91Ó°ÊÓ

The sample mean is a fundamental statistic used to estimate the central tendency of a dataset. In this study, after collecting the service times for 40 customers, we're interested in determining the average time it took the cashier to complete each transaction. To find the sample mean, sum up all the observed service times and then divide by the total number of data points, which is 40.

Mathematically, this is expressed as:

• The sum of all observed values: \( X_1 + X_2 + \ldots + X_{40} \)
• Divided by the total number of observations \( n = 40 \)

This gives us the sample mean, denoted as \( \bar{x} \). This mean serves as a point estimate for the population mean, which represents the average service time for all customers served by the cashier.

A confidence interval provides a range of values which is likely to contain the population mean. After calculating the sample mean, it's important to establish how "confident" we are in this estimate by calculating a confidence interval. This interval is calculated using a chosen level of confidence, often 95% or 99%.

To calculate a confidence interval, you need:

• The sample mean \( \bar{x} \)
• An estimated standard deviation \( s \)
• The size of your sample \( n \), which is 40
• A z-value corresponding to the chosen confidence level (e.g., 1.96 for 95% confidence)

The confidence interval is expressed as:\[\bar{x} \pm z \cdot \frac{s}{\sqrt{n}}\]This means that we can say, with a certain level of confidence, that the true population mean falls within this calculated range.

The population mean is the average of a group, where the group consists of every member of a defined population. In this exercise, it would be the true average service time of all customers ever served by the cashier at the supermarket. Since collecting data on every single customer is impractical, statisticians use sample data to estimate the population mean.

The calculated sample mean allows us to make such estimations. We use the concept of the confidence interval to express how close we believe the sample mean is to the actual population mean. By understanding the population mean, the supermarket can evaluate cashier performance and make informed decisions on staffing and operational improvements.

Data collection is a vital step in statistical inference as it lays down the foundation for all subsequent analysis. In our task, we are determining the time taken for a cashier to serve 40 customers. It's crucial to ensure randomness in selecting these customers to avoid bias.

Key considerations during data collection include:

• Random Timing: Choose customers at different times to avoid bias from only busy or slow periods.
• Accuracy: Ensure that the timing is precise to reduce errors in the analysis.
• Sufficient Sample Size: The sample of 40 customers should be enough to provide a reliable estimate of the population mean.

Accurate and diverse data makes the statistical analysis credible, ensuring that the final confidence interval and estimated population mean are as accurate as possible.