Population and samples：練習問題

A researcher intends to take a simple random sample of size \(n\) from a finite population. Which of the following statements must be true for the sample to be considered a simple random sample?

A researcher wishes to obtain a random sample of size \(n\) from a finite population. According to the Pearson Edexcel Statistics specification, which of the following is a necessary and sufficient condition for a sample to be considered a simple random sample?

A marketing firm uses quota sampling to survey 200 people about a new product. They instruct interviewers to find 100 men and 100 women, with 50 in each group being over the age of 40.

Which of the following statements best describes the difference between this quota sample and a stratified random sample with the same proportions?

A large college has 2,400 students: 1,400 are studying STEM subjects and 1,000 are studying Humanities. A researcher wants to take a 5% stratified sample based on the faculty. How many STEM students should be included, and what is a primary advantage of using proportional stratification in this context?

A factory produces 10,000 lightbulbs a day in 20 different batches. A quality control manager decides to select 2 batches at random and test every lightbulb in those selected batches.

Which sampling method is being used, and what is a specific risk associated with this method regarding the population variance?

Explain the difference between a simple random sample and an unrestricted random sample in terms of the sampling process.

A local charity wants to survey its volunteers about a new policy. The database of volunteers is indexed by ID numbers from 001 to 850. The researcher uses a random number generator to select a starting point between 001 and 017, and then selects every \(50^{th}\) volunteer on the list.

Identify this sampling technique and explain one specific practical constraint that might prevent this method from being used if the charity only has access to a paper filing cabinet ordered by the date the volunteers joined.

A large corporation has 5,000 employees distributed across three branches: London (3,000), Manchester (1,500), and Cardiff (500). A researcher intends to take a sample of 200 employees using disproportional stratified sampling to ensure the Cardiff branch has at least 40 representatives.

Calculate the sampling fraction for the Cardiff branch and explain one statistical disadvantage of using disproportional ratios compared to proportional stratification when estimating the mean company-wide salary.

A large international bank wishes to investigate the spending habits of its 500,000 customers. The bank’s database lists customers alphabetically by their last name. A researcher decides to select a sample of 2,000 customers by choosing every 250th name on the list, starting from a randomly selected position between 1 and 250.

(a) Identify the sampling technique being used by the researcher.

(b) Explain whether this sampling technique satisfies the requirements of a simple random sample as defined in the Pearson Edexcel Statistics specification.

(c) The researcher later discovers that certain surnames (e.g., those starting with 'Mc' or 'O'') are clustered in specific geographic regions. Discuss one potential bias that could arise from using this sampling method if there is a periodic pattern in the customer list that aligns with the sampling interval.

A wildlife conservationist is conducting a study on the health of a rare species of mountain goat. The total population is estimated to be 1,200, spread across three distinct mountain ranges: Range A (600 goats), Range B (400 goats), and Range C (200 goats).

(a) The conservationist intends to take a stratified sample of 120 goats using proportional allocation. Calculate the number of goats that should be sampled from each range.

(b) Due to the extreme terrain in Range C, the conservationist finds it nearly impossible to locate goats without assistance. They decide to capture one goat in Range C, tag it, and then use local tracking data from that goat's herd to find other goats. Identify this specific non-random sampling technique and justify its use in this context.

(c) Compare the use of stratified sampling with cluster sampling for this study. Specifically, explain which method would be more appropriate if the health characteristics are very similar within each mountain range but differ significantly between the three ranges.