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The population mean, denoted by \( \mu \), is a fundamental concept in statistics. It represents the average of all data points in a population. A population encompasses all possible outcomes or individuals of interest, making \( \mu \) a specific fixed value, even though it is often unknown to us in practice. Understanding the population mean is crucial because it provides us with the central tendency of the entire population. Imagine wanting to know the average height of all adults in a country; the population mean would represent this specific average height. In real-world scenarios, calculating the exact population mean is often challenging, if not impossible, due to the size and accessibility of populations. That's where the concept of a sample mean comes into play, allowing us to make educated guesses about the population mean.

The sample mean, symbolized by \( \bar{x} \), is our best estimate of the population mean (\( \mu \)) when dealing with large populations. It is calculated by taking the average of a randomly selected sample from the population. In the context of our exercise, the sample consists of 50 observations, and the sample mean is given as 56.4.The sample mean helps statisticians make inferences about the population without having to measure every single individual. Here are key reasons why the sample mean is used:

• It is a practical solution when collecting data from the entire population is not feasible.
• It provides a point estimate of the population mean, aiding critical decision-making in fields like marketing, healthcare, and public policy.

By understanding the sample mean, we can predict the population mean and estimate other population parameters through various statistical methods.

The margin of error quantifies the uncertainty or potential error in our estimation of the population mean. It tells us how much we can expect our sample mean to vary from the true population mean. In statistical practice, the margin of error is crucial because it gives context to the sample mean, thereby improving our confidence in the estimates we make.To calculate the margin of error, the formula used is:\[ ME = z \cdot \left( \frac{s}{\sqrt{n}} \right) \]where:- \( z \) is the z-score corresponding to the confidence level- \( s \) is the standard deviation of the sample- \( n \) is the sample sizeIn our example, using a 95% confidence level, a z-score of 1.96 was used, and the calculated margin of error was approximately 0.74. This indicates that the true population mean is likely within 0.74 units of our sample mean, reinforcing the reliability of our estimate.

A confidence interval provides a range within which we can expect the true population mean to lie, given a specified level of confidence (typically 95% or 99%). It is made up of two components: the sample mean and the margin of error.The confidence interval can be expressed as:\[ (\bar{x} - ME, \bar{x} + ME) \]In our problem, the sample mean is 56.4, and the margin of error is approximately 0.74. Therefore, the 95% confidence interval is:\[ (56.4 - 0.74, 56.4 + 0.74) \]which simplifies to approximately \( (55.66, 57.14) \).This interval suggests that we are 95% confident that the population mean falls within this range. Confidence intervals are essential as they provide more informative insights than a single point estimate, allowing us a better understanding of the variability and reliability of our statistics.