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In statistics, the population mean is denoted as \( \mu \). It represents the average of a set of values across an entire population. Understanding the population mean helps give context when comparing sample data. When we talk about a population mean, we're speaking of a true average, although the exact value is often unknown. That's why we estimate it using a sample mean. The example given involves the mean cost per month to lease a one-bedroom apartment.The sample mean, denoted as \( \bar{x} \), is taken from a random sample and serves as an estimate of the population mean. By examining the mean cost from our sample of apartments, we use this sample mean and look to predict the actual mean for the entire population of available apartments.

The standard deviation is a measure of how spread out numbers are in a set. More specifically, the sample standard deviation, represented as \( s \), indicates the extent of variation in our sample data. Knowing this allows us to understand how much variation or "spread" is present compared to our sample mean.For instance, in the given exercise, we have a sample standard deviation of 50. This shows us that the costs in our sample don't deviate wildly from one another by much -- on average, they vary by $50 from the mean. Larger standard deviations suggest greater variability in the data, and smaller ones indicate less spread.Employing sample standard deviation helps when calculating the confidence interval, giving us an idea of the true variability in the population when the population standard deviation is unknown.

In the context of confidence intervals, the critical value is crucial because it sets the bounds of our estimate's reliability. For a confidence interval, this value, often denoted as \( z_{\alpha/2} \), is taken from the statistical distribution that applies to our data.The critical value is chosen based on our desired level of confidence. In the example, a 98% confidence level requires a critical value of approximately 2.33 from the standard normal distribution. This indicates that we're using 98% of the central values from a normal distribution to establish our confidence interval.Selecting the right critical value is essential for accurately estimating the range in which we expect the population mean to fall with our selected certainty level.