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When you need to find how many different groups you can form from a larger set, you use the combination formula. This formula is useful because it provides a way to count options without actually listing them all. The formula to calculate the number of combinations is:

• \( C(n, k) = \frac{n!}{k!(n-k)!} \)

Here, \( n \) represents the total number of items in the set, and \( k \) represents the number of items to choose. The exclamation mark (!) signifies a factorial, which is the product of all positive integers up to that number.
For example, in the exercise, you want to know how many ways you can choose 2 tellers from 5. By using \( C(5, 2) \), you calculate that there are 10 different possible combinations, which represent the different "samples" of 2 tellers.

The population mean is a measure of central tendency that tells you the average value of an entire set of items. It's the sum of all values in the population divided by the number of values.
In the exercise, the population is the number of errors made by 5 tellers: 2, 3, 5, 3, and 5. To find the population mean, add these values together and divide by 5.

• Population Mean \( = \frac{2 + 3 + 5 + 3 + 5}{5} \)
• Which simplifies to \( \frac{18}{5} = 3.6 \).

The population mean of 3.6 represents the average number of errors made by the tellers. By comparing this mean to other calculations, you understand how well sample statistics estimate the population values.

A statistical sample is a subset of items from a larger set, known as a population. By studying samples, we can make inferences or generalizations about the entire population. In this exercise, we're considering all possible pairs of tellers as samples.
Listing all possible samples of size 2 from the tellers results in groups such as (2,3), (2,5), and so on. Calculating the mean for each of these pairs gives us insight into the characteristics of the population. Sampling in this way allows us to choose a manageable number of observations to analyze, instead of attempting to study every detail of the population.
By using each sample, you compute the sample mean, which is the average of the values in a specific sample. These sample means can be used to approximate or estimate the population mean.

Mean comparison involves analyzing the relationship between the mean of sample means and the population mean. It provides a way to assess the accuracy and reliability of sample statistics as estimators of population parameters.
In this exercise, after computing the means for all possible pairs, you find the mean of these sample means:

• Mean of Sample Means \( = \frac{36}{10} = 3.6 \).

This result is then compared to the population mean of 3.6. The fact that they are equal demonstrates the power of statistical sampling: even though it's often impractical to measure an entire population, well-constructed samples can offer precise estimates.