For a population of 4 members, taking a sample of size 2 with replacement, how many possible samples are there?
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Answer: C. 16
Explanation
Sampling with replacement means every member has an equal chance of selection every single time. Raising the population size to the power of the sample size gives total possible outcomes. Calculating four raised to the power of two yields exactly sixteen possible sample combinations. Combinatorics provides formulas to count possible outcomes based on whether replacement is allowed. The notation N represents population size while lowercase n denotes the chosen sample size. Multiplication principles dictate that independent sequential selections multiply their individual choices together.
Polling organizations use sampling replacement models when simulating repeated voter preference surveys. Computer scientists rely on these counting rules to determine password combination spaces and security. Geneticists calculate possible allele combinations using similar exponential sampling formulas in laboratory studies. Choosing four simply reflects the population size rather than the total sample space permutations. Selecting eight represents an incorrect arithmetic addition instead of the required exponential calculation. Total thirty two incorrectly multiplies population and sample size rather than applying proper powers.
- Subject
- Mathematics
- Category
- Statistics
- Topic
- —
- Difficulty
- medium
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