Step 1

Concept and reason

Estimator: It is a rule that tells you how to calculate the estimate based on the sample.

Parameter space: The set of all possible value is called parameter space.

Unbiasedness: An estimator ${T}_{n}=T({x}_{1},\text{}{x}_{2},\cdots ,{x}_{n})$ is said to be an unbiased estimator of $\gamma \left(\theta \right)$ if $E\left({T}_{n}\right)=\gamma \left(\theta \right)$ for all $\theta \in \mathrm{\Theta}$

Baisedness: An estimator ${T}_{n}=T({x}_{1},\text{}{x}_{2},\cdots ,{x}_{n})$ is said to be an unbiased estimator of $\gamma \left(\theta \right)$ if $E\left({T}_{n}\right)\ne \gamma \left(\theta \right)$ for all $\theta \in \mathrm{\Theta}$

Fundamentals

The formula for sample mean is,

$\stackrel{\u2015}{x}=\frac{\sum x}{n}$

The formula for sample variance is,

$s}^{2}=\frac{1}{n}\sum _{i=1}^{n}{({x}_{i}-\stackrel{\u2015}{x})}^{2$

$S}^{2}=\frac{1}{n-1}\sum _{i=1}^{n}{({x}_{i}-\stackrel{\u2015}{x})}^{2$

The formula for standard deviation is,

$s={\sqrt{s}}^{2}$

Step 2

From the known information sample median, ranges and standard deviation are biased estimators.

The sample mean, variance and the proportion are unbiased estimators of population parameters.

Step 2

From the known information $E\left(\stackrel{\u2015}{x}\right)=\mu ,\text{}E\left(\hat{p}\right)=p$ and $E\left({s}^{2}\right)={\sigma}^{2}$. So, the unbiased estimators are sample mean, variance and the proportion.

Sample variance used to estimate a population variance.

Sample proportion used to estimate a population proportion.

Sample mean used to estimate a population mean.

The sample variance is an unbiased estimator of population variance. The sample proportion is an unbiased estimate of the population proportion and the sample mean is an unbiased estimate of the population mean.

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