Confidence - Confidence Interval

Inferential statistical analysis is the process of sampling characteristic from large populations, summarizing those characteristics or attributes, and drawing conclusions or making predictions from the summary or descriptive information. When inferences are made based on sample data, there is always a chance that a mistake will be made. The probability that the inference will be correct is referred to as the degree of confidence with which the inference can be stated. There are two types mistakes that can occur: type I error and type II error.

Type I error:
-. It made if H0 is rejected when H0 is true. The probability of a type I error is denoted by alpha.
-. Probability of incorrectly rejecting the null hypothesis – in most cases, it means saying a difference or correlation exists when it actually does not. Also termed alpha. Typically levels are 5 or 1 percent, termed the .05 or .01 level
-. Stating that the result of sampling are unacceptable when in reality the population from which the sample was taken meets the stated requirements

The probability of a type I error – rejecting what should be accepted – is known as the alpha risk, or level of significance. A level of significance of 5 percent corresponds to a 95 percent chance of accepting what should be accepted. In such an instance, the analyst would have 95 percent confidence in the conclusions drawn or the inferences made. Another interpretation would be that there is a 95 percent chance that the statements made are correct.

Type II error:
-. It made if H1 accepted when H0 is true. The probability of a type II error is denoted by beta.
-. Probability of incorrectly failing to reject the null hypothesis – in simple terms, the chance of not finding a correlation or mean difference when it does exist. Also termed beta, it is inversely related to type I error. The value of 1 minus the type II error (1 – beta) is defined as power.
-. Stating the results of sampling are acceptable when in reality the population from which the sample was taken does not meet the stated requirements

The probability of a type II error – accepting what should be rejected – is known as the beta risk. It is important when acceptance sampling plans are developed and used.

It is possible to estimate population parameters, such as the mean or the standard deviation, based on sample values. How good the predictions are depends on how accurately the sample values reflects the values for the entire population. If a high level of confidence in the inferences is desired, a large proportion of the populations should be observed. In fact, in order to achieve 100 percent confidence, one must sample the entire population. Because of the economic considerations typically involved in inspection, the selection of an acceptable confidence interval is usually seen as a trade-off between cost and confidence. Typically, 90, 95, 99 percent confidence levels are used, with the 99.73 percent leve used in certain quality control applications.

A confidence interval is a range of values that has a specified likelihood of including the true value of a population parameter. It is calculated from sample calculations of the parameters.

There are many types of population parameters for which confidence interval can be established. Those important in applications include means, proportions (percentages) and standard deviations.

Confidence intervals are generally presented in the following format:
Point estimate of populations parameter +/- (Confidence factor)x(Measure variability)x(Adjusting factor)


Source:
-. Fundamentals of Industrial Quality Control, 3rd edition, Lawrence S. AFT, St. Lucie Press, London, 1998
-. Mathematical Statistics with Application, William Mendenhall, Richard L. Sceaffer, Dennis D. Wackerly,
-. Multivariate Data Analysis, Joseph F. Hair, Jr.; William C. Black; Barry J. Babin; Rolph E. Anderson; Ronald L. Tatham, Person Education International, 2006, Singapore