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  Introduction to Estimation and confidence intervals

Basic Definitions:

Confidence Interval: The estimating interval with a specific level of confidence is termed as confidence interval.

Confidence Level: The percentage of time in which the true mean will lie in the interval estimate given is termed as confidence level.

Consistent Estimator: It is the estimator that gets closer to the value of parameter as the sample size rises.

Degrees of Freedom: It is the number of data values which are permitted to differ, once a statistic has been recognized.

Estimator: It is a sample statistic that is used to estimate a population parameter. It should be unbiased, consistent and quite efficient.

Interval Estimate: To estimate a parameter, a range of values are used. These are termed as Interval estimate.

Maximum Error of the Estimate: It is the maximum difference between the point estimate and the real parameter. The Maximum Error of Estimate is 0.5 the width of confidence interval for proportions and means.

Point Estimate: In point estimate, to estimate a parameter a single value is used.

Relatively Efficient Estimator: It is estimator for a parameter with smallest variance.

T distribution: Whenever the population variance is unknown then this distribution is used.

Unbiased Estimator: It is an estimator whose predicted value is the mean of the parameter being estimated.

Introduction to Estimation:

One area of concern in the inferential statistics is the estimation of population parameter from the sample statistic. Here it is significant to realize the order. The sample statistic is computed from the sample data and the population parameter is inferred (or predicted) from this sample statistic. We can also state that Statistics are computed and parameters are estimated.

The other area of inferential statistics is sample size recognition. That is, how big of a sample must be taken to make a precise estimation. In such cases, the statistics cannot be used as the sample has not been taken yet.

Point Estimates:

There are two kinds of estimates: Point Estimates and Interval Estimates. The point estimate is the best single value.

A good estimator should satisfy three conditions:

a) Unbiased: The predicted value of estimator should be equivalent to the mean of the parameter
b) Consistent: The value of estimator approaches the value of parameter as the sample size rises.
c) Relatively Efficient: The estimator has the minimum variance of all estimators that could be employed.

Confidence Intervals:

The point estimate is going to be distinct from the population parameter as due to sampling error, and there is no way to identify who is close to the actual parameter. For this reason, statisticians like to provide an interval estimate that is a range of values employed to estimate the parameter.

Confidence interval is the interval estimate with some specific level of confidence. The level of confidence is the probability which the interval estimate will hold the parameter. The level of confidence is 1-alpha. 1-alpha area lies in the confidence interval.

Maximum Error of Estimate:

The maximum error of estimate is symbolized by E and is one-half the width of confidence interval. The fundamental confidence interval for a symmetric distribution is set-up to point estimate minus the maximum error of estimate is less than the true population parameter that is less than the point estimate plus the maximum error of estimate. This formula will work for proportions and means as they will use the Z or T distributions that are symmetric. Afterward, we will talk regarding variances, that do not use a symmetric distribution and the formula will be distinct.

Area in Tails:

As the level of confidence is 1-alpha, the amount in tails is alpha. There is a notation in statistics that means the score that has the specified region in the right tail.

Illustrations:

Z(0.05) = 1.645 (the Z-score that has 0.05 to the right, and 0.4500 between 0 and it).
Z(0.10) = 1.282 (the Z-score that has 0.10 to the right, and 0.4000 between 0 and it).

In short-hand notation, the () are generally dropped, and the probability is written as a subscript. The greek letter alpha is used to signify the area in both the tails for confidence interval, and therefore alpha/2 will be the area in one tail.

Given below are some common values:

Confidence Level    Area between 0 and z-score    Area in one tail (alpha/2)    z-score
50%                        0.2500                                      0.2500                                0.674
80%                        0.4000                                      0.1000                                1.282
90%                        0.4500                                      0.0500                                1.645
95%                        0.4750                                      0.0250                                1.960
98%                        0.4900                                      0.0100                                2.326
99%                        0.4950                                      0.0050                                2.576

Note that in the above table, the area between 0 and z-score is simply one-half of the confidence level. Therefore, when there is a confidence level that isn't given above, all you require to do is to find it and divide the confidence level by two, and then look-up the area in the inner part of the Z-table and look-up the z-score at outside.

Also note that, if you look at student's t distribution, the top row is a level of confidence and the bottom row is z-score. However, this is where I got the extra digit of precision from.

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