This question aims to select the **best answer** from the given **statements** provided that the statistic is the **unbiased parameter estimator**.

We have to check whether a statistic is calculated from a random sample or the **value of the statistic** is equal to the value of the parameter in a single sample. If a statistic is the unbiased estimator of a parameter, then the values of statistics are **very close** to the value of the parameter. It can also be assumed that the values of the statistics are **centered** at the parameter value or the distribution of the statistic has an **approximately normal** shape in many samples.

## Expert Answer

The **bias estimators** of a parameter are the ones whose sample mean is **not centered** and they are not distributed properly. It is the mean of the difference of $ d (X) $ and $ h (\theta) $.

\[ b _ d ( \theta ) = E _ \theta d ( X ) – h ( \theta ) \]

Here, **d ( X )** is the distribution of samples and $ \theta $ is the value of the parameter with an **estimator** $ h ( \theta ) $

If $ b _ d ( \theta ) $ becomes zero, then the biased estimator will be equal to the sample distribution and it will be called the **unbiased estimator** of the parameter. It is represented in the following way:

\[ 0 = E _ \theta d ( X ) – h ( \theta ) \]

\[ E _ \theta d ( X ) = h ( \theta ) \]

The sampling distribution of the statistics is **centered** when the sample has an **estimated value** equal to the parameter. According to the given information, Statistics is the unbiased estimator of a parameter, meaning the sample distribution will be centered.

## Numerical Results

From the given statement, we can conclude that the statement **“values of the statistics are centered at the value of the parameter when observing many samples”** is the best answer.

**Example**

A **survey** is done to calculate the number of **non-vegetarian** people in a **small classroom**. The numbers were reported as:

\[ 8 , 5 , 9 , 7 , 7 , 9 , 7 , 8 , 8 , 10 \]

Mean of these numbers $ = \frac { sum (x) } { 10 } $

\[ Mean = 7 . 8 \]

It means that the mean of the sample is not **underestimated** or **overestimated** as its value is **close to 8.** The mean according to the **binomial distribution** is given as:

\[ \mu = n p \]

Here $ \mu $ represents the **standard deviation** and **np** is the average number of successes so according to the given example,

\[ \mu = 16 \times 0.5 = 8 \]

The mean of the sample is also 8 which is demonstrated below:

\[ E X = \frac { 1 } { 10 } ( 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 ) \]

\[ E X = \frac { 80 } { 10 } \]

The **sample mean is 8** which shows the unbiased estimator of a parameter.

*Image/Mathematical drawings are created in Geogebra. *