Question

a simple random sample of n measurements from a population is a subset of the population selected in a manner such that which of the following is/are true? (select all that apply.)
every sample of size n from the population has an equal chance of being selected.
each subset of the population has an equal chance of being included in the sample.
the simplest method of selection is used to create a representative sample.
every sample of size n from the population has a proportionally weighted chance of being selected.
every member of the population has an equal chance of being included in the sample.

Explanation:

• For "Every sample of size \(n\) from the population has an equal chance of being selected": This is a fundamental property of simple random sampling. In simple random sampling, all possible samples of size \(n\) have the same probability of being chosen.
• For "Each subset of the population has an equal chance of being included in the sample": This is not a correct statement. In simple random sampling, we focus on samples (sub - sets of size \(n\)), not all subsets (which can be of various sizes).
• For "The simplest method of selection is used to create a representative sample": While simple random sampling is a relatively straightforward method, the key is about equal probability of sample selection, not just simplicity for representativeness. Representativeness is a consequence (if the sample is large enough) of the equal - probability sampling, not the simplicity of the method.
• For "Every sample of size \(n\) from the population has a proportionally weighted chance of being selected": This is incorrect. In simple random sampling, it is an equal - chance (not proportionally weighted) selection for samples of size \(n\).
• For "Every member of the population has an equal chance of being included in the sample": If every sample of size \(n\) has an equal chance of being selected, then every member of the population has an equal chance of being in the sample. For example, if we have a population of size \(N\) and we are taking a sample of size \(n\), the probability that a particular element is in a randomly selected sample of size \(n\) is \(\frac{n}{N}\) (derived from combinatorial counting of the number of samples that include the element divided by the total number of samples of size \(n\)).

Answer:

• Every sample of size \(n\) from the population has an equal chance of being selected.
• Every member of the population has an equal chance of being included in the sample.