# 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:

1. Point estimation
2. 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

1. Unbiasedness and minimum variance
2. Consistency
3. Efficiency
4. 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:

1. The sample mean  is an unbiased estimator for population mean Î¼

i.e.,
2. The sample variance  is not an unbiased estimator for population variance

i.e.E(s2) â‰  Ïƒ2
3.  is an unbiased estimator for population variance Ïƒ2
E(s2) = Ïƒ2
A statistic T is known to be a minimum variance unbiased estimator (MVUE) of ÆŸ.

If (i) T is unbiased for ÆŸ.

(ii) T has the minimum variance among all the unbiased estimator of ÆŸ.

Example:

1. The Sample mean  is an MVUE for population mean Î¼.
2. The Sample proportion â€˜pâ€™ 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 (Tâ†’ 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.

Example
Consider sample observations 12, 7, 28, 15, 18 taken from a population containing 50 units. Find an estimate of the population mean.
Solution
Let Î¼ be the estimate of .

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

Example
A random sample of size 6 is taken from a population containing 150 units. If the sample observations are 10, 15, 13, 18, 21 and 24, find an estimate of the standard error of sample mean.
Solution

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

Example
A random sample of 150 problems from a book contained 12 problems that had typo errors. What is the estimate of sample proportion of the problems with errors?
Solution

# 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,

Example
Consider a college with 1500 students. A sample of 100 students is randomly chosen and their average weight is found to be 62 kg. Let the population standard deviation be Ïƒ = 10 kg. Find the confidence interval for a confidence level of 95% and 99%.
Solution
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.

If the sample proportion p is known, then we can calculate the confidence interval using the relation p Â± Z. S.Ep

Example
A company produces 15000 light bulbs every day. From a sample of 300 bulbs 5% were found to be defective. Estimate the number of bulbs that can be expected to be spoilt in the daily production process at 99% level of confidence.
Solution
Given N = 15000, n = 300, p = 5% = 0.05

Thus the number of bulbs that can be reasonably expected to be spoiled in the entire production on a daily basis at 99% level of confidence is (0.01785 Ã— 15000, 0.08215 Ã— 15000) = (268, 1233)

# 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

Example
The standard deviation of mark was found to be 12. What should be the size of the sample in order to be 95% confident that the error of estimates of mean would not exceed 2 marks?
Solution

If the value of standard deviation is not given then the sample size can be found using the relation

Example
In a school, 20% of the students perform poorly in academics. What size of the sample should be taken so as to ensure that the error of estimation of the proportion should not be more than 5% with 99% confidence?
Solution