Basic Concepts of Statistics

A video game company wants to know what the favorite genre of students at your school is. In order not to ask all $800$ students, they decide to survey $10$ students from each class in the school.

Identify the following elements of the statistical study:

• Population:

• Sample:

• Individual:

Sometimes it is impossible to study the entire population, which is why we use a sample. Explain in which of these cases you think it is compulsory to use a sample instead of asking the entire population:

1. a) We want to find the mean mark of the last math test in your 1st ESO class.
2. b) A light bulb factory wants to know how many hours they stay lit before blowing.
3. c) The headteacher of the school wants to know how many students came to school today.
4. d) A hospital wants to analyse the blood group of all residents of an autonomous community.

To improve the quality of the food in the cafeteria of a large secondary school with $1200$ students, the school council decides to ask the opinion of the first $5$ students from each class who enter the school on Monday morning. Identify the following elements of this statistical study:

• a) Population.

• b) Sample.

• c) Individual.

A famous influencer wants to know what Mexican teenagers think about the new time limit on a social media platform. To do this, they post a poll on their Instagram "stories" for 24 hours. $15,400$ people take part.

1. a) Identify the population under study and the sample used.
2. b) Do you think the sample is representative of the population? Reason your answer by analyzing two possible factors that could invalidate the results (biases).
3. c) Propose a more reliable method for choosing a sample of $500$ individuals that truly represents all teenagers in Mexico.

Classify the following statistical variables as qualitative (if they express a quality or characteristic) or quantitative (if they are expressed by a number):

1. a) The names of the most used apps on your friends' cell phones.
2. b) The number of siblings each student in your class has.
3. c) The brand of sneakers worn by the players in a team.
4. d) The maximum temperature reached in your city each day of this week.
5. e) The favorite sport of the members of an athletic club.

Within quantitative variables, we distinguish between discrete (when they take isolated values, usually whole numbers) and continuous (when they can take any value in an interval, such as decimals). Classify these variables:

1. a) The number of goals scored by a team in each league game.
2. b) The exact time taken by a swimmer to cover $50$ meters.
3. c) The number of books in the students' backpacks.
4. d) The height of the players on the Mexican national basketball team.

A well-known streamer wants to know what their followers' favorite video game in Mexico is. To do this, they conduct a quick poll during a livestream in which $1500$ people participate. Classify the following variables collected in the study as qualitative or quantitative:

• a) Title of the favorite video game.

• b) Number of hours they play per week.

• c) Console or platform used (PC, PS5, Switch, etc.).

• d) Amount of money spent on "skins" or battle passes per month.

• e) Rating of the livestream from 1 to 10.

An online clothing store wants to know the level of satisfaction of its customers (Very dissatisfied, Dissatisfied, Neutral, Satisfied, Very satisfied).

• a) Classify this variable. Is it quantitative or qualitative?

• b) Although words are used, sometimes numbers ($1$ to $5$) are assigned to calculate the mean. Do you think this turns the variable into a quantitative one? Reason your answer.

• c) If the store sends the survey to $500$ customers who bought something yesterday, who is the "individual" in this study?

Classify the following statistical variables as qualitative or quantitative. In the case of quantitative variables, reflect on whether it makes sense to perform arithmetic operations (such as the mean) with their values:

1. a) The postal code of the residence of students at a middle school.
2. b) The serial number of the computers in the library.
3. c) The daily mean temperature recorded in the school playground during the month of January.
4. d) The jersey number of the players on a basketball team.
5. e) The time, expressed in seconds, taken by a group of swimmers to cross a $50$-meter swimming pool.

Note: Think about whether the "number" acts as a label or as a real measurement.

A group of friends is discussing which film to go and see at the cinema. They have written down their preferences:

Adventure, Comedy, Horror, Adventure, Adventure, Horror, Comedy, Adventure, Comedy.

Answer the following questions on the basic concepts:

1. a) What is the statistical variable being studied?
2. b) What type of variable is it?
3. c) What is the absolute frequency (number of times it appears) of the category "Adventure"?

Look at the following bar chart representing the pets of the students in a class and answer based on the concepts learned:

1. a) What is the sample size (total number of students surveyed)?
2. b) What is the variable studied and what type is it?
3. c) What is the absolute frequency of the variable value "Cat"?

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

Look at the following diagram representing a study on tree health in a nature reserve containing $5000$ trees. The forestry team has analysed $150$ of them.

• a) What is the population size?

• b) What is the sample size?

• c) If the researchers record whether the tree has a pest or not, what type of variable are they studying?

In a study on fruit consumption in a class of $30$ students, some data from the frequency table have been lost. We know that the relative frequency ($h_i$) of students who eat "3 pieces a day" is exactly $0.2$ and that the absolute frequency ($f_i$) of those who eat "1 piece a day" is three times that of those who eat "none".

If $2$ students do not eat any pieces of fruit a day, complete the following data with reasoning:

1. a) How many students eat $3$ pieces of fruit a day?
2. b) How many students eat $1$ piece of fruit a day?
3. c) What is the absolute frequency of those who eat "2 pieces a day", knowing that all the above options add up to the class total?

A survey was conducted on the number of children per family in a housing development of $200$ families. It is known that $15\%$ of the families have $3$ or more children.

1. a) What is the absolute frequency of the event "having 3 or more children"?
2. b) If the angle in a circle graph for families with "1 child" is $144^\circ$, how many families have exactly one child?
3. c) Calculate the relative frequency of the group of families that have "1 child".

Many things happen in our daily lives, but not all of them depend on chance. State which of the following situations are random experiments and which are deterministic:

1. a) Tossing a coin and seeing whether it lands on heads or tails.
2. b) Calculating how much $3$ movie tickets will cost if each one is worth $160$ pesos.
3. c) Asking a classmate what their favorite color is.
4. d) Dropping a stone from a balcony and seeing if it falls downward.
5. e) Drawing a card from a Spanish deck without looking.

State whether the following phenomena are random or deterministic, briefly justifying your answer:

• a) Knowing whether it will rain in your city on the exact day of your birthday next year.

• b) Calculating the price of $5$ movie tickets if you know that each one costs $170$ pesos.

• c) Predicting the number that will come up when rolling a 10-sided die in a role-playing game.

• d) Knowing the grade on a math test if you have left all the questions blank.

• e) Drawing a card from a Spanish deck of cards and getting the Ace of Coins.

We are carrying out an experiment on the growth of some tomato plants under different types of light. We have $200$ plants in total.

1. a) If we decide to analyze only the plants in the row closest to the window, are we studying the population or a sample?
2. b) Define the individual in this study.
3. c) We want to measure three variables: the height of the plant in $cm$, the color of the leaves (dark green, light green, or yellowish), and the number of tomatoes produced by each plant. Classify each variable.
4. d) Is this a random or a deterministic experiment? Justify your answer.

Identify whether the following phenomena are random or deterministic, explaining why in each case. For random phenomena, describe their sample space:

1. a) Dropping a stone from a balcony and timing how long it takes to reach the ground (assuming there is no wind and the height is fixed).
2. b) The number that will win the top prize in the national lottery draw.
3. c) Drawing a card from a Spanish deck of cards and observing whether it is a "picture card" (Jack, Knight, or King).
4. d) Mixing an acid with a base in a chemistry laboratory following an exact ratio to produce a salt.
5. e) The exact time it will take for the next bus to arrive at the stop.

Imagine that you want to carry out a study on the reading habits of teenagers in your city. Give an example of:

1. A question whose answer is a qualitative variable.

2. A question whose answer is a discrete quantitative variable.

3. A question whose answer is a continuous quantitative variable.

In a study on the reading habits of teenagers in a city, the statistical variable is defined as: "Number of books read during last summer".

• a) What type of variable is this?

• b) If the study is carried out on a sample of $200$ young people and the sum of all their read books is $450$, what does the value $2.25$ represent in this context?

• c) Come up with three possible values that an individual could take for this variable.

A cell phone manufacturer wants to test the screen durability of a new model. The factory produces $10.000$ phones a day. For the study, $50$ phones are selected and dropped repeatedly from a height of $1.5$ meters until the screen breaks, recording how many drops each one withstands.

• a) Explain why in this case it is strictly necessary to use a sample instead of studying the entire population.

• b) Identify the variable studied and classify it.

• c) What would happen to production if the manufacturer decided not to use a sample and analyzed every individual in the population?

Identify the population and the individual in each of the following statistical studies:

• 1. We want to find the mean weight of the $250$ g loaves of bread produced by a bakery in one day.

• 2. We want to know the favorite subject of 1st ESO students across Spain.

• 3. A vet analyses the type of diet of cats living in an urban colony in their neighbourhood.

• 4. A light bulb company wants to know how many hours their low-energy bulbs stay lit before blowing.

Imagine that you want to carry out a statistical study to convince the headteacher of your school that lessons should start 15 minutes later to improve performance.

1. a) Clearly define the population and an appropriate sample that is not biased.
2. b) Design three different variables that you could measure: one qualitative, one quantitative that is a count (whole numbers), and another quantitative that is a measurement (with decimals).
3. c) State which data collection instrument you would use for each (survey, direct observation, or measuring device).

Analyze the following situation: A municipality wants to know whether citizens are happy with the new bike lane. To save costs, they decide to ask only people using the bike lane on a Saturday morning.

1. a) Is the sample random?
2. b) What type of bias do you think this study will show? Will the results overestimate or underestimate general satisfaction in the city?
3. c) Propose a way of choosing the sample that includes people who do not use bicycles so that the study is objective.
4. d) If the variable is "Level of satisfaction (Not at all, A little, A lot)", what type of variable is it? Could the arithmetic mean of these data be calculated? Justify why.