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Calculating the sample mean provides a snapshot of the average waiting time from the selected customers. Simple yet effective, this calculation starts by summing all individual waiting times and dividing by the number of customers sampled. For instance, if you have waiting times such as 5, 10, and 15 minutes, your sample mean calculation would involve:

• Add: 5 + 10 + 15 = 30
• Divide by Total Observations: 30/3 = 10

This results in a sample mean of 10 minutes. In statistics, the term used to denote each observation is typically \(X\), where \(X_1\), \(X_2\), ..., \(X_{45}\) represent individual waiting times for 45 customers. Your formula becomes \[ \bar{X} = \frac{(X_1 + X_2 +...+ X_{45})}{45} \] This sample mean endeavors to accurately reflect the average waiting time for the broader customer base.

A confidence interval provides a range that likely contains the true population mean. Once you have your sample mean, the next step is to create this interval, which helps you gauge the reliability of your sample mean.Choosing a confidence level, such as 95%, means that you are 95% sure the interval contains the population mean. This requires calculating the standard error, which combines the standard deviation and sample size.The formula to obtain the confidence interval is \[ \bar{X} \pm z \times \frac{s}{\sqrt{n}} \] where

• \(\bar{X}\) is the sample mean
• \(z\) is the z-score linked to the selected confidence level (e.g., 1.96 for 95%)
• \(s\) is the sample standard deviation
• \(n\) is the sample size, in this case, 45 customers

This interval reflects the reliability of your sample mean and offers a more comprehensive picture of the population mean, providing essential context for your manager's decision-making.

Random selection is key to ensuring your sample truly represents the general customer base. By selecting customers randomly, you remove biases that might skew the results, such as only choosing customers during certain times of day or those visiting specific branches. This process ensures that

• Each customer has an equal chance of being selected
• The sample data will better imitate the diversity of the entire customer pool

Imagine placing all customer names into a hat and drawing 45 names at random. This illustration mirrors the essence of random selection. This randomness is elemental in statistics, as it enhances the validity of the results, providing a more genuine reflection of the overall population.

Estimating the population mean is the final goal of conducting a sampling study. Using your calculated sample mean and confidence interval, you make an informed prediction about the average waiting time for the whole customer base. The underlying assumption here is that your sample is a microcosm of the entire population. Thus, data gathered from your sample allows you to estimate this broader "true" mean. Keep in mind:

• Sample size affects accuracy. Larger samples usually provide more reliable estimates.
• Randomness in selection boosts credibility and relevance of your estimates.

In this particular bank study, the estimate presents potential average customer waiting times. A precise estimate becomes a valuable tool in service improvement strategies and future decision-making processes.