Quantitative Business Methods for Decision Making Estimation and Testing of Hypotheses

Lecture Outlines Estimation n n Confidence interval for estimating means n n Confidence interval for predicting a new observation n n Confidence interval for estimating proportions

Lecture Outlines (con’t) Hypothesis Testing n n Null and alternative hypotheses n n Decision rules (Tests) and their level of significance n n Type I and Type II errors n n Tests of hypotheses for comparing means n n Tests of hypotheses for comparing proportions

Estimating a Population Mean n n Population mean is estimated by, the sample mean n n Standard error of, i.e. will decrease as n gets large.

Confidence Interval for Estimating if is known With a 95% degree of confidence is estimated within ( ) written as Or more accurately by

Confidence Interval for if is not known Use instead of, remember, and “t” is 95th% percentile of the t distribution with (n-1) degrees of freedom.

An Illustration Suppose n= 26. Then degrees of freedom (d.f.) = n-1 = 25. A two-sided degree of C.I. is computed by But, for a one-sided 95% C.I., t = instead of 2.064

Assumptions and Sample Size for Estimation of the mean Assumptions and Sample Size for Estimation of the mean The population should be normally (at least close to) distributed. If skew, then median is an appropriate measure of the center than the mean. To estimate mean with a specified margin of error (m.e.), take a random sample of size n large size.

Prediction Interval for a New Observation on X Prediction Interval for a New Observation on X Prediction Interval for a new observation is given by

Confidence Interval for a Population Proportion Let denote the proportion of items in a population having a certain property An estimate of is the binomial proportion:, What is ? For a C.I. for, use

Confidence Intervals for the Proportion (con’t) For estimating,“t” is the percentile of the t-distribution with (equivalently, percentile of the standard normal distribution), and s.e. of p is

Hypotheses Testing n n The hypothesis testing is a methodology for proving or disproving researcher’s prior beliefs. n n Statements that express prior beliefs are framed as alternative hypotheses. n n Complementary statement to an alternative hypothesis is called null hypothesis.

Null and Alternative H a : Researcher’s belief that are to be tested (alternate hypothesis) H 0 : Complement of H a (Null hypothesis)

Statistical Decision A decision will be either: Reject H 0 (H a is proved) or Do not reject H 0 (H a is not proved)

Hypothesis Testing Methodology for the mean Depending upon what an investigator believes a priori, an alternative hypothesis is formulated to be one of the followings: one-sided

A Test Statistic Regardless of what an alternative hypothesis about the mean is formulated, the decision rule is defined by a t- statistic:

Decision Rules for Testing Hypotheses About the Mean

Type I and Type II Errors Type I and Type II Errors

Comparing Two Means The reference number  is a specified amount for comparing the difference between two means. There are two distinct practical situations resulting in samples on X and Y.

Two Sampling Designs Paired Sample Two independent Samples

Paired Sample n n Two variables X and Y are observed for each unit in the sample to measure the same aspect but under two different conditions. n n Thus, for n randomly selected units, a sample of n pairs (X, Y) is observed. n n Compute differences: X 1 -Y 1 = d 1, X 2 -Y 2 = d 2, etc. and then mean n n Compute S d of differences

Paired Samples (con’t) Compute t-statistic:

Paired Samples (con’t) Reject H 0 if absolute value of t-statistic is more than the desired percentile of the t-distribution. Alternatively, find the p value of the t-statistics and reject H 0 if the p value is less than the desired significance level.

Two Independent (Unpaired) Samples n n Populations of variables X and Y (for example, males salary X and females salary Y). n n Take samples independently on X and Y. n n Compute n n Compute pooled standard deviation

Unpaired Samples (con’t) n n Compute n n Finally, compute t-statistic= Use p value to reach a decision

Comparing Proportions To estimate in a 95% C.I., compute,

Comparing Two Proportions n n For testing hypothesis about the difference, compute and t-statistic=