Exercises: Introduction

To find out the musical preferences of the students at a secondary school of $850$ students, the music teacher asked the $30$ students in his 4th ESO class. Identify the population, the sample, the sample size and the individuals of this statistical study.

Classify the following statistical variables into qualitative, discrete quantitative or continuous quantitative:

• The eye colour of an optician's patients.

• The number of books read in a year by the members of a library.

• The maximum daily temperature recorded in a city during the month of August.

• The brand of mobile phone used by the teachers at a school.

State in which of the following cases it is better to study the entire population and in which it is necessary to select a sample. Give brief reasons for your answer.

• Checking the state of the brakes on all cars manufactured by a brand before selling them.

• Studying the average lifespan of light bulbs produced by a factory (taking into account that to test them they must be left on until they blow).

• Knowing the age of the $25$ students in a class.

• Knowing the cholesterol level of the inhabitants of a country.

In a shopping centre car park there are $450$ vehicles. An employee notes the type of vehicle (car, motorbike, van, 4x4) of $50$ of them chosen at random. Identify the statistical variable being studied and state its type.

A courier company delivers $5\,000$ packages a day. To assess the average delivery time, the supervisor randomly selects $100$ packages and records the minutes they take to reach their destination. Determine the size of the population and the sample size.

Classify the following quantitative variables as discrete or continuous:

• The number of goals scored by a football team in each league match.

• The time taken by runners to complete a marathon.

• The amount of water, in litres, consumed by a family per month.

• The number of passengers travelling on an aeroplane.

Imagine a statistical study is carried out on the laptops owned by students in your city. Give an example of a qualitative variable, a discrete quantitative variable, and a continuous quantitative variable that could be investigated in this study.

A study is to be carried out on customer satisfaction at a pizzeria. To do this, the manager decides to survey only people who order pizza with pineapple. Do you think the chosen sample is representative of all customers? Justify your answer.

In an investigation into healthy lifestyle habits at a secondary school, Lucas collects four pieces of data from each student: their postal code, the number on their futsal jersey, the time in minutes it takes them to get to class, and their rating of the cafeteria menu ("insufficient", "acceptable", "good", or "excellent"). When classifying the variables, Lucas writes in his notebook: "The postal code and the jersey number are numbers, so they are both discrete quantitative variables; in addition, the menu is nominal qualitative because it cannot be measured with an instrument." Identify the two conceptual errors made by Lucas, explain in detail why his reasoning is flawed using the definition of each type of variable, and correctly classify the four analysed variables.

The management team of an educational institution with $720$ enrolled students wishes to conduct two different statistical investigations: first, to determine the level of satisfaction of the $45$ students who use the school transport service daily regarding the punctuality of the bus; second, to determine the average number of weekly hours that students across the entire school spend on recreational screen time. For this second study, a teacher proposes placing an interviewer at the entrance of the library at 8:00 in the morning and surveying the first $60$ students who enter. For each of the two cases, determine with justification whether it is more appropriate to conduct a census or to select a sample. In addition, critically analyse the teacher's proposal for the second study, explaining what type of bias it introduces and why that sample would not be representative.

In a hospital, the following data are collected from patients admitted to A&E: blood group, heart rate in beats per minute, body temperature in degrees Celsius and the number of times they have visited A&E in the past year. Classify each of these four statistical variables and justify why body temperature and heart rate belong to different categories.

A technology company employs $300$ programmers and $200$ graphic designers. The human resources department wants to carry out a survey on the working environment by selecting a sample of $50$ employees using proportional stratified random sampling. How many programmers and how many designers should be in the sample?

A national television show launches a survey via its Twitter account asking: "Do you think the use of social networks should be banned for under-16s?". A total of $15\,000$ people respond. Identify the target population of the show and explain in detail why the results of this sample may be biased and not representative.

When filling out a form, citizens are asked to enter their national identity number (DNI) and the postcode of their municipality. Although both pieces of data consist of numerical digits, from a statistical point of view they are not considered quantitative variables. Explain what type of variables they are and why it makes no sense to calculate the arithmetic mean of postcodes.

The headteacher of a secondary school has an alphabetical list of the $600$ enrolled students. He wants to select a sample of $30$ students using systematic random sampling.

• Calculate the sampling interval (the step size between one individual and the next).

• If the first student chosen at random is the one in position number $12$ on the list, state which positions the next three students selected for the sample will occupy.

Your local council wants to study the reading habits of citizens. Design a statistical study plan indicating:

• What the population would be.

• A suitable method for choosing a representative sample of $200$ people, avoiding surveying only people leaving the municipal library.

• Two qualitative statistical variables and two quantitative variables you would include in the questionnaire.

The money a person carries in their purse or wallet is measured in euros and cents (for example, $12,50\text{ \euro}$). Discuss whether this statistical variable should be considered discrete quantitative or continuous quantitative. Provide arguments in favour of both classifications and conclude which is the most mathematically rigorous.

In a city of $120\,000$ inhabitants, a telephone survey is conducted with $600$ people.

• What percentage of the total population was surveyed?

• It is known that $15\,\%$ of the city's inhabitants are young people aged between $18$ and $25$ years old. For the sample of $600$ people to be representative regarding age, how many young people in that age group should be included in the sample?

An athletics club has $1\,500$ registered athletes distributed across three categories according to their age: U14 ($600$ athletes), U16 ($525$ athletes), and U18 ($375$ athletes). The board of directors wishes to select a representative sample of $120$ athletes using stratified sampling with proportional allocation to consult them about the new training track schedules. Without performing exact calculations yet, roughly estimate what percentage of the total sample should correspond to the U14 category and explain between which two whole tens the number of selected U14 athletes should lie. Next, calculate with mathematical precision how many athletes must be chosen from each of the three strata. Finally, check whether the exact value for the U14 category fits your previous estimate and verify numerically that the sum of the three groups equals the total size required for the sample.

In a software development company with $500$ employees distributed across three offices (Madrid, Barcelona, and Valencia), a stratified sampling with proportional allocation has been carried out for an employee satisfaction survey, selecting a total sample of $80$ people. When reviewing the study file, the original census data for each office were deleted, but the extracted sample sizes have been preserved: $44$ employees from Madrid and $24$ from Barcelona participated in the sample. Determine how many employees from Valencia were part of the sample and reconstruct the total number of employees ($N_i$) that the company has in each of the three offices, justifying each step mathematically. Finally, if instead of stratified sampling it had been decided to apply systematic sampling to the complete list of the $500$ employees to obtain the $80$ respondents, calculate the interval $k = N/n$, interpret why this quotient is not an integer, and reflect on what practical difficulty this poses for the researcher.

A municipality is divided into three districts: the North district has $4\,500$ inhabitants, the South district has $2\,700$ inhabitants and the East district has $1\,800$ inhabitants. The council needs to select a sample of $200$ people using stratified sampling with proportional allocation. Calculate step by step how many respondents must be selected from each of the three districts so that the sample is representative.

A newspaper publishes the following headline: "$80\,\%$ of traffic accidents occur within $20\text{ km}$ of the driver's home. Therefore, driving near home is much more dangerous than making long journeys". Critically analyse this statement from a statistical perspective. Identify the flaw in the interpretation of the data and explain what additional information about the driver population would be needed to draw a valid conclusion.

In a cinema auditorium there are $1\,200$ seats numbered from $1$ to $1\,200$. The manager wants to survey $40$ spectators.

• Explain how you would select the sample using systematic sampling. If the first seat chosen is seat $17$, which will be the last seat in the sample?

• If the seats are divided into Stalls ($800$ seats) and Circle ($400$ seats), explain how you would make the selection using proportional stratified sampling.

In a health study, the weight (in $\text{kg}$) and height (in $\text{m}$) of individuals are measured. From these data, the Body Mass Index (BMI) is calculated using the formula $\text{BMI} = \frac{\text{Weight}}{\text{Height}^2}$. Subsequently, patients are classified into three categories: "Underweight" (BMI $< 18,5$), "Normal weight" (BMI between $18,5$ and $24,9$) and "Overweight" (BMI $\geq 25$). Identify and classify the type of statistical variable in each phase of the study: weight, height, the numerical value of BMI, and the final category assigned to the patient.

In a secondary school of $800$ students, $60\,\%$ are girls. In addition, it is known that $25\,\%$ of the school's total students are in 4th ESO. We wish to select a representative sample of $40$ students using stratified sampling that takes into account both sex and year group simultaneously. Assuming that the proportion of girls in 4th ESO is the same as in the rest of the school, calculate how many girls in 4th ESO should be included in the sample.

During a military conflict, military engineers analysed the bomber planes that returned to base with hits from enemy fire. They observed that most of the impacts were on the wings and fuselage, so they proposed reinforcing the armour in those areas. A statistician opposed this decision, arguing a serious bias in the sample. Identify what the total population of planes was, what the analysed sample was, and explain statistically why the engineers' conclusion was flawed (what is known as "survivorship bias").

The council of a large city is evaluating the construction of a new cycle lane that would eliminate parking spaces for cars. Propose three sample design or data collection methods that would yield intentionally biased results: one that favours "Yes" to the cycle lane, another that favours "No", and explain why the bias occurs in each case. Finally, propose a rigorous method to obtain a truly representative sample of the city's opinion.

Two polling companies want to predict the outcome of a national election. Company A places a survey on the homepage of a well-known online sports newspaper and gets $50\,000$ voluntary responses. Company B makes phone calls to $1\,500$ people selected using stratified random sampling based on age, sex and electoral roll province. Argue in detail, based on the concepts of population, sample, size and representativeness, which of the two surveys will provide a more reliable prediction despite the huge difference in sample size.

An association of municipalities with $18\,000$ working-age inhabitants commissions a study on the impact of remote working in the region. The active population is distributed across three economic sectors: Services ($10\,800$ people), Industry ($5\,400$ people), and Agriculture ($1\,800$ people). The technical team has resources to survey $n = 600$ people and is considering two methodological proposals: Proposal 1 consists of applying stratified sampling with proportional allocation by employment sector using the municipal register; Proposal 2 consists of sharing an open questionnaire on local social media and regional news portals until the first $600$ responses received are recorded. Model mathematically the sample for Proposal 1 by calculating how many workers must be selected from each sector and show that the sample proportion of each stratum matches its actual weight in the population. Subsequently, evaluate Proposal 2 by identifying at least two severe selection biases based on the employment sectors analysed, and argue why, in statistics, a large sample does not correct for the shortcomings of a biased sampling design.

In a logistics distribution warehouse, pallets of $N = 1\,400$ packages moving along a continuous automated conveyor belt are inspected. The boxes are loaded onto the belt in a cyclical, repeating order of $7$ product types: the 1st is fragile, the 2nd textile, the 3rd food, the 4th electronics, the 5th footwear, the 6th household cleaning, and the 7th stationery (and the 8th is fragile again, always repeating the same sequence). The quality inspector must draw a sample of $n = 50$ packages from a pallet and is evaluating two methods: Method A (systematic sampling with interval $k = N/n$ and random start $a$ between $1$ and $k$) and Method B (stratified sampling with proportional allocation considering each product type as a stratum). Calculate the interval $k$ for Method A and determine the position numbers of the first four packages that would be inspected if the resulting random start were $a = 15$. Prove what type of goods absolutely all the packages selected with that start in Method A would contain, and explain what periodic phenomenon ruins the representativeness. Finally, compare both methods and justify conclusively which one must be chosen to ensure rigorous quality control.