Populations, Samples, and Sampling Methods

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

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

A software company wants to know its users' opinions on a new update of its application. Briefly explain in what situation the company would carry out a census and in what situation it would carry out a sample study.

$3400$ deer live in a nature reserve. A team of biologists decides to take a sample equivalent to $15\%$ of the population to carry out a health study. Calculate the sample size they must select.

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

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

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

• The fur color of a fox species.

• The number of natural satellites of the planets in the solar system.

• The mass in pounds of postal shipments from a parcel company.

• The brand of the computers in a computer lab.

State whether the following quantitative variables are discrete or continuous:

1. a) The volume of water in a reservoir measured in cubic feet.
2. b) The number of heartbeats 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.

A team of nutritionists measures the amount of sugar, in grams, contained in different brands of cereals from a supermarket. Identify what the statistical variable of the study is and what type it is.

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 feet.
3. c) Number of offspring it has had throughout its life.
4. d) Extinction risk level (Low, Vulnerable, Endangered).

Classify each statistical variable in detail. In the case of qualitative variables, state explicitly whether they are ordinal or nominal.

In an astronomical observatory, the following characteristics of an exoplanet catalog are recorded:

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

B) Planet type (Rocky, Gas giant, Ice giant).

C) Number of confirmed moons.

D) Estimated mean temperature (in Kelvin).

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

In a high school there are $120$ students in 9th grade, $100$ in 10th grade, $90$ in 11th grade, and $50$ in 12th grade. A sample of $72$ students is to be selected using stratified random sampling with proportional allocation. Calculate how many 11th-grade students must 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 extracted using stratified sampling with proportional allocation. Determine the sample size that corresponds to each of the three sheds.

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 is $160$ individuals, determine how many individuals of each color make up the sample.

In a municipality divided into three districts (North, South and Center), a survey has been carried out using stratified sampling with proportional allocation. The total sample is $300$ people, of which $120$ 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.

In a technology 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 factory has two machines, A and B, for the production of gears. It is known that machine A produces $65\%$ of the daily total. The quality department takes a stratified sample with proportional allocation of $180$ gears to evaluate the cutting precision. If machine B produces $1400$ gears a day, calculate the total daily population size 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, whose workforce sizes are given by algebraic expressions in terms of $x$. A sample is taken using stratified random sampling with proportional allocation of size $130$.

Knowing that there are exactly $50$ 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 the sample size corresponding to the Marketing and Development departments.

On an assembly line, $1200$ cell phones have been manufactured in one day. A quality control 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 occupying position $14$, determine the sampling interval, and the positions of the first three and the last phone selected in the sample.

A database contains the records of $N = 2520$ clients. It is desired to extract a sample of size $n = 45$ using systematic sampling. The random start determines that the first client selected is the one occupying 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 what position in the database is occupied by the client who was selected in the $32$-th place in the sample.

The educational authorities of a region want to assess the English level of 10th-grade students. In the region there are $25$ high schools, each with exactly $120$ students enrolled in that year. A total sample of $600$ students is required. Two sampling methods are proposed:

• Method A: Choose $5$ high schools at random and assess all their 10th-grade students.

• Method B: Choose $24$ 10th-grade students at random from each of the $25$ high 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.

A city council wants to know citizens' opinions on the construction of a new bike lane downtown. To do this, a pollster stands outside a bicycle shop and interviews $150$ people leaving it. Analyze whether the selected sample is representative of the city's population and explain what type of bias is being committed in the study design.

It is desired to estimate the mean daily time spent on social media by teenagers across an entire state. To do this, $200$ students are randomly selected from a private high school located in the most affluent neighborhood of the capital city. Justify why this sample is not representative of the population and propose a more appropriate sampling method.

A national television channel wants to predict the outcome of an important legislative referendum. To do this, during a prime-time debate program, 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 probably not representative of the general population.