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Inferential Statistics Dr. Dennis S. Mapa Prof. Manuel Leonard F. Albis UP School of Statistics Decision Making Under Uncertainty A credit investigator recommends favorable action to an application for credit using certain characteristics of the client (credit scoring). Using regression analysis, a financial analyst concludes that a financial ratio is significantly correlated with a set of business variables. Decision Making Under Uncertainty An economist concludes that a set of variables can be considered as “deep” determinants of economic growth using the results of his/her econometric model from a sample of cross-country data. Decision Making Under Uncertainty These are examples of decision making that employ statistical methods. The statistical methods used are under the area of statistical inference or inferential statistics. What is Inferential Statistics? Statistical inference refers to methods by which one uses sample information to make inferences or generalizations about a population. Deals with inferences about some characteristic(s) of the population. Recall the other area of statistics is descriptive statistics which is composed of methods concerned with collecting, describing and analyzing a set of data without drawing inferences on the population to which the data belong. Point Estimation An estimator is any statistic whose value is used to estimate an unknown parameter. A realized value of an estimator is called an estimate. Two Important Statistics the mean the standard deviation The Arithmetic Mean the most common “average” the population mean is denoted by the greek letter µ (“mu”), N Xi i 1 N the sample mean (estimator) is denoted by n X Xi i 1 n Standard Deviation the positive square root of the variance Population Standard Deviation N Sample Standard Deviation X i 2 i 1 N X i X n s i 1 2 n 1 The standard deviation is known as volatility. Proportions In a binomial experiment where the outcomes can be considered as “success” or “failure” (example: having a positive or negative return), the parameter of interest is the proportion (p) of sucess. A success is usually denoted by 1 and failure 0. A (point) estimator of the proportion (p) is, X pˆ n where X represents the number of successes in a sample of n observations Statistical Inference Statistical inference refers to methods by which one uses sample information to make inferences or generalizations about a population. The two areas of statistical inference are interval estimation and hypothesis testing. Statistical Inference Confidence Interval Estimation uses sample data to calculate the lower and upper bound of an interval such that there is a specified probability that the true parameter value lies within the interval. This is in contrast to finding a single value in point estimation. Hypothesis Testing is the process of making decisions on whether or not the sample agrees with the researcher’s assertion regarding some characteristic of the population. Sampling Distributions An important concept in statistical inference is the notion of sampling distributions. The probability distribution function of a statistic is called its sampling distribution. Recall that statistics (such as the sample mean) are used to estimate the unknown parameters of the population (such as the population mean) Sampling Distributions A statistic is a random variable whose value depends only on the observed sample and may vary from sample to sample. The sampling distribution of a statistic will depend on the size of the population, the size of the sample, and the method of choosing the sample. The standard deviation of the sampling distribution is called the standard error of the statistic. It tells us the extent to which we expect the values of the statistic to vary from different possible samples. What is the Standard Error? It is a measure of error of estimation. It is the positive square root of the variance of the estimator. It measures, on the average, the dispersion of each possible value of the estimator from the actual value being estimated.