Exercises: Sampling

A well-known streamer wants to find out his followers' opinion about the new design of his channel. To do so, he selects the first $250$ users who write in the chat during a live stream and sends them a form.

In this context, identify the population under study and the sample selected. Do you think the sample is representative of all his followers? Justify your answer.

A company manufacturing screen protectors for mobile phones wants to test the resistance of its new "Gorilla-X" model. To do this, it subjects the screens to increasing pressure until they break. Explain with reasons why in this case it is strictly necessary to resort to sampling rather than conducting a census of the entire production.

In an Instagram giveaway with $1200$ participants, a random sample of $5$ winners is to be drawn using simple random sampling. If we assign each participant a number from $1$ to $1200$, use the following list of three- and four-digit random numbers to determine the position numbers of the $5$ winners:

$0845, 1520, 0023, 3412, 0911, 1199, 0567$

A music festival has a list of $4000$ festival-pass buyers ordered by purchase date. A satisfaction survey is to be conducted on a sample of $100$ people using systematic random sampling.

1. a) Calculate the sampling interval $h$.
2. b) If the first element chosen at random is number $27$, state the position numbers of the next three selected for the sample.

A secondary school has $600$ students divided into three levels:

• Junior Cycle: $350$ students.

• Leaving Certificate: $200$ students.

• Post-Leaving Certificate (PLC) courses: $50$ students.

A sample of $60$ students is to be obtained using stratified random sampling with proportional allocation. Calculate how many students should be selected from each educational level for the sample to be proportional.

To study the sleep hours of teenagers in Ireland, a psychologist decides to interview $100$ young people leaving a nightclub at $6:00$ in the morning on a Sunday. Do you think this method of selecting the sample is appropriate for obtaining valid conclusions about the general teenage population? Identify the potential bias in this choice.

In an e-sports club there are $1500$ members. It is known that $60\%$ are PC gamers, $30\%$ are console gamers and $10\%$ are mobile gamers. A sample of $120$ members is to be chosen to test a new peripheral using proportional stratified sampling. Determine the number of members that will make up each stratum in the sample.

A public library has its books registered in a database numbered from $1$ to $12500$. The librarian wants to inspect the condition of the books using systematic sampling of size $n = 250$.

1. a) Determine the value of the sampling interval $h$.
2. b) If the first book selected is number $14$, what will be the number of the book that occupies position $10$ in the selected sample?

A secondary school has $N = 960$ enrolled students. Cian, a 6th Year student, wants to conduct a survey on reading habits by selecting a sample of $n = 40$ students using systematic random sampling. In his notebook, he writes the following procedure:

«I calculate the sampling interval: $h = 960 / 40 = 24$. Since the sample must be random, I choose a number at random between $1$ and $960$, getting $850$. Starting from it, I select students $850, 874, 898, 922, 946$, and since the list ends at $960$, I start over from the beginning by adding $24$ to the first numbers on the list until completing the $40$ students».

1. a) Identify the conceptual error made by Cian regarding the random start and explain why his method violates the definition and properties of systematic sampling.
2. b) State the exact interval in which the first individual should have been selected and write down the numbers of the first four students that would make up the sample if the random start had been the number $11$.

The public library network of a municipality has three branches serving populations of different sizes: the Central Library has $15\,000$ registered users; the North Library, $4\,500$ users; and the South Neighbourhood Library, $750$ users. The department of culture is planning a survey of $n = 300$ users to assess general satisfaction with the digital catalogue and is considering two options:

• Method A: Stratified sampling with equal allocation ($100$ users chosen randomly from each branch).

• Method B: Stratified sampling with allocation proportional to the size of each branch.

1. a) Calculate how many users should be surveyed in each library if Method B is applied (round to the nearest integer if necessary).
2. b) If the primary goal is to obtain an overall, unbiased estimate of user satisfaction across the municipality, justify which method is preferable. What serious distortion would using Method A cause in the overall mean, and in what specific situation would it make sense to use Method A?

A streaming platform wishes to conduct a study on documentary consumption among its subscribers in a city. The subscriber population is divided into three districts: the North District has $12\,500$ users, the Central District has $18\,200$ and the South District has $9\,300$.

If it is decided to take a sample of $800$ people using stratified random sampling with proportional allocation, how many subscribers should be selected from each district?

The organisers of an electronic music festival have issued $4\,500$ digital tickets numbered sequentially from $1$ to $4\,500$. To carry out a security and satisfaction check, a sample of $75$ attendees is to be selected using systematic random sampling.

1. a) Calculate the sampling interval $h$.
2. b) If the first attendee randomly selected is the one with ticket number $28$, state the numbers of the next three tickets that will be part of the sample.

A social media analyst wants to study the impact of a viral video that has received $150\,000$ comments. To avoid analysing all of them, they decide to draw a sample of $120$ comments. They propose two methods:

1. Analyse the first $120$ comments that appear on the post.

2. Assign a number to each comment and use a random number generator to choose the $120$.

The following diagram represents the population of comments (each square stands for a group of them) and the ones each method chooses.

Justify which of the two methods is more appropriate from a statistical point of view and explain what bias problem the method you have rejected has.

In a sports club there are $1\,200$ members. It is known that $60\%$ are men and $40\%$ are women. Furthermore, $20\%$ of the members are minors, and the rest are adults. It is desired to choose a sample of $150$ members for a health survey.

If stratified random sampling with proportional allocation is used considering only sex (man/woman):

1. a) How many men and how many women should make up the sample?
2. b) If it is later decided that the sample should also be proportional to age (minors/adults), how many adults should be selected in total for the sample of $150$ people?

A shopping mall wants to carry out a study on the average spending per customer on the last Saturday of the month. It is estimated that $8\,400$ customers will come, numbered from $1$ to $8\,400$ in order of arrival. The aim is to choose a sample of $140$ people using systematic sampling.

1. a) Determine the selection interval.

1. b) If chance picks customer number $15$ as the starting point (first element of the sample), what will be the order number of the customer in position $50$ of the sample collected?

A cosmetics company has $3$ factories in Ireland: one in Dublin with $450$ employees, another in Cork with $350$ and another in Galway with $200$. A sample of $100$ employees is to be taken for a study on work-life balance.

Compare the number of employees that would be selected from the Galway factory if using:

1. a) Stratified random sampling with equal allocation.
2. b) Stratified random sampling with proportional allocation.

It is desired to study the level of microplastic pollution on a $2$-kilometre-long beach. The researchers divide the beach into $200$ equal plots, each $10$ metres wide. For the study, they choose a plot at random from the first $10$ and, from that point on, select a plot every $10$ positions.

1. a) Identify the type of sampling used.
2. b) If the first plot chosen is number $7$, write the numbers of the plots that would occupy the $10$ and $20$ positions in the sample.
3. c) Is this a probability sampling method? Justify your answer.

A secondary school has $600$ students distributed as follows: $1^{\circ}$ Year Bachillerato ($180$ students), $2^{\circ}$ Year Bachillerato ($140$ students) and ESO ($280$ students). A sample of $60$ students is to be obtained using proportional stratified random sampling.

1. a) Determine the number of students from each educational level that should be selected.
2. b) If within the $2^{\circ}$ Year Bachillerato stratum it is decided to select the sample students systematically from an alphabetical list, what will be the sampling interval for that specific stratum?

According to a survey by the Central Statistics Office (CSO), the daily time that Irish young people aged 16 to 24 spend on social media follows a distribution with mean $\mu = 160$ minutes and standard deviation $\sigma = 40$ minutes. A sociology team selects a simple random sample of $n = 100$ young people from that age group.

1. a) Estimate: Before working with tables, calculate the standard error of the mean ($\sigma_{\bar{X}} = \sigma / \sqrt{n}$) and reason how many standard deviations away from the population mean the value of $168$ minutes is. Based on this, make an intuitive, qualitative estimate of whether the probability $P(\bar{X} \ge 168)$ will be high (greater than $40\%$), moderate (between $10\%$ and $40\%$), or very low (less than $5\%$).
2. b) Check: Formally justify the probabilistic model followed by the sample mean $\bar{X}$ and calculate the probability $P(\bar{X} \ge 168)$ precisely.
3. c) Compare the exact numerical result with your initial estimate and assess whether the observed behaviour is consistent with the properties of the sampling distribution.

A report on urban mobility in the Greater Dublin Area indicates that $65\%$ of daily journeys are made by private vehicle ($p = 0.65$). The National Transport Authority takes a random sample of $n = 200$ commuters on a working day and analyses the sample proportion $\hat{p}$ of private vehicle users.

1. a) Model the sampling distribution of $\hat{p}$: explicitly check whether the conditions to approximate it using a normal distribution are met and specify the parameters (mean and standard deviation, rounding the latter to four decimal places) of that distribution.
2. b) Calculate the probability that, in the sample, the proportion of people using private vehicles is between $60\%$ and $70\%$ (both inclusive). Apply the continuity correction to either the number of people or the proportion in your calculation.
3. c) In light of the result, argue whether it would be plausible or suspicious if an external audit of $200$ randomly selected commuters found fewer than $55\%$ private vehicle users.

In a prestigious training academy for professional esports players, students are divided into three categories based on their skill level: Beginner, Advanced, and Elite. It is known that the number of Beginner students is three times that of Elite students, and that Advanced-level students represent $40\%$ of the academy's total.

To conduct a study on cognitive performance, it is decided to draw a sample using proportional stratified random sampling. If after making the selection it is observed that exactly $18$ Elite-level students have been chosen:

1. a) Determine the size of the selected sample.
2. b) How many students from the Beginner and Advanced levels make up the selected sample?

An automated logistics company uses a systematic random sampling system to inspect the quality of packages leaving its warehouse. Packages are arranged on a conveyor belt and the daily production is known to be $4500$ units. A sample of $125$ packages is to be inspected.

1. a) Determine the sampling interval $h$ and explain the selection process if the first element randomly chosen is number $14$.
2. b) Critical analysis: If packages are produced in cycles of $36$ units per machine shift, and it is suspected that the last unit of each cycle often has seal defects, explain whether the systematic sampling designed in part a) is suitable for detecting this problem or if it could introduce a significant bias into the results.

The daily time that teenagers aged between 16 and 18 spend on a popular short-video social network follows a normal distribution with mean $\mu = 110$ minutes and standard deviation $\sigma = 25$ minutes.

A simple random sample of $64$ young people is selected. Calculate the probability that:

1. a) The mean daily time spent using the social network in this sample is greater than two hours.
2. b) The total sum of the daily times of the $64$ young people in the sample is less than $112$ hours.

A smartphone battery manufacturing company claims that the lifespan of its products follows a normal distribution with a standard deviation of $120$ charge cycles. A consumer laboratory suspects that quality has dropped and takes a random sample of $100$ batteries, obtaining a sample mean of $1180$ cycles.

If the probability of obtaining a sample mean less than or equal to $1180$ cycles is only $0.0062$ (assuming the company's claim about the standard deviation is true):

1. a) What is the mean lifespan $\mu$ that the company claims in its advertising?
2. b) If the laboratory decides to double the sample size to gain precision while maintaining the same sample mean, how would the above probability change? Justify your answer without needing to carry out all the calculations.

It is desired to study the proportion of households in a metropolitan area that subscribe to more than three streaming platforms. No prior data is available, so the worst-case scenario is assumed for calculating the sample size ($p=q=0.5$).

1. a) Calculate the minimum sample size required to estimate this proportion with a maximum error of $4\%$ and a confidence level of $96\%$.
2. b) If, after conducting the study with a sample of $200$ households, it is found that $84$ of them satisfy the condition, calculate the probability that, in another different sample of equal size, the sample proportion is greater than $0.45$.

In a study on university students' monthly leisure expenditure, a simple random sample of $100$ students is taken. It is known that the variable "monthly expenditure" has a population standard deviation of $\sigma = 30$ euros.

1. a) Calculate the probability that the modulus of the difference between the sample mean and the population mean is less than $5$ euros.
2. b) If we want the probability calculated in part a) to increase to $0.99$, by how much should we increase the sample size?

A health centre claims that $25\%$ of its regular patients follow a vegetarian diet. To test this claim, a consumers' association selects a sample of $150$ patients.

1. a) Describe the distribution of the sample proportion $\hat{p}$ of vegetarian patients if the health centre's claim is true.
2. b) Using the normal approximation (with continuity correction), calculate the probability that there are between $30$ and $45$ vegetarian patients in the sample (inclusive).

We want to compare the mean spending on mobile applications between users of two different operating systems, System A and System B. It is known that spending on System A follows a $N(\mu_A, 12)$ distribution and on System B a $N(\mu_B, 15)$ distribution.

Two independent samples are taken: one of $40$ System A users and another of $50$ System B users.

1. a) Describe the distribution of the difference between sample means $\bar{X}_A - \bar{B}_B$.
2. b) Assuming that the population mean spending is the same for both systems ($\mu_A = \mu_B$), calculate the probability that the sample mean for System A exceeds that for System B by more than $4$ euros.

In a public university with $N = 12\,000$ students, the population is divided into three faculties: Humanities ($H$), Health Sciences ($S$), and Social Sciences ($C$). A study using stratified sampling with proportional allocation has been designed, but part of the original records have been lost. Only the following confirmed information is available:

1. The number of students enrolled in Social Sciences is exactly twice that of Humanities ($N_C = 2\times N_H$).

2. The Faculty of Health Sciences has $N_S = 3\,600$ enrolled students.

3. In the final selected sample, the Social Sciences stratum contains exactly $70$ more students than the Humanities stratum ($n_C - n_H = 70$).

1. a) Algebraically reconstruct the full population composition of the university by calculating the size of each stratum ($N_H$, $N_S$, and $N_C$).
2. b) Determine the total sample size $n$ that was drawn and the exact number of students allocated to each faculty ($n_H$, $n_S$, and $n_C$).
3. c) The dean of Health Sciences requests expanding the research and demands that, in a new sampling that strictly maintains proportional allocation relative to the entire university, her faculty has at least $150$ surveyed students to ensure internal representativeness. Calculate what the minimum overall size $n'$ of this new university sample must be, and how many students in total would be surveyed between the other two faculties.

The emergency department of a local hospital treats an average of $N = 600$ patients daily. From historical accumulated records, the waiting time until first medical attention is known to have a mean of $\mu = 140$ minutes and a standard deviation of $\sigma = 36$ minutes. The medical management wishes to audit the service quality by drawing a daily sample of $n = 50$ patients and is evaluating three alternative sampling designs:

• Design 1: Systematic sampling selecting 1 patient out of every $12$ according to their arrival order over the 24 hours of the day ($h = 600 / 50 = 12$).

• Design 2: Convenience or accidental sampling by surveying the first $50$ patients arriving at the emergency department starting at 08:00 in the morning.

• Design 3: Simple random sampling without replacement from the full computerised record of the day's $600$ patients.

1. a) Critically analyse Design 2: explain why it does not meet the criteria of a probability sampling and identify two specific selection biases that would prevent extrapolating its conclusions to the overall operation of the emergency department.
2. b) For Design 3, calculate the standard error of the sample mean $\sigma_{\bar{X}}$ in two ways: assuming an infinite population ($\sigma / \sqrt{n}$) and applying the finite population correction factor $\sqrt{\frac{N - n}{N - 1}}$. Calculate the percentage reduction introduced by the correction factor and justify whether it is methodologically necessary to apply it in this case based on the sampling fraction $n / N$.
3. c) Compare Design 1 with Design 3: if patient influx and severity presented a cyclical pattern linked to medical staff shift changes every 8 or 12 hours, explain what specific statistical risk would invalidate systematic sampling compared to simple random sampling.