Question

what role does probability play in sampling?
a. it determines which sampling method to use
b. it helps in choosing the sample size
c. it allows researchers to make inferences from a sample to the whole population
d. it reduces the need for sampling

Explanation:

Probability sampling ensures that each element in the population has a known non - zero probability of being selected. This random selection allows researchers to use statistical theory. By applying probability concepts, they can calculate the likelihood that the sample statistics (like the sample mean) accurately represent the population parameters. For example, with a simple random sample (a type of probability sampling), the Central Limit Theorem (which is based on probability) can be used. It states that the sampling distribution of the sample mean will be approximately normal for a large enough sample size. This enables researchers to construct confidence intervals and perform hypothesis tests, which are ways of making inferences from the sample to the population.

Option a: The sampling method (e.g., simple random sampling, stratified sampling) is a design choice. While probability is involved in probability - based sampling methods, it's not that probability "determines" the method. The research question, nature of the population, etc., also play roles.

Option b: Sample size determination involves factors like the margin of error, confidence level, and an estimate of the population variance (in the case of estimating a population mean). Probability is related to the confidence level (e.g., a 95% confidence level corresponds to a certain probability in the normal distribution), but it's not the sole or main factor in choosing sample size. There are formulas \(n=\frac{z^{2}\sigma^{2}}{E^{2}}\) (for estimating a population mean) where \(z\) is related to probability (from the standard normal distribution), but other elements (\(\sigma\), \(E\)) are also crucial.

Option d: Probability is fundamental in sampling, especially in probability - based sampling. It does not reduce the need for sampling. Sampling is still required to study a population when it's not feasible to study every element, and probability helps in making the sampling process valid for inference.

Answer:

C. It allows researchers to make inferences from a sample to the whole population