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When we talk about population variance, we're referring to a measure that expresses the diversity or spread of all possible outcomes in a population. One way to think about it is by considering it as the average of the squared differences from the mean of the population. Population variance, symbolized by \(\sigma^{2}\), gives us insight into how similar or different each data point is relative to the population mean.In the context of confidence intervals, such as the one mentioned \((51.47, 261.90)\), this interval gives us a range within which the true population variance \(\sigma^{2}\)is expected to fall a specific percentage of the time (e.g., 90%). These intervals are a powerful tool because they help in making statistical inferences about the populationbased on the sample data.

The sample standard deviation, denoted by \(s\), is a measure of how much the observations in a sample differ from the sample mean.It is essentially an estimate of the population standard deviation based on sample data. To calculate it, we first need the sample variance \(s^{2}\) and then take its square root.Here’s a quick breakdown:

• Start from the sample variance, \(s^{2}\).
• The sample standard deviation \(s\) is simply the square root of this sample variance.

This measure is crucial because it helps us understand how much actual variation the sample data exhibits. It's particularly useful when making predictions or assessing the confidence in data results.

The sample variance \(s^{2}\) serves as a concrete method to approximate the population variance from a sample.This value tells us how much the values in the sample differ from their mean. It is calculated by taking the differences of each data point from the sample mean, squaring these differences, and then averaging them.To connect with the exercise, once we've identified the confidence interval of the population variance \(\sigma^{2}\),we use the method of estimating \(s^{2}\) by leveraging these bounds. Specifically, in the original exercise, \(s^{2}\) is estimated as the square root of the product of the confidence interval boundaries \(51.47\) and \(261.90\). Once the sample variance is known, it can be used in further calculations, such as determining the sample standard deviation.Understanding sample variance is key to grasping the variability and reliability of the sample as a representative of the broader population.