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The combination formula is a mathematical way to find out how many different groups can be formed from a larger set. When you want to choose a group of items but the order doesn't matter, you use combinations. This is different from permutations, where the order does matter.

To calculate combinations, we use the formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where:

• \( n \) is the total number of items.
• \( r \) is the number of items you want to choose.
• The exclamation mark \(!\) represents a factorial, which means multiplying the number by all the positive integers below it.

Let’s consider the exercise: You have 5 tellers and you want to sample 2 of them. Plugging these values into the formula gives us \( \binom{5}{2} = \frac{5!}{2!(5-2)!} = 10 \). This means there are 10 possible ways to choose 2 tellers from a group of 5.

Sample means are calculated by averaging the samples you take from a larger population. In statistics, it's crucial because it gives a point of reference about the population based on smaller, manageable subsets.

For instance, in the exercise, we have several samples of 2 tellers and for each pair, we calculate the average number of errors they made. This results in a list of sample means:

• (2, 3): Mean = \((2+3)/2 = 2.5\)
• (2, 5): Mean = \((2+5)/2 = 3.5\)
• (3, 5): Mean = \((3+5)/2 = 4.0\)
• and so on...

Each mean gives a mini snapshot of the population, helping us get a clearer understanding of the data’s range and tendencies, without analyzing every single data point.

The population mean is the average of all data points in an entire population. This is important because it provides a complete picture of the data as opposed to the sample mean, which only focuses on portions of it.

To find the population mean, you add up all the data points and divide by the total number of points. For the exercise, this is done by summing the tellers' errors: \(2 + 3 + 5 + 3 + 5 = 18\). You then divide this sum by the number of tellers, which is 5. So, the population mean is \(\frac{18}{5} = 3.6\).

Having the population mean allows us to verify if our sample means are accurately reflecting the larger dataset. In this exercise, both the population mean and the mean of the sample means were 3.6, showing accurate sampling.

Sampling with replacement means that after choosing an item from the population, it is returned before the next selection. This ensures each sample selection is independent and maintains the population size across different sample selections.

In the context of the exercise, each time you pick two tellers and calculate the mean errors, the tellers go back into the pool for the next sample selection. This keeps the sampling process fair and consistent with how probability works.

Sampling with replacement allows statisticians to gather numerous samples and calculate accurate statistics, which increases the reliability of results when sampling a population. It can sometimes lead to repeated data points in samples, but it also reflects the randomness and variability realistically encountered in larger populations.