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Quantitative Business Methods for Decision Making Estimation and Testing of Hypotheses Lecture Outlines Estimation Confidence interval for estimating means Confidence interval for predicting a new observation Confidence interval for estimating proportions 403.6 2 Lecture Outlines (con’t) Hypothesis Testing Null and alternative hypotheses Decision rules (Tests) and their level of significance Type I and Type II errors Tests of hypotheses for comparing means Tests of hypotheses for comparing proportions 403.6 3 Estimating a Population Mean Population mean is estimated by x , the sample mean s Standard error of x , i.e. s x n will decrease as n gets large. 403.6 4 Confidence Interval for Estimating if n is known x is With a 95% degree of confidence estimated within ( X 2 X 2 ) x x written as X 2 x Or more accurately by X 1.96 x 403.6 5 Confidence Interval for is not known n if x Use x ts x instead of x 1.96 x , remember s x s n , and “t” is 95th% percentile of the t distribution with (n-1) degrees of freedom. 403.6 6 An Illustration Suppose n= 26. Then degrees of freedom (d.f.) = n-1 = 25. A two-sided 95% degree of C.I. is computed by x 2.064s x But, for a one-sided 95% C.I. , t = 1.711 instead of 2.064 403.6 7 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 z 2 2 n 2 m.e. 403.6 8 Prediction Interval for a New Observation on X Prediction Interval for a new observation is given by 1 x ts 1 n 403.6 9 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 X proportion: p , What is X? n For a C.I. for , use p ts p 403.6 10 Confidence Intervals for the Proportion (con’t) For estimating ,“t” is the percentile of the t-distribution with df (equivalently, percentile of the standard normal distribution), and s.e. of p is sp p(1 p) n 403.6 11 Hypotheses Testing The hypothesis testing is a methodology for proving or disproving researcher’s prior beliefs. Statements that express prior beliefs are framed as alternative hypotheses. Complementary statement to an alternative hypothesis is called null hypothesis. 403.6 12 Null and Alternative Ha: Researcher’s belief that are to be tested (alternate hypothesis) H0: Complement of Ha (Null hypothesis) 403.6 13 Statistical Decision A decision will be either: Reject H0 (Ha is proved) or Do not reject H0 (Ha is not proved) 403.6 14 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: Ha : 0 1. 2. Ha : 0 3. Ha : 0 one-sided 403.6 15 A Test Statistic Regardless of what an alternative hypothesis about the mean is formulated, the decision rule is defined by a t- statistic: X 0 t Sx 403.6 16 Decision Rules for Testing Hypotheses About the Mean Hypotheses 1. Ho: = o Ha: o 2. Ho: o Ha: o 3. Ho: o Ha: o Decision Rule At level of significance, reject Ho in favor of Ha if either t-statistic t 2 or -t 2 At level of significance, reject Ho in favor of Ha if t-statistic t At level of significance, reject Ho in favor of Ha if t-statistic t 403.6 17 Type I and Type II Errors Decision taken is Accept H0 Accept H1 ----------------------------------------------------------------------Suppose correct Type I Error H0 is true decision (wrong decision) -----------------------------------------------------------------------Suppose Type II Error correct H1 is true (wrong decision) decision Type I error : reject H0 if H0 is true. Type II error : Do not reject H0 when H0 is false. 403.6 18 Comparing Two Means X population X , X Y population Y , Y Consider 1. H0: X - Y = 2. H0: X - Y Ha: X - Y , Ha: X - Y , 3. H0: X - Y Ha: X - Y . 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. 403.6 19 Two Sampling Designs •Paired Sample •Two independent Samples 403.6 20 Paired Sample Two variables X and Y are observed for each unit in the sample to measure the same aspect but under two different conditions. Thus, for n randomly selected units, a sample of n pairs (X, Y) is observed. Compute d differences: X1-Y1= d1, X2-Y2= d2, etc. d and then mean Compute Sd of differences 403.6 21 Paired Samples (con’t) Compute Sd sd n d t-statistic: t - statistic Sd 403.6 22 Paired Samples (con’t) •Reject H0 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 H0 if the p value is less than the desired significance level. 403.6 23 Two Independent (Unpaired) Samples Populations of variables X and Y (for example, males salary X and females salary Y). Take samples independently on X and Y. Compute X, S , Y, S x y Compute pooled standard deviation S n1 1S2x n2 1S2y n1 n2 2 403.6 24 Unpaired Samples (con’t) Compute SE of X Y S x y Finally, compute t-statistic= 1 1 S n1 n 2 X Y Sx y Use p value to reach a decision 403.6 25 Comparing Proportions 1 and 2 To estimate 1 2 in a 95% C.I., compute, p1 1 p1 p2 1 p2 p1 p2 1.96 n2 n1 403.6 26 Comparing Two Proportions For testing hypothesis about the difference 1 2 , compute n1 p1 n2 p2 n n1 n2 , p n and t-statistic= p1 p2 p (1 p ) n 403.6 27