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A statistic is an unbiased estimator of a parameter. Select the best answer.

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. 

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