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.

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.

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.

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.

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 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.

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 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.

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.

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.

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.