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The population mean, often symbolized by the Greek letter µ, is the average of all data points in a population. It provides a central value which is representative of the entire population. In statistics, knowing the population mean is crucial for making predictions and assessing differences between groups.

For example, in our exercise, the population mean is given as 90. This tells us that if you could catalog every single individual's value in the population, on average, you would get 90.

The population mean acts as a benchmark to compare sample means against and is often used in various statistical analyses and tests. Understanding the population mean can help in interpreting how typical or atypical a sample or an outcome might be.

Standard deviation, denoted by the Greek letter σ, measures the amount of variability or dispersion within a set of data points. A lower standard deviation means that the values tend to be close to the mean, while a higher standard deviation means the values are spread out over a wider range.

In our scenario, the population standard deviation is 18. This tells us that individual data points can deviate from the mean by 18 units on average.

Understanding standard deviation is essential for gauging how varied the data points are, which can offer insights into the reliability and predictability of data.

• High standard deviation: Data is spread out, less predictable.
• Low standard deviation: Data is clustered closely around the mean, more predictable.

The sample mean, represented by the symbol \(\bar{x}\), refers to the average of a sample taken from a larger population. When you take a sample from a population and calculate its mean, it serves as an estimate of the population mean.

According to the problem, since the population mean is 90, the sample mean for both sample sizes (10 and 35) is also expected to be 90. This is because, in theory, and especially with large sample sizes, the sample mean approaches the population mean.

The sample mean is particularly useful because it allows us to make inferences about the population without needing to collect data from everyone. It's like a snapshot that can give us information about the bigger picture.

Sample size, symbolized by \(n\), crucially affects the precision of the sample mean and standard deviation of a dataset. It's the number of observations or data points collected in a sample.

A larger sample size generally means a more accurate estimate of the population parameters.

In the given exercise, two different sample sizes are considered: 10 and 35.

• For a sample size of 10, we calculated the sample standard deviation as \(5.70\) using the formula \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\).
• At a sample size of 35, this variation reduces to \(3.04\), depicting how larger sample sizes result in a smaller standard deviation of the mean.

Larger sample sizes lead to lower variability in the sample mean estimates, making them more reliable in representing the population.