# Estimation: Point and Interval Estimation

## Estimation

The procedure of making judgement or decision about a population parameter is referred to as statistical estimation or simply estimation.  Statistical estimation procedures provide estimates of population parameter with a desired degree of confidence. The degree of confidence can be controlled in part, (i) by the size the sample (larger sample greater accuracy of the estimate) and (ii) by the type of the estimate made. Population parameters are estimated from sample data because it is not possible (it is impracticable) to examine the entire population in order to make such an exact determination.The statistical estimation of population parameter is further divided into two types, (i) Point Estimation and (ii) Interval Estimation

## Point Estimation

The objective of  point estimation is to obtain a single number from the sample which will represent the unknown value of the population parameter. Population parameters (population mean, variance etc) are estimated from the corresponding sample statistics (sample mean, variance etc).
A statistic used to estimate a parameter is called a point estimator or simply an estimator, the actual numerical value obtained by estimator is called an estimate.
Population parameter is denoted by θ which is unknown constant. The available information is in the form of a random sample x1,x2, … , xn of size n drawn from the population. We formulate a function of the sample observation x1,x2, … , xn. The estimator of θ is denoted by $\hat{\theta}$. Different random sample provide different values of the statistics $\hat{\theta}$. Thus $\hat{\theta}$ is a random variable with its own sampling probability distribution.

## Interval Estimation

A point estimator (such as sample mean) calculated from the sample data provides a single number as an estimate of the population parameter, which can not be expected to be exactly equal to the population parameter because the mean of a sample taken from a population may assume different values for different samples. Therefore we estimate an interval/ range  of values (set of values) within which the population parameter is expected to lie with a certain degree of confidence. This range of values used to estimate a population parameter is known as interval estimate or estimate by confidence interval, and is defined by two numbers, between which a population parameter is expected to lie. For example, $a<\bar{x}<b$ is an interval estimate of the population mean μ, indicating that the population mean is greater than a but less than b. The purpose of an interval estimate is to provide information about how close the point estimate is to the true parameter.

Note that the information developed about the shape of a sampling distribution of the sample mean i.e. Sampling Distribution of $\bar{x}$ allows us to locate an interval that has some specified probability of containing the population mean $\mu$.

## Which of the two types of estimation do you like the most, and why?

• Point estimation is nice because it provides an exact point estimate of the population value. It provides you with the single best guess of the value of the population parameter.
•  Interval estimation is nice because it allows you to make statements of confidence that an interval will include the true population value.

# Advantages of using the Interval and Point Estimation

The problem with using a point estimate is that although it is the single best guess you can make about the value of a population parameter, it is also usually wrong.

• A major advantage of using interval estimation is that you provide a range of values with a known probability of capturing the population parameter (e.g., if you obtain from SPSS a 95% confidence interval you can claim to have 95% confidence that it will include the true population parameter.
• An interval estimate (i.e., confidence intervals) also helps one to not be so confident that the population value is exactly equal to the single point estimate. That is, it makes us more careful in how we interpret our data and helps keep us in proper perspective.
• Actually, perhaps the best thing of all to do is to provide both the point estimate and the interval estimate. For example, our best estimate of the population mean is the value $32,640 (the point estimate) and our 95% confidence interval is$30,913.71 to \$34,366.29.
• By the way, note that the bigger your sample size, the more narrow the confidence interval will be.
• If you want narrow (i.e., very precise) confidence intervals, then remember to include a lot of participants in your research study.