Exercises: Concepts

To find out the favourite video game of the students at a secondary school of $850$ students, a survey was given to $40$ students chosen at random during break. Identify the following elements of the study:

• Population

• Sample

• Individual

Classify the following statistical variables as qualitative or quantitative:

1. a) Your friends' favourite colour.
2. b) The number of WhatsApp messages you receive per day.
3. c) The brand of trainers worn by the students in your class.
4. d) The amount of money you save each month.

An animal shelter wants to conduct a study on the health of its dogs. To do so, the vet records the weight of each of the $35$ dogs living there.

State the statistical variable studied and classify it (qualitative, discrete quantitative or continuous quantitative).

Classify these quantitative variables as discrete or continuous:

1. a) The number of cousins each student has.
2. b) The exact time it takes you to get from home to school.
3. c) The temperature of a drink can when coming out of the vending machine.
4. d) The number of books in your classroom library.

In a survey carried out in a class of $25$ students about which social network they prefer (TikTok, Instagram or Twitter), the following answers were obtained:

Social network	Number of students
TikTok	$12$
Instagram	$9$
Twitter	$4$

What is the name of the value that is repeated the most in this study? State what that value is in this survey.

A study is to be carried out to find out the favourite ice cream flavour of the inhabitants of a town of $50,000$ people. To do this, the first $100$ customers entering a town centre ice cream shop on a Monday morning are asked.

Do you think this sample is representative of the entire population of the town? Briefly explain your answer.

Look at the following statistical variables related to a school trip and classify them:

1. Destination country.

2. Duration in number of days.

3. Distance travelled in kilometres.

4. Level of satisfaction (Very low, Low, Medium, High).

If in a statistical study the population consists of all the cars in a town and the variable is "car colour", identify:

1. a) The individual.
2. b) The type of variable (Qualitative or Quantitative).
3. c) Suggest three possible values this variable can take.

Hugo has classified four statistical variables collected among his $2.º$ de ESO classmates, but he has made a conceptual error in each of them:

1. T-shirt size ($\text{XS}$, $\text{S}$, $\text{M}$, $\text{L}$, $\text{XL}$): "It is quantitative continuous because body measurements increase progressively."

2. Number of siblings: "It is qualitative ordinal because siblings can be ordered from oldest to youngest."

3. Daily time spent getting to high school: "It is quantitative discrete because we always say whole minutes, like $15$ minutes or $20$ minutes."

4. Home postal code: "It is quantitative discrete because it consists of a $5$-digit number."

For each of the four variables:

1. a) Clearly explain the reasoning error that Hugo makes.
2. b) Write the correct classification and justify it using the definitions of qualitative variable (nominal or ordinal) and quantitative variable (discrete or continuous).

A youth magazine wants to prepare a report on the leisure and lifestyle habits of teenagers in your town. To do this, they ask you to design the survey questions yourself.

1. a) Write a different statistical variable for each of the following four conditions:

• Variable A: It must be qualitative nominal.

• Variable B: It must be qualitative ordinal.

• Variable C: It must be quantitative discrete.

• Variable D: It must be quantitative continuous.

1. b) For the quantitative continuous variable you proposed, state three possible numerical values that a student could answer and explain why it would not make sense to consider it discrete.
2. c) Precisely define who would constitute the population of this study and who would be an individual.

A city council wants to know the digital leisure habits of teenagers aged between 12 and 16. To do this, it randomly selects 400 students from three different secondary schools and asks them how many hours per day on average they spend on social media (TikTok, Instagram, etc.).

Identify the following elements of the statistical study:

1. a) The population.
2. b) The sample.
3. c) The individual.
4. d) The statistical characteristic studied and its classification.

An animal shelter has recorded the age (in years) of the dogs adopted over the last month. The data obtained is as follows:

$$2, 1, 5, 3, 2, 4, 1, 2, 3, 2, 5, 1, 4, 2, 3$$

Construct a frequency table that includes:

• Variable ($x_i$)

• Absolute frequency ($n_i$)

• Relative frequency ($f_i$)

• Cumulative absolute frequency ($N_i$)

• Percentage ($\%$)

Look at the following graph representing the number of books read by a group of students during the summer holidays:

Based on the graph above, answer:

1. a) How many students were asked in total?
2. b) What is the most frequent value of the variable? What is this statistic called?
3. c) What percentage of students read exactly 2 books?

An e-sports club measured the reaction time (in milliseconds) of its players. As it is a continuous variable, the data has been grouped into the following intervals:

Time (ms)	Frequency ($n_i$)
$[150, 170)$	$6$
$[170, 190)$	$12$
$[190, 210)$	$18$
$[210, 230)$	$4$

1. a) Calculate the class midpoints of each interval.
2. b) Represent the data using a histogram.
3. c) In which interval is the reaction time of most players found?

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

A student obtained the following marks in the Mathematics tests during the second term:

$$7.5; \quad 8.2; \quad 9.0; \quad 2.1; \quad 8.7$$

1. a) Calculate the mean ($\bar{x}$) and the median ($Me$).
2. b) If the teacher decides that the term mark will be the median instead of the mean so that a "bad day" does not hurt as much, what would their mark be?
3. c) Explain why the median is more representative than the mean in this case.

A survey was conducted in a Year 9 class on preferred music genre by gender. The results are shown in this two-way table:

	Pop	Rock	Trap	Total
Girls	$8$	$3$	$4$	$15$
Boys	$5$	$6$		$15$
Total		$9$		30

1. a) Complete the missing data in the table.
2. b) What percentage of the whole class prefers Rock?
3. c) If we choose a boy at random, what is the probability that his preferred music is Pop? Use Laplace's rule.

Two basketball teams scored the following points in their last 5 matches:

• Team A: $72, 75, 74, 73, 76$

• Team B: $50, 95, 60, 110, 55$

1. a) Calculate the mean and the range of both teams.
2. b) Looking at the results, which of the two teams is more consistent in its scoring? Justify your answer by connecting the mean with the spread of the data.

In a study on the eye colour of students at a secondary school, the following data was obtained:

Colour	$n_i$
Brown	$120$
Blue	$30$
Green	$50$

1. a) What type of variable is "Eye colour"?
2. b) Can the mean of this data be calculated? Why?
3. c) If we wanted to represent this data in a pie chart, calculate the size of the central angle (in degrees) that would correspond to the "Green eyes" sector.

In a local hospital that serves $35\,000$ people, two different statistical investigations are proposed:

• Study 1: Check the operation, calibration, and disinfection of the $18$ mechanical ventilators in the intensive care unit (ICU).

• Study 2: Determine the overall satisfaction level of users with the emergency department over the course of an entire year.

1. a) For each of the two studies, choose with justification whether a census should be conducted or a sample should be selected. In each case, compare factors such as group size, cost, time required, and the risk of omitting items.
2. b) If in Study 2 it were decided to select a sample of $200$ patients by interviewing only those leaving the emergency department on a Tuesday between $10:00$ and $13:00$, reason why this selection method would not yield a representative sample and explain which groups of patients would be excluded or underrepresented.

In a secondary school, $500$ students are enrolled, distributed as follows: $150$ in 1st Year, $125$ in 2nd Year, $125$ in 3rd Year, and $100$ in 4th Year. The Student Council wants to conduct a survey about the cafeteria food by selecting a representative sample of $40$ students.

The Student Council representative, Aoife, proposes randomly selecting exactly $10$ students from each year group so that "all levels have the same voice."

1. a) Estimate: Before performing exact calculations, reason whether the representative's proposal equitably represents the entire population of the school or if it favors students of some year groups over others.
2. b) Check: Calculate exactly how many students from each year group should be selected so that the sample is proportional to the actual number of students in each level.
3. c) Assess: Based on the numerical results obtained, explain whether your initial estimate was correct and reason what methodological problem the allocation proposed by the representative presents.

A city council wants to conduct a survey on cycle lane usage among young people. To do this, it selects $450$ students from three different secondary schools proportionally to the total number of students in each school. There are $600$ students in School A, $400$ students in School B and $500$ students in School C.

1. a) State the population, the sample and the individual in this statistical study.
2. b) How many students should be selected from each school for the sample to be representative and proportional?
3. c) Classify the variable "daily time (in minutes) each young person uses a bicycle" according to its nature.

A streaming platform analyses data from its teenage users to improve its algorithm. Classify the following variables collected in the study:

1. a) Number of devices connected simultaneously in the household.
2. b) Favourite film genre (horror, comedy, action, etc.).
3. c) Total weekly viewing time (in hours and minutes).
4. d) Rating given to a series (1 star, 2 stars, 3 stars, 4 stars or 5 stars).
5. e) Brand of the main device used to watch content.

Explain whether any of these quantitative variables could be treated as qualitative in a given context.

A survey was conducted on the number of hours Year 9 students sleep on a weekday. We have the following incomplete data from the frequency table, where $x_i$ is the number of hours, $n_i$ the absolute frequency, $h_i$ the relative frequency and $H_i$ the cumulative relative frequency:

$x_i$	$n_i$	$h_i$	$H_i$
6	4		0.16
7		0.24
8	10
9
10	2		1.00

1. a) Calculate the total sample size ($N$).
2. b) Fill in all the gaps in the frequency table.
3. c) What percentage of students sleep 8 hours or more?

The following histogram represents the time (in minutes) it takes students to get from home to school. The data has been grouped into intervals of the form $[a, b)$.

1. a) Construct the absolute frequency table from the graph, identifying the class midpoints of each interval.
2. b) Calculate the mean time it takes students to get to school using the class midpoints.
3. c) In which interval is the median of the data found? Explain your reasoning.

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

The mean mark of a group of 12 students in a mathematics test was $6.5$. After reviewing the tests, the teacher realizes that he made a mistake with two students: he gave one student a $4$ when they actually had a $7$, and he gave another a $9$ when they actually had a $6$.

1. a) What is the total sum of the marks of the 12 students before the correction?
2. b) What is the new mean of the group after correcting the errors?
3. c) Can we know if the median has changed with this data? Justify your answer.

Look at the following two graphs representing the price evolution of a model of trainers in two different shops over the same 4-month period.

1. a) Explain why the graph for Shop B appears to show a much steeper price rise than the graph for Shop A, even though the actual price data is identical in both.
2. b) What is this visual effect called in statistics and why is it considered a misleading practice?
3. c) Describe how the vertical axis of Shop B should be set up so that the comparison is objective.

In a talent show, a contestant receives the following scores from 5 judges: $8, 7, 9, 8$ and an unknown fifth score ($x$).

1. a) If the mean of the five scores is $8.2$, what is the score of the fifth judge?
2. b) If the competition rules state that to calculate the final score, the highest and lowest scores must be removed and the mean of the remaining three calculated, what would the contestant's final score be with the value of $x$ found previously?
3. c) How would it affect the median of the five original scores if the fifth judge had given them a $10$ instead of the value $x$?

A study on digital leisure in 13-year-old teenagers yields the following data on the number of daily hours spent in front of a screen:

$$3, 5, 2, 4, 3, 2, 8, 3, 1, 4, 3, 2, 3, 5, 3, 10, 2, 3, 4, 1$$

1. a) Calculate the three measures of central tendency: mean, median and mode.
2. b) Looking at the values of the mean and median, would you say the distribution is symmetrical or that there are extreme values skewing the mean? Justify your answer by identifying these values.
3. c) If we remove the two users with the highest screen time, what would the new median be? Explain whether this measure is sensitive to extreme values in the same way as the mean.

The Department of Education in a region of Ireland wants to model the time that Junior Cycle students spend on screen time and studying. In that region, there are $120\,000$ Junior Cycle students, enrolled in three types of schools: community and comprehensive schools ($65\,\%$), voluntary secondary schools ($30\,\%$), and fee-paying private schools ($5\,\%$). It is decided to draw a sample of $1\,200$ students.

1. a) Model the sampling design: define the population, the individual, and calculate how many students must be selected from each type of school for the sample to be representative and proportional to the educational reality.
2. b) Propose four questions for the questionnaire that collect, respectively: a qualitative nominal variable, a qualitative ordinal variable, a quantitative discrete variable, and a quantitative continuous variable related to the object of the study. Justify the classification of each variable.
3. c) A technician proposes reducing costs by sending a link to the questionnaire via social media and analyzing the first $1\,200$ responses that arrive voluntarily. Reason two mathematical or statistical reasons why this sample would cease to be representative and would generate bias in the results.

The organizing committee of a provincial school mathematics olympiad publishes a report with the following three facts about an internal study:

1. A census was conducted rather than sampling, even though students from $15$ different secondary schools participated.

2. The variable $X$ analyzed was quantitative continuous, and the values obtained ranged between $42.5$ and $118.4$.

3. The variable $Y$ analyzed was qualitative ordinal, and no individual achieved the highest possible category.

1. a) Reconstruct a realistic and coherent context for this study: define who the population is, who the individual is, and what variables $X$ and $Y$ could measure exactly for all three statements to be true at the same time.
2. b) Explain why in this specific situation it was feasible to conduct a census instead of taking a sample, and what would change if the goal were to find out those same variables for all the youth in the province.
3. c) A provincial newspaper runs the headline: "The youth of the province obtain an outstanding average score in logical reasoning." Identify the serious conceptual error the journalist makes by confusing the population studied in the census with the general population of young people, and explain what bias invalidates that conclusion.