Exercises: Tables

To find out opinions on the use of cell phones in the classroom, a survey was conducted among $45$ students at a school with a total of $600$ students. State the key elements of this statistical study:

1. a) The population under study.
2. b) The selected sample.
3. c) The individual.

Classify the following statistical variables as qualitative, discrete quantitative or continuous quantitative based on data from a music application:

1. a) The musical genre of the songs (Pop, Rock, Trap...).
2. b) The number of "likes" on a playlist.
3. c) The duration of a song in seconds.
4. d) The name of the artist or band.
5. e) The storage space occupied by the app on the phone (in MB).

A group of friends write down the number of times they have been to the cinema in the last month. The data obtained are:

$2, 1, 0, 3, 2, 1, 1, 0, 2, 2, 1, 4, 1, 0, 2, 1, 1, 2, 3, 2$.

Organise these data into a frequency table that includes:

• The tally (using tally marks).

• The absolute frequency ($f_i$).

In a 3rd ESO class, students were asked about their favorite sport. Complete the missing values in the following frequency table:

Sport	Absolute frequency ($f_i$)	Relative frequency ($h_i$)	Percentage (%)
Basketball	$6$	$0.24$	$24\%$
Football	$10$
Tennis	$4$	$0.16$
Swimming		$0.20$	$20\%$
Total	$25$	$1.00$	$100\%$

A study has been conducted on the number of books read by members of a book club during the summer. Look at the table and answer the questions using cumulative frequencies:

Books read ($x_i$)	Students ($f_i$)	Cumulative frequency ($F_i$)
$0$	$2$	$2$
$1$	$5$	$7$
$2$	$8$	$15$
$3$	$6$	$21$
$4$	$4$	$25$

1. a) How many students have read exactly $2$ books?
2. b) How many students have read $2$ books or fewer?
3. c) What is the total number of students participating in the study?

For a study on the weight of students' backpacks, the data have been grouped into intervals of $1$ kg. Calculate the class mark ($x_i$) for each of the intervals in the table:

Weight (kg)	Class mark ($x_i$)	Frequency ($f_i$)
$[2, 3)$		$4$
$[3, 4)$		$12$
$[4, 5)$		$9$
$[5, 6)$		$5$

A survey was conducted on the number of pets families in a neighbourhood have. Due to an error, some data in the table were erased. Knowing that a total of $40$ families were surveyed, calculate the missing values:

No. of pets	Families ($f_i$)	Relative Frequency ($h_i$)
$0$	$12$	$0.30$
$1$		$0.45$
$2$	$7$
$3$	$3$	$0.075$
Total	$40$	$1.00$

A courier company studies the weight of the parcels it sends daily. It has organised the information in the following table of cumulative relative frequencies ($H_i$):

Weight (kg)	$f_i$	$h_i$	$H_i$
$[0, 2)$	$20$	$0.40$	$0.40$
$[2, 4)$	$15$	$0.30$	$0.70$
$[4, 6)$	$10$	$0.20$	$0.90$
$[6, 8)$	$5$	$0.10$	$1.00$

Answer the following questions based on the table:

1. a) What percentage of packages weigh less than $4$ kg?
2. b) What relative frequency is represented by packages weighing $6$ kg or more?
3. c) What is the total number of packages analysed in this study?

A study is to be conducted on the type of content most watched on a popular streaming platform by teenagers in a city. To do so, $150$ students in $3^\text{o}$ of secondary school from three randomly selected secondary schools are surveyed. Identify the following elements of the statistical study:

1. a) Population.
2. b) Sample.
3. c) Individual.
4. d) Type of statistical variable.

A group of $25$ young people recorded the number of times they went to the cinema in the last quarter, obtaining the following data:

$2, 1, 0, 3, 2, 1, 2, 0, 1, 4, 2, 1, 0, 2, 1, 2, 3, 2, 0, 1, 2, 1, 0, 2, 1$

Organise this information into a frequency table that includes: absolute frequency ($n_i$), relative frequency ($h_i$) and percentage frequency ($\%$).

In a survey on the number of pets that families in a neighbourhood have, the results shown in the following absolute frequency table have been obtained:

No. of pets ($x_i$)	No. of families ($n_i$)
$0$	$12$
$1$	$18$
$2$	$14$
$3$	$6$

Calculate the column of cumulative absolute frequencies ($N_i$) and the cumulative relative frequencies ($H_i$). What percentage of families have at most $2$ pets?

The time, in minutes, taken by $20$ students to get from home to school has been measured. The data are:

$5, 12, 18, 24, 7, 15, 30, 10, 22, 14, 8, 11, 28, 5, 19, 13, 16, 25, 12, 9$

Construct a frequency table by grouping the data into $5$ intervals of equal width, starting at $5$ and ending at $30$. Include in the table the class marks ($x_i$) and the absolute frequencies ($n_i$).

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

Fill in the blanks in the following frequency table for a discrete quantitative variable:

Data value ($x_i$)	Abs. frequency ($n_i$)	Rel. frequency ($h_i$)	Percentage frequency ($\%$)
$1$	$8$	$0.20$
$2$		$0.35$
$3$	$12$
$4$			$15\%$
Total	$N=$	$1.00$	$100\%$

In a Physical Education class, students were asked about their favorite sport outside school hours. The results are:

• Soccer: $12$ students.

• Basketball: $8$ students.

• Swimming: $6$ students.

• Tennis: $4$ students.

Create a table showing the absolute frequencies and calculate the angles required to represent these data in a circle graph (central angle size).

A nutritionist analyses the amount of sugar (in grams) per $100\text{ ml}$ in $40$ different soft drinks. The results have been grouped in the following table:

Amount of sugar ($g$)	Class mark ($x_i$)	Abs. frequency ($n_i$)
$[0, 4)$		$5$
$[4, 8)$		$12$
$[8, 12)$		$18$
$[12, 16]$		$5$

1. a) Complete the column of class marks.
2. b) Calculate the column of relative frequencies ($h_i$).
3. c) State how many soft drinks have $8$ grams of sugar or more.

A survey was conducted among office employees about the number of languages they speak. We know that $40\%$ speak only one language, $35\%$ speak two languages, $20\%$ speak three languages and the rest speak four languages.

If a total of $60$ people work in the office, construct the complete frequency table (absolute, relative and percentage). How many people speak exactly four languages?

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

In a video game store, a study was conducted on the number of peripherals (controllers, keyboards, mice) that customers purchase in a single visit. However, part of the information in the frequency table has been lost. We know that the study was carried out on a total of $N = 80$ customers.

Complete the missing data in the table, knowing that the cumulative relative frequency of buying 3 peripherals or fewer is $0.875$.

Peripherals ($x_i$)	Absolute freq. ($f_i$)	Relative freq. ($h_i$)	Cumulative absolute freq. ($F_i$)	Cumulative relative freq. ($H_i$)
1	24
2
3	16
4
5	6

A technology laboratory is testing the lifespan in hours of a new drone battery. They have grouped the results in a continuous variable frequency table, but they have only provided us with the class marks ($x_i$) and some frequencies.

It is known that all intervals have the same width and that the lower limit of the first interval is $1.2$ hours. Reconstruct the complete table (intervals, class marks and frequencies), knowing that the cumulative absolute frequency of the third interval is $42$ and that the total number of drones tested is $N = 50$.

Class mark ($x_i$)	Abs. frequency ($f_i$)	Cumulative abs. frequency ($F_i$)
$1.45$	8
$1.95$	14
$2.45$
$2.95$
$3.45$	3

To organise an esports tournament, the organisation analyses the weekly hours that young members of a club spend training. The following cumulative relative frequency table ($H_i$) shows the distribution, but the data for the middle categories are missing.

Knowing that there is the same number of young people who train "From 5 to 10 hours", "From 10 to 15 hours", and "From 15 to 20 hours", and that the club has a total of $120$ members, calculate exactly how many members train in each of the intervals in the table.

Training hours	Cumulative relative freq. ($H_i$)
$[0, 5)$	$0.15$
$[5, 10)$
$[10, 15)$	$0.65$
$[15, 20)$
$[20, 25]$	$1.00$

In a study on water consumption per household in a housing estate, the following cumulative frequency data have been obtained, but the individual absolute frequencies have been omitted.

Consumption ($m^3$)	Cumulative abs. frequency ($F_i$)
$[0, 10)$	$12$
$[10, 20)$	$30$
$[20, 30)$	$65$
$[30, 40)$	$90$
$[40, 50]$	$100$

It has been decided that households consuming more than $30 m^3$ will pay a surcharge. However, after a review, it is discovered that $10$ households in the interval $[20, 30)$ actually consumed $32 m^3$. Calculate the new corrected percentage relative frequency table ($\%$) and determine what percentage of the total number of households will ultimately have to pay the surcharge.

A school wants to conduct a representative survey on the use of digital textbooks. The school has $400$ students in the 1st year of secondary, $350$ in the 2nd year of secondary, $300$ in the 3rd year of secondary, and $250$ in the 4th year of secondary. They want to draw a sample of $130$ students using stratified random sampling with proportional allocation.

Draw the table for the total population and, from it, create the frequency table for the sample indicating how many students from each grade should be surveyed. What is the relative frequency of the 3rd year of secondary students in the sample compared to that in the total population?

A survey has been conducted on the number of pets in $200$ families. The results are presented in the following incomplete relative frequency table ($h_i$).

No. of pets	$h_i$
0	$0.35$
1	$x$
2	$y$
3	$0.12$
4 or more	$0.08$

Calculate the values of $x$ and $y$, knowing that the mean number of pets is exactly $1.25$ (for the interval "4 or more", use the value $4$ as the class mark). Once found, construct the complete absolute frequency table.

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

A local municipality analyzes the age of the users of a sports center. The data for the first $160$ users are summarized in this cumulative frequency table:

Age	$F_i$
$[0, 18)$	$40$
$[18, 35)$	$95$
$[35, 60)$	$140$
$[60, 85]$	$160$

If during the afternoon $40$ new users register, all of them belonging to the age group $[18, 35)$, how does the relative frequency table change? Calculate the resulting percentage relative frequency table ($\%$) after the new registrations and comment on how this change has affected the relative importance of the group over $60$ years old.

You are given the following frequency table for a discrete quantitative variable $X$ representing the number of times a group of students went to the cinema this month. However, the absolute frequencies $f_2$ and $f_3$ have been erased.

$x_i$	$f_i$	$F_i$	$H_i$
0	5	5	$0.125$
1			$0.400$
2		31
3	9		$1.000$

Calculate the total number of data values ($N$) and the missing frequencies to complete the table. Justify each step using the properties of cumulative and relative frequencies.