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In statistical analysis, an unbiased estimator is crucial because it ensures that, on average, the estimator equals the parameter it is estimating. In simpler terms, the expected value of an unbiased estimator matches the true value of the parameter. In our exercise, we have two estimators, \( \bar{X}_1 \) and \( \bar{X}_2 \), aiming to estimate the population mean \( \mu \).

The concept of unbiasedness in this context means:

• The average calculated using the sample size of \( 2n \) for \( \bar{X}_1 \).
• The average calculated using the sample size of \( n \) for \( \bar{X}_2 \).

For both, the expected value of the sample mean equals \( \mu \), confirming that both \( \bar{X}_1 \) and \( \bar{X}_2 \) are unbiased estimators. This quality is critical because it determines the legitimacy of these estimators in accurately reflecting the true population mean.

Variance measures the dispersion of a set of data points. In the context of estimators, variance denotes how much the estimations will fluctuate when sampling is repeated.

For our exercise, we know:

• The variance of \( \bar{X}_1 \) is \( \frac{\sigma^2}{2n} \).
• The variance of \( \bar{X}_2 \) is \( \frac{\sigma^2}{n} \).

These variances highlight the difference in precision between the two estimators. Since the variance of \( \bar{X}_1 \) is less than that of \( \bar{X}_2 \), it signifies that \( \bar{X}_1 \) provides more consistent and reliable estimates of \( \mu \). Lower variance means that the estimates are less spread out and thus closer to \( \mu \) on average, imparting \( \bar{X}_1 \) with a higher level of precision.

Estimating the population mean \( \mu \) involves using sample data to deduce the average value of the entire population.

Here's a brief overview:

• \( \bar{X}_1 \) and \( \bar{X}_2 \) are two different approaches with sample sizes \( 2n \) and \( n \), respectively.
• A larger sample size, as used by \( \bar{X}_1 \), generally results in a more accurate estimation of \( \mu \).
• This is because a larger sample size typically captures more variability in the data, leading to better representation.

In the estimation of the population mean, it is not only the unbiased nature of an estimator that counts but also its variance. In our case, since both \( \bar{X}_1 \) and \( \bar{X}_2 \) are unbiased and \( \bar{X}_1 \) has a lower variance, \( \bar{X}_1 \) is the preferred choice, making it a better estimator of \( \mu \). Understanding these principles is essential for effectively interpreting statistical data and making informed decisions based on analyses such as these.