Sampling Distribution

 Course Outline

·           Parameter and statistics: Sampling distribution of mean, Variance and chi-square; Standard error of statistics (concept only); Central limit theorem (concept only).

·           Estimation of parameter; Confidence interval for mean (Difference between means) & Variance.

Introduction

·     Estimate and Estimators: When the data are collected by sampling from a population, the most important objective of statistical analysis is two draw inferences or generalizations about that population from the information embodied in the sample data. Statistical estimation is concerned with the methods by which population characteristics are estimated from sample information. An estimator is a sample statistic used to estimate a population parameter. In the other words, it is a rule that tells how to calculate an estimate based on the measurements contained in a sample. For example if we use a value of to estimate the population mean m, or a value of s² to estimate the population variance s², then we are in each case using a point estimate. A point estimate is a single static that is used to estimate an unknown population, where as interval estimate sues the range of values to estimate the population parameter.

·     Point and Interval Estimate: If an estimate of a population parameter is given by a single value (a specific observed value of statistic) then the estimate is called a point estimate of parameter.

          In the case of interval estimation instead of assigning a single value to a population parameter, an interval is constructed around the point estimate and then probabilistic statement that is interval contains the corresponding population parameter is made.

·     Confidence Level: The probability that we associate with an interval estimate is called the confidence level. This probability indicates how confident we are that the interval estimate will include the population parameter. A higher probability means more confidence. In estimation, the most commonly used confidence levels are 90%, 95% and 99%. 95% confidence level indicates that 95% probability that the value of randomly drawn items lies within the limits indicated. There is thus risk of only 5% that it will fall outside the limits so indicated.

·     Confidence Interval: In point estimation, statistics which is calculated from observed sample is represented as estimator of parameter of population. This value of statistics may or may not be proper representation of the parameter. Therefore two limits are considered within which the parameter lies with certain confidence probability. This two limits are called confidence limits and the internal is called confidence interval.

          In an interval estimation of the population parameter q, if we can find two quantities t₁ and t₂ based on sample observation drawn from the population such that the unknown parameter q is included in the interval [t₁, t₂] in a specified percentage of cases, then this interval is called a confidence interval for the parameter q.

·     Parameter and Statistic: Statistic (or sample statistic) is a statistical measure based only on all the units selected in a sample, i.e. sample mean, sample S.D. etc are called statistic whereas parameter is a statistical measure based on all the units in the population and defined as true characteristic of total population.

·     Sampling Distribution: Number of random samples each containing r units (or r observations) from population containing n units = c(n, r) = K (say) (If r is less than n, the number of samples will be more than n but if r = n, then value will be only one sample equal to population) Each sample will give its own value for any statistic (mean, S.D. etc). All such values of statistic together with their corresponding frequencies constitute the sampling distribution of the statistics. The values of statistic varies from sample to sample. This variation in the values of the statistic is known as sampling variation.

·     Sampling distribution of mean: Let a population be infinitely large and having the population mean m and population variance s². If x is a random variable denoting the measurement of the characteristics, then,

          expected value of X, E(X) = m and variance of X, Var (X) = s²

^{Exercise 6.2}