Confidence Interval PDF

CONFIDENCE INTERVAL  Introduction to Inferential Statistics Statistical inference refers to the methods by which we arrive at a conclusion about a population on the basis of the information contained in a sample drawn from that population. The...

CONFIDENCE INTERVAL  Introduction to Inferential Statistics Statistical inference refers to the methods by which we arrive at a conclusion about a population on the basis of the information contained in a sample drawn from that population. The sample is drawn from a population randomly and hence a random sample of size n considers a method of selecting a sample with predetermined probability. Sampling distribution provides the underlying probability distribution of the statistics by which we try to generalize for population. As the sampling distribution is based on all the possible samples from a population of size N, the expected value, variance, and other population characteristics can be obtained theoretically from the statistics of a random sample. Statistical inference is comprised of two broad components: (i) estimation and (ii) test of hypothesis.  Estimation Estimation refers to the methods of finding the value of statistic from a sample corresponding to its population value such that it satisfies some good properties to represent a parameter. On the other hand, once we obtain a value to estimate the corresponding population value or parameter, it is needed to know whether the sample value used as an estimate is close to the true value. This issue is of great importance in statistics and there are various techniques to make comment or to make decision about generalizing the estimates obtained by estimating parameters for the population as a whole. As the estimates are obtained from randomly drawn samples, it is likely that the values we consider to be population characteristics may differ from the true value. It means that although drawn from the population, a sample estimate may not be exactly equal to the corresponding parameter. Hence, as a statistician or biostatistician, it is necessary to provide procedures to make decision about the population values such that with the help of underlying sampling distribution, we may come up  Ways of Estimation This estimation can be made in two ways:  CONFIDENCE INTERVAL FOR A POPULATION MEAN () An interval estimate of  is an interval (L, U) containing the true value of  with a probability of 1-, where 1- = is called the confidence coefficient, L = lower limit of the confidence interval, and U = upper ofprobability limit= of  not the confidence lying in this interval. It is interval the interval between Low (L) and upper (U) limit within which the µ population mean will lie If Population standard deviation (σ) is known CI= FOR 95% CONFIDENCE INTERVAL Z=2 and t will depend on sample size but would be close to 2 If Population standard deviation (σ) is not known CI= s= Degree of freedom= CONFIDENCE INTERVAL FORMULA IN GENERAL CONFIDENCE INTERVAL FOR THE DIFFERENCE BETWEEN TWO POPULATION MEANS Sometimes there arise cases in which we are interested in estimating the difference between two population means. From each of the populations an independent random sample is drawn and, from the data of each, the sample means ( and 2 ) are calculated If Population standard deviations (σ) are known CI= ( 2 ) If Population standard deviations (σ) are not known CI= ( 2 ) Degree of freedom= CONFIDENCE INTERVAL FOR A POPULATION PROPORTION To estimate a population proportion we proceed in the same manner as when estimating a population mean. A sample is drawn from the population of interest, and the sample proportion , is computed. This sample proportion is used as the point estimator of the population proportion. A confidence interval is obtained by the general formula sample proportion n-sample size CONFIDENCE INTERVAL FOR THE DIFFERENCE BETWEEN TWO POPULATION PROPORTIONS Interpretin g Confidence Intervals Reference Chapter 6 Estimation WAYNE W. DANIEL, BIOSTATISTICS A Foundation for Analysis in the Health Sciences