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The sample mean is a crucial concept in statistical estimation. It is simply the average of a set of data points collected from a sample. In this case, you would collect the waiting times for 45 randomly selected customers at the bank. The sample mean offers an estimate of the population mean, serving as a point estimate.

To calculate the sample mean, you add up all of the observed waiting times and divide by the number of customers in your sample, which is 45. Mathematically, it is expressed as:

• \( Sample \ mean = \frac{1}{N}\sum_{i=1}^{N} X_{i} \)

where \( N \) represents the total number of observations, and \( X_{i} \) signifies each individual waiting time.

The sample mean is important because it provides a summary statistic that represents the center of your data set. However, since the sample is just a subset of the entire population, there will always be some degree of variability or error, which is why we also consider the confidence interval.

A confidence interval gives a range of values, derived from the sample, that is likely to contain the population mean. When you calculate the confidence interval, you also set a confidence level, such as 95%, which indicates how confident you are that the interval includes the true mean.

The formula for a confidence interval is as follows:

• \( Confidence \ Interval = Sample \ Mean \pm (Z \times \frac{Standard \ Deviation}{\sqrt{N}}) \)

Here, the \( Z \) represents the Z-score associated with your chosen confidence level. For a 95% confidence interval, the Z-score is typically 1.96.

This method allows you to understand the reliability of your sample mean. If your confidence interval is small, you feel more certain about the population mean. Conversely, a wide interval indicates more uncertainty.

Reporting a confidence interval alongside the sample mean provides a more complete picture, explaining both your point estimate and the potential error inherent in sampling.

The population mean is what you ultimately seek to estimate in a statistical study like the one described. It is the average of a particular characteristic—in this case, waiting times—across the entire population.

Since it is often impractical or impossible to observe the whole population, we estimate it using our sample data. It is important to remember that while the sample mean gives us an estimate, it is not guaranteed to be exactly equal to the population mean due to potential sampling errors.

The strength of statistical estimation techniques is that they allow you to infer the population mean with an associated level of confidence through both the sample mean and the confidence interval.

Understanding the population mean and its estimation ties directly into making informed decisions. In this scenario, the bank manager can better understand the likely average waiting times for future customers based on your analysis.

Random sampling is a fundamental method used in statistics to ensure that the sample used in a study is representative of the population. This technique involves selecting individuals based purely on chance, reducing bias in your data collection process.

In the bank scenario, random sampling helps ensure that the waiting times you record from 45 customers are as reflective of the overall customer base as possible. It's a simple but powerful approach because:

• It minimizes the risk of selection bias, where certain types of people might be unintentionally included or excluded.
• It increases the reliability and validity of your results, making your findings more generalizable to the entire population.

Using random sampling helps ensure that any patterns or trends you observe, such as average waiting times, are not skewed by the data collection method itself but are instead reflective of the real situation.