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An estimator is termed as biased if its expected value, when repeatedly calculated, does not equal the true value of the parameter being estimated. In simple terms, an estimator is biased if it consistently overestimates or underestimates the parameter in expectation. In our exercise, the given estimator \( \hat{\Theta} = \frac{1}{c} \sum_{i=1}^{n} (X_i - \bar{X})^2 \) for estimating the population variance \( \sigma^2 \) is a prime example of a biased estimator.The bias of this estimator is crucial because it helps us understand the systematic error involved in the estimation process. The formula for the bias, derived in the solution, is given by:

• \( \text{Bias}(\hat{\Theta}) = \sigma^2 \left( \frac{n-1}{c} - 1 \right) \)

This formula reveals how the bias will change depending on the value of \( c \). If \( c = n-1 \), the bias becomes zero, making the estimator unbiased. However, for any other value, the estimator will overestimate or underestimate \( \sigma^2 \) depending on whether \( c \) is less or more than \( n-1 \).

Unbiased estimation is a desirable property of statistical estimators, where the expected value of the estimator matches the true parameter value it estimates. For the variance estimation, the unbiased estimator commonly used is the sample variance:

• \( S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2 \)

This formula divides by \( n-1 \) instead of \( n \) because it corrects the bias introduced when estimating the population variance from a sample. This division creates an unbiased estimate of \( \sigma^2 \), as it doesn't consistently overshoot or undershoot the actual variance.The idea behind unbiased estimation is to remove systematic errors, giving results that average out to be correct over many samples. This concept is key in statistical analysis because it ensures the reliability and accuracy of the inferences made from sample data. Unbiased estimators are preferred in practice to maintain validity in statistical conclusions.

Sample variance is a statistical measure that quantifies the variability or spread of sample data points around their mean. It is denoted by \( S^2 \) and calculated using the formula

• \( S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2 \)

Here, \( n \) is the number of observations, and \( \bar{X} \) is the sample mean. This formula provides an unbiased estimator for the true population variance.Sample variance plays a critical role in hypothesis testing and confidence interval estimation. By measuring the extent to which each data point deviates from the mean, sample variance offers a snapshot of data dispersion. This snapshot helps in understanding the spread of the data, which is crucial for statistical inference.In summary, sample variance is a fundamental metric that helps describe the consistency or variability within a dataset, serving as a building block for other statistical procedures.

Population variance is a statistical measure that describes the dispersion of a complete dataset. It tells us how data points in the population are spread out around the mean. Represented as \( \sigma^2 \), population variance is calculated as

• \( \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (X_i - \mu)^2 \)

where \( N \) is the total number of elements, and \( \mu \) is the population mean.Unlike sample variance, population variance considers the entire dataset, which provides a comprehensive view of the distribution of values. However, in practice, it is often impractical to have data for every member of a population, making sample variance more commonly used.Understanding population variance is essential because it gives complete insights into how broadly or narrowly data values are distributed across the whole population. Measuring this variance accurately is crucial for various fields, including economics, psychology, and engineering, to make informed decisions based on complete population data.