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 collected 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 the joint frequencies coincides with the total number of students surveyed.

$10$ people were asked about the number of times they go to the movie theater per month ($X$) and the number of books they read per month ($Y$). The answers 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, indicating the joint absolute frequencies.

Given the following joint frequency table that relates the math grades ($X$) and the physics grades ($Y$) 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 marginal distribution table for the variable $X$ (math grades), indicating each value of $X$ and its corresponding marginal frequency $f_{i\bullet}$.

From the same table as the previous exercise, which relates the math ($X$) and physics ($Y$) grades:

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

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

Complete the marginal totals (rows and columns) of the following contingency table that relates the number of 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 notebook 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 they spend playing sports per week ($Y$):

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

Write down the conditional distribution of the number of hours of sports ($Y$) 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 ($Y$) have been recorded. We know that the variable $X$ takes the values $0, 1, 2$ and the 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 $(2,1)$.

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

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

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

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

$X \setminus Y$	$0$	$1$	$2$	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 fertilizer in ounces ($X$) and the height of the plant in inches ($Y$) 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 the height of the plants ($Y$) and find its marginal arithmetic mean $\bar{y}$.

Using the table from the previous exercise on fertilizer ($X$) 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 the height of the plants knowing that exactly $10\text{ g}$ of fertilizer was applied to them (that is, calculate $\bar{y}_{X=10}$).

An economic study relates the monthly income level in thousands of dollars ($X$) and the leisure expenditure in hundreds of dollars ($Y$) of 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 surveyed families has an income of $2$ thousand dollars and spends $3$ hundred dollars on leisure?
2. b) What percentage of the families that earn $3$ thousand dollars spend $2$ hundred dollars on leisure?

The heights in inches ($X$) and the number of medals won ($Y$) by a group of athletes have been recorded in the following table, with the 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 the 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 the variable $Y$ (traffic accidents).

From the same data as 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 graph 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 of test $X$.

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 the variable $X$.

The following table relates the weekly training hours ($X$) and the time in a race 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 there are $N=40$ athletes in total 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 the covariance is $\sigma_{xy} = \frac{\sum x_i y_j f_{ij}}{N} - \bar{x}\bar{y}$.

An economic study groups both the variable $X$ (advertising expenditure in thousands of dollars) and the variable $Y$ (sales in thousands of dollars) 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 two-dimensional distribution.

Prove 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 $\bar{y}_{X=6}$. Then, calculate the marginal mean $\bar{y}$ and check that the weighted mean relationship is satisfied: $\bar{y} = \frac{\sum \bar{y}_{X=x_i} \cdot f_{i\bullet}}{N}$.

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 is 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 equiprobable (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 the joint absolute frequencies $f_{ij}$.

In a class, the scores obtained on the first exam ($X$) and on the second exam ($Y$) are recorded 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 one-dimensional variable $Z = X + Y$ is defined, representing the sum of the scores from both exams. From the two-dimensional table, construct the absolute frequency table of the variable $Z$ and calculate its arithmetic mean $\bar{z}$.

A packaging machine classifies products according to their weight in ounces ($X$) and their length in inches ($Y$). Instead of the 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 $100\text{ oz}$ is $f_{100\bullet} = 45$, determine the total number of evaluated products $N$ and reconstruct the table of joint absolute frequencies $f_{ij}$.