Exercises: Population and sample

An electronic components factory produces a batch of $5000$ resistors. To check their quality, the quality control department selects and analyses $150$ resistors. Identify the population, the sample, the individual and the sample size in this statistical study.

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

• The coat colour of a species of fox.

• The number of natural satellites of the planets in the Solar System.

• The mass in kilograms of postal parcels from a courier company.

• The brand of computers in an IT room.

A study on reading habits is to be carried out in a municipality with $24000$ inhabitants. To do this, $800$ citizens are interviewed. Determine the study population, the sample and the sample size.

A software company wants to know its users' opinions about a new app update. Briefly explain in which situation the company would conduct a census and in which situation it would conduct a sample survey.

A nature reserve is home to $3400$ deer. A team of biologists decides to take a sample equal to $15\%$ of the population to carry out a health study. Calculate the size of the sample they should select.

State whether the following quantitative variables are discrete or continuous:

1. a) The volume of water in a reservoir measured in cubic hectometres.
2. b) The heart rate (number of beats per minute) of an athlete after a race.
3. c) The maximum daily temperature recorded in a city.
4. d) The number of passengers on a commercial flight.

To carry out quality control in a soft drinks factory, $45$ cans are taken from a total daily production of $1800$ cans. What percentage of total production does the selected sample represent?

A team of nutritionists measures the amount of sugar, in grams, contained in different cereal brands at a supermarket. Identify the statistical variable of the study and state its type.

Carlos wants to conduct a survey on recycling habits at his secondary school, where $600$ students are enrolled across three educational stages: ESO ($360$ students), Bachillerato ($180$ students), and Vocational Training ($60$ students). To carry out his Statistics project, he must select a sample of $60$ students. Carlos states in his report: "Since there are $3$ educational stages, for the study to be fair and representative I must survey exactly $20$ students from each one, since $60 : 3 = 20$".

1. a) Identify and explain in detail the reasoning error that Carlos has made.
2. b) Calculate how many students from each stage should actually be chosen if stratified random sampling with proportional allocation is applied.

During the month of July, the Directorate-General for Traffic (DGT) launches an intensive campaign to monitor the use of seat belts and child restraint systems on the secondary roads of an autonomous community. During that week, patrols stop and inspect a total of $14\,400$ vehicles out of the approximately $480\,000$ planned journeys. At each checkpoint, officers record whether the occupants are wearing seat belts in accordance with regulations (yes/no) and the total number of occupants travelling in the vehicle.

1. a) Identify precisely the population, the sample, an individual, and the sample size $n$ of this study.
2. b) Justify why the DGT resorts to a sample rather than conducting a census in this type of roadside campaign.
3. c) Classify the two statistical variables analysed in each vehicle according to their typology (qualitative nominal/ordinal or quantitative discrete/continuous).

In a secondary school there are $120$ students in 1st ESO, $100$ in 2nd ESO, $90$ in 3rd ESO and $50$ in 4th ESO. A sample of $72$ students is to be selected using stratified random sampling with proportional allocation. Calculate how many 3rd ESO students should be part of the sample.

A city council wants to know the opinion of citizens regarding the construction of a new cycle lane in the city centre. To do this, a surveyor stands at the entrance of a bicycle shop and interviews $150$ people leaving it. Analyse whether the selected sample is representative of the city's population and explain what type of bias is being introduced in the design of the study.

On an assembly line, $1200$ mobile phones are manufactured in one day. A quality control check is to be carried out by selecting a sample of $40$ phones using systematic sampling. If the first phone selected at random is the one in position $14$, determine the sampling interval, and the positions of the first three phones and the last phone selected in the sample.

A poultry farm has three sheds with laying hens: shed A has $4500$ hens, shed B has $3000$ and shed C has $2500$. A sample of $200$ hens is to be selected using stratified sampling with proportional allocation. Determine the sample size for each of the three sheds.

It is desired to estimate the mean daily time spent on social media by teenagers across an entire autonomous community. To do this, $200$ students from a private secondary school located in the wealthiest neighbourhood of the capital are selected at random. Justify why this sample is not representative of the population and suggest a more suitable sampling method.

In a marine ecology study, the following characteristics of a dolphin population are recorded:

1. a) Species to which it belongs.
2. b) Body length in metres.
3. c) Number of calves it has had in its lifetime.
4. d) Extinction risk level (Low, Vulnerable, Endangered).

Classify each statistical variable in detail. For qualitative variables, state explicitly whether they are ordinal or nominal.

A polling company has selected a sample of $630$ people for a market research study. Given that this sample represents exactly $3,5\%$ of the target population, calculate the total size of the study population.

The following diagram represents a stratified sampling process with proportional allocation.

If the total population consists of $2400$ individuals, calculate the percentage and the population size of the orange stratum. If the total sample size is $160$ individuals, determine how many individuals of each colour make up the sample.

According to data from the National Statistics Institute (INE), a rural district in Spain has a total population of $24\,000$ inhabitants, distributed across three age groups: young people aged 0 to 19 ($4\,800$ people), adults aged 20 to 64 ($14\,400$ people), and elderly people aged 65 or older ($4\,800$ people). A sample of $120$ inhabitants is to be selected using stratified sampling with proportional allocation for a study on district healthcare.

1. a) Before performing any exact calculations, make an intuitive estimate: reason what fraction or percentage of the total population the adult group (20 to 64 years) approximately represents, and estimate how many adults should be part of the sample of $120$ people.
2. b) Apply the formula for stratified sampling with proportional allocation ($n_i = n \cdot \frac{N_i}{N}$) to calculate the exact number of individuals that must be selected in each of the three age strata.
3. c) Check whether your initial estimate was accurate and assess whether the resulting distribution respects the representativeness of the district's population structure.

The Town Hall of a coastal municipality of $60\,000$ inhabitants wishes to conduct a public consultation to decide whether to pedestrianise the historic centre. For this purpose, a budget is available that allows for a sample of $600$ in-person or telephone surveys. In the municipal technical team, three different proposals arise to select the participants:

• Proposal 1: Station survey takers on Saturday morning at the doors of the three main supermarkets in the historic centre and interview the first $600$ people who leave.

• Proposal 2: Use the municipal register (padrón) ordered alphabetically (from 1 to $60\,000$). Calculate the systematic jump $k = 60\,000 / 600 = 100$, randomly choose a starting number $a$ between 1 and 100, and survey the residents at positions $a, a+100, a+200, \dots$

• Proposal 3: Open an online survey on the municipal website for one month and promote it via social media; they estimate that more than $3\,000$ residents will participate with this method at virtually no cost.

Reasonably analyse the advantages, disadvantages, and potential biases of each of the three options. Justify which of the methods is the most appropriate to guarantee that the sample is representative, and explain why a larger sample size (Proposal 3) does not by itself guarantee better representativeness.

In a municipality divided into three districts (North, South and Centre), a survey was carried out using stratified sampling with proportional allocation. The total sample size is $300$ people, of whom $120$ belong to the North district and $75$ to the South district. If the total population of the municipality is $18000$ inhabitants, calculate the population of each of the three districts.

A database contains the records of $N = 2520$ customers. A sample of size $n = 45$ is to be drawn using systematic sampling. The random start determines that the first customer selected is the one in position $a = 23$.

Find a general algebraic expression for the position of the $i$th individual in the sample (for $1 \le i \le 45$) and determine the position in the database of the customer who was selected $32$th in the sample.

In a tech company, the workforce is divided into three categories: $50\%$ are programmers, $30\%$ are designers and the rest are managers. A work climate study is carried out by selecting a sample using stratified sampling with proportional allocation. If there are exactly $16$ managers in the sample, what is the total sample size and how many programmers and designers make it up?

A national television channel wants to predict the result of an important legislative referendum. To do this, during a debate program in prime time, it asks viewers to call a premium-rate phone number or send an SMS to vote for their preferred option. More than $100000$ responses are received.

Identify and explain at least three sources of bias (systematic errors) in this sampling design and reason why, despite the huge sample size, the results are unlikely to be representative of the general population.

A factory has two machines, A and B, for producing gears. Machine A is known to produce $65\%$ of the daily total. The quality department takes a stratified sample with proportional allocation of $180$ gears to evaluate cutting precision. If machine B produces $1400$ gears per day, calculate the size of the total daily population and the number of gears from each machine that make up the sample.

In a study on the adoption of a new technology, the population of a company is divided into three departments: Sales, with $3x$ employees; Marketing, with $x + 40$; and Development, with $2x - 20$. A sample of size $130$ is taken using stratified random sampling with proportional allocation.

Knowing that there are exactly $60$ employees from the Sales department in the sample, set up an equation to determine the value of $x$, calculate the total population of the company, and find the sample size corresponding to the Marketing and Development departments.

At an astronomical observatory, the following characteristics of a catalogue of exoplanets are recorded:

A) Distance to the host star (in Astronomical Units).

B) Type of planet (Rocky, Gas giant, Ice giant).

C) Number of confirmed moons.

D) Estimated mean temperature (in Kelvin).

Rigorously classify each of these variables. For quantitative variables, determine in which cases it makes statistical sense to calculate the arithmetic mean, and in which cases the median might be a more representative measure of central tendency if extreme outliers existed.

The educational authorities of a region want to assess the English level of 4th ESO students. In the region, there are $25$ secondary schools, each with exactly $120$ students enrolled in that year group. A total sample of $600$ students is required. Two sampling methods are proposed:

• Method A: Choose $5$ schools at random and assess all their 4th ESO students.

• Method B: Choose $24$ 4th ESO students at random from each of the $25$ schools.

Identify the type of sampling applied in each method and argue which of the two designs guarantees greater representativeness of the total population, explaining the possible disadvantages of the method you discard.

In a district hospital with $450$ admitted patients distributed among the Surgery, Traumatology, and Internal Medicine wards, the medical management conducts a quality audit using stratified random sampling with proportional allocation on a total sample of $90$ patients.

In the final report, the initial census data by ward were misplaced, but the following reliable data remain:

• Exactly $36$ patients from Surgery and $24$ from Traumatology were included in the audited sample.

• To select the patients corresponding to Internal Medicine from their own internal list of admitted patients (ordered from 1 to $N_{\text{interna}}$), the team applied systematic sampling with a fixed jump $k$. In that list, the first patient randomly selected was in position $a = 3$, and the last patient selected from that same ward was in position $148$.

1. a) Deduce how many Internal Medicine patients were part of the sample of $90$ patients.
2. b) Knowing that the Internal Medicine patients occupy positions $a, a+k, a+2k, \dots$, set up the relationship between the first selected patient ($a = 3$), the last one ($148$), and the number of patients from that ward in the sample to calculate the jump $k$ and the total number of patients admitted to Internal Medicine ($N_{\text{interna}}$).
3. c) Using the proportionality condition ($n_i = n \cdot \frac{N_i}{N}$) and knowing the hospital's total population ($N = 450$), reconstruct how many patients were admitted to Surgery ($N_{\text{cirug\u00eda}}$) and how many to Traumatology ($N_{\text{trauma}}$).

Based on data from the INE's Economically Active Population Survey (EPA) for a Spanish province with $320\,000$ employed persons, the distribution by employment sector is as follows: Services with $224\,000$ workers, Industry with $48\,000$, Construction with $32\,000$, and Agriculture with $16\,000$. A social research institute has the budget to survey a sample of $n = 500$ workers.

A team member proposes: "If we apply strict proportional allocation, the sample of agricultural workers will be so small that it will be impossible to study their working conditions reliably. We should survey $125$ workers from each of the four sectors ($500 : 4 = 125$)". The study director replies that this distribution would not be valid for obtaining general data for the province.

1. a) Model the proportional sample by calculating the exact number of workers ($n_i$) that would correspond to each sector to respect provincial representativeness with $n = 500$.
2. b) Compare the actual distribution against the proposal of $125$ surveys: calculate the percentage that Agriculture represents in the actual employed population and the percentage it would represent in the sample proposed by the researcher. Perform the same comparative calculation for the Services sector.
3. c) Write a reasoned statistical justification: what distortion would the non-proportional sample of $125$ people per sector cause if one wanted to estimate the average salary of workers across the entire province? What methodological alternative would you propose to thoroughly study the agricultural sector without distorting the overall conclusions for the province?