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A finite population refers to a population with a limited number of elements from which samples can be drawn. In our exercise, the finite population is \(\{6, 1, 3, 2\}\), consisting of exactly four elements. When dealing with finite populations, it's important to understand that the characteristics derived from samples may be different from those of the population. This distinction occurs because not all possible variations are represented if the population size is small. Finite populations are typical in real-world scenarios, such as class sizes or small study groups.

• Limited number of elements.
• Possible variation between sample characteristics and population characteristics.
• Common in real-world finite systems.

We need to be careful when making conclusions based on samples from a finite population, as the sample might not perfectly replicate the population structure. The calculations of probabilities and means can be affected due to this limitation.

The sample mean is an average of the sample values and is a crucial component in statistics for estimating population parameters. To compute the sample mean from a sample, sum up all the sample items, then divide by the number of items in the sample.In the context of the given exercise with samples of size 2, the sample mean, denoted as \(\bar{x}\), is calculated by taking each pair of items from our population:

• (6,1): \(\bar{x} = \frac{6+1}{2} = 3.5\)
• (6,3): \(\bar{x} = \frac{6+3}{2} = 4.5\)
• (6,2): \(\bar{x} = \frac{6+2}{2} = 4\)
• (1,3): \(\bar{x} = \frac{1+3}{2} = 2\)
• (1,2): \(\bar{x} = \frac{1+2}{2} = 1.5\)
• (3,2): \(\bar{x} = \frac{3+2}{2} = 2.5\)

Calculating the sample mean helps us understand the distribution of data and can provide insights on the population mean, although it may not perfectly match it.

The population mean, represented as \(\mu\), is the average of all elements within a finite population, providing a single value that epitomizes the entire group.To find the population mean in the given exercise, you add all elements of the population and divide by the number of elements:\[\mu = \frac{6+1+3+2}{4} = \frac{12}{4} = 3\]

• Summarizes central tendency of the entire population.
• A key reference point for comparing sample means.

The population mean is incredibly valuable as it offers a baseline for comparison with sample means. In our problem, it helps determine whether samples align with or differ from the overall population tendencies.

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. This fundamental concept helps determine how many ways we can choose items from a set, particularly relevant in calculating the number of possible samples from a finite population.To calculate the number of possible samples without replacement from our given population of four elements, we use the formula for combinations:\[C(n,k) = \frac{n!}{k!(n-k)!}\]Applying this to our population of size 4, taking samples of size 2:\[C(4,2) = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6\]

• Helps in finding the number of ways to form samples.
• Uses factorials to determine combinations without regard to order.

Understanding combinatorics is key to solving problems involving finite populations, as it provides the groundwork for the possible arrangements or selections and is essential in calculating probability distributions based on sample data.