Introduction to statistics

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.

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.

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.

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.

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.

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 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.

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.

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.

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").

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.