The sample mean is an MVUE for population mean

- The sample mean is an unbiased estimator for the population mean. This means that on average, the sample mean will equal the population mean.
- The minimum variance unbiased estimator (MVUE) is an estimator that achieves the lowest possible variance among all unbiased estimators. In other words, it is the most efficient estimator.
- The sample mean satisfies both of these properties, making it an MVUE for the population mean.
- To understand why the sample mean is an MVUE, we need to consider the properties of unbiased and efficient estimators.

The sample mean is a consistent estimator for population mean

- A consistent estimator is one that converges to the true population parameter as the sample size increases.
- The sample mean satisfies this property because as the sample size increases, the sample mean becomes more and more representative of the population mean.
- This is known as the law of large numbers, which states that as the sample size approaches infinity, the sample mean converges to the population mean.
- Therefore, the sample mean is a consistent estimator for the population mean.

The sample mean is a sufficient estimator for population mean

- A sufficient estimator is one that contains all the information about the population parameter that is available in the sample.
- The sample mean is a sufficient estimator for the population mean because it uses all the available information in the sample to estimate the population mean.
- It takes into account every observation in the sample and combines them to provide an estimate of the population mean.
- Therefore, the sample mean is a sufficient estimator for the population mean.

The sample mean satisfies all the properties of an MVUE

- The sample mean is an unbiased estimator for the population mean, as it is equal to the population mean on average.
- It is a consistent estimator, as it converges to the population mean as the sample size increases.
- It is a sufficient estimator, as it contains all the available information in the sample to estimate the population mean.
- Therefore, the sample mean satisfies all the properties of an MVUE for the population mean.
- In conclusion, the correct answer is option 'D' - all of these. The sample mean is an MVUE, a consistent estimator, and a sufficient estimator for the population mean.