Q 8.2. - Elementary Statistics

Question

What is a confidence interval estimate of a parameter? Why is such an estimate superior to a point estimate?

Step-by-Step Solution

Answer

The interval estimate of the confidence interval is defined as the sample statistic $\pm$ margin of error which describe the precision of the estimator and the confidence level gives the uncertainty of the statistics an estimator of the population parameter.

So, confidence intervals is preferred against point estimate for the above two advantages of confidence interval estimate.

Step 1 :

Suppose we have to infer the unknown value of the population parameter $\theta$( for example population mean).

In the internal estimation of $\theta$ we find two limits, say ${\theta }_1 and {\theta }_2 ({\theta }_1<{\theta }_2)$ from the sample observations such that $\theta$ lies between ${\theta }_1 and {\theta }_2$ with a certain degree of confidence ( measured in terms of probability ) in notation we write,

$P\left[{\theta }_1\le {\theta }_2\le {\theta }_3\right]=1-\alpha$for all $\theta$ where $\alpha$is independent of $\theta and 0\le \alpha \le 1.$

The limits ${\theta }_1 and {\theta }_2$ are called confidence limits the interval $\left[{\theta }_1 ,{\theta }_2\right]$ is called confidence interval with confidence coefficient $1-\alpha$ in the words we can say,

The$100(1-\alpha )\%$ confidence interval to $\theta be [{\theta }_1,{\theta }_2].$

Step 2

Generally $\alpha$ is taken as very small (close to zero) for instance $\alpha =0.01 or \alpha =0.05.$

This interval $\left[{\theta }_1,{\theta }_2\right]$ is the interval estimate of unknown population parameters $\theta$ together with the $100(1-\alpha )\%$ confidence level.

In point estimation, we estimate the unknown parameter by a single value. Now instead of the mean of the estimator is equal to ( or close to) an unknown parameter, it may happen that the standard deviation of the estimator is very high i,e. value of the estimators are largely deviated from the population parameter and hence large sampling error may occur.

Step 3:

So, it is customary to give, together with the estimate, the standard error ( standard deviation) of the estimator. This idea is actually used in interval estimation we use a confidence interval to express the precision and uncertainty associated with a particular estimator.

The interval estimate of the confidence interval is defined as the sample statistic  margin of error which describe the precision of the estimator and the confidence level gives the uncertainty of the statistics an estimator of the population parameter.

So, confidence intervals is preferred against point estimate for the above two advantages of confidence interval estimate.