ x + z = 50 \end{cases}$$

and solve it by substitution.

1. a) Obtain the values of $x$, $y$, and $z$. Compare both methods and justify which of the two is more efficient and convenient in practice for two-way tables, evaluating the number of operations and the risk of error.
2. b) Suppose that the marginal total of the «Conventional» row ($60$) had also been erased in the report. Reason whether it would still be possible to reconstruct the entire table using either method or if information would be missing, justifying your answer based on the relationship between the marginal totals and the total sample size $N$.