# Theory of Estimation

Estimation deals with the methods and techniques adopted for finding likely value of a population parameter using statistics from a sample drawn from the population. There are two types of estimation. They are:- Point estimation
- Interval estimation

# Point estimation

While estimating an unknown parameter, if a single value is proposed as the estimate, such estimation is called*point*

*estimation*. Thus, based on the sample mean if we conclude that population mean is 85 g, it is point estimation.

Î¼ is said to be a point estimator of if it estimates and is denoted by,

**Note:** The point estimators of population mean and population variance are the corresponding sample statistics.

Criteria of good estimator:

There are four main properties associated with a good estimator. They are

- Unbiasedness and minimum variance
- Consistency
- Efficiency
- Sufficiency

**(i) Unbiasedness and Minimum Variance:**

An estimator *T* is said to be an unbiased estimator for the parameter ÆŸ

If E(*T*) = ÆŸ *i.e.*, The mean of sampling distribution of statistic *T* is the population parameter ÆŸ

If E(*T*) â‰ ÆŸ then *T* is said to be a biased estimator of ÆŸ.

If E(*T*) > ÆŸ then *T* is said to be a positively biased estimator of ÆŸ.

If E(*T*) < ÆŸ then *T* is said to be a negatively biased estimator of ÆŸ.

** **

** Examples:**

- The sample mean
*is an unbiased estimator for population mean Î¼**i.e.*, - The sample variance is not an unbiased estimator for population variance
*i.e.*,*E*(*s*^{2}) â‰*Ïƒ*^{2} - is an unbiased estimator for population variance
*Ïƒ*^{2}

^{}*E*(*s*^{2}) =*Ïƒ*^{2}A statistic T is known to be a minimum variance unbiased estimator (MVUE) of ÆŸ.^{}If (i)*T*is unbiased for ÆŸ.*T*has the minimum variance among all the unbiased estimator of ÆŸ.

** Example:**

- The Sample mean is an MVUE for population mean Î¼.
- The Sample proportion â€˜
*p*â€™ is an MVUE for population*P*.

**Consistency:**

An estimator *T* is known to be consistent estimator of the parameter ÆŸ if the difference between *T* and ÆŸ can be made smaller and smaller by taking the sample size *n* larger and larger.

Mathematically, *T* is consistent

*E* (*T*) â†’ ÆŸ and

*V* (*T*) â†’ 0 as *n* â†’ âˆž

** **

**Example:**

The sample mean, sample S.D. and sample proportion are all consistent estimators for the corresponding population parameters.

**Efficiency:** An estimator *T* is known to be an efficient estimator of ÆŸ if *T* has the minimum standard error among all the estimators of ÆŸ when the sample size is kept fixed.

**Example:** The sample mean is an efficient estimator for the population mean Î¼.

**Sufficiency:** An estimator *T* is known to be a sufficient estimator of ÆŸ if *T* contains all the information about ÆŸ.

**Example:** The sample mean is a sufficient estimator for the population mean Î¼.

Hence, the estimate of the population mean is 16.

The estimate of standard error of sample mean (when population S.D. is unknown) is given by

- If the sample proportion
*p*is known, then we can calculate the estimate by the relation as given below:

# Interval estimation

In point estimation, we propose a single value as the estimate of the unknown parameter. In most of the situations, the value so proposed is unlikely to be the actual value of the parameter. Instead, if we propose a small interval around the point estimate as the likely interval to contain the parameter, our proposition would be stronger. This interval which is likely to contain the parameter is called interval estimate.

The table given below gives you the values of confident coefficient for different confidence levels.

The values of confidence coefficient corresponding to the confidence levels are very important while calculating the confidence interval.

The confidence interval for population mean is given by,

** **

*N*= 1500,

*n*= 100, = 62 kg, Ïƒ = 10 kg.

Let

*Î¼*be the population mean. Then, 95% confidence interval for

*Î¼*is

So with 95% confidence we can say that the students on an average weigh between 60.04 kg and 63.96 kg.

Similarly, 99% confidence interval for

*Î¼*is

So, with 99% confidence we say that, the students on an average weight between 59.42 kg and 64.58 kg.

*p*is known, then we can calculate the confidence interval using the relation

*p*Â±

*Z.*S.E

_{p}

*N*= 15000,

*n*= 300,

*p*= 5% = 0.05

# Determination of Sample Size

The size of a random sample can be determined by the formula,Where, *E* is the admissible error while estimating the population mean