Business Statistics CONTENTS Comparing two samples Comparing two - PowerPoint PPT Presentation

TWO 𝜈 S OR MEDIANS: COMPARISONS Business Statistics

CONTENTS Comparing two samples Comparing two unrelated samples Comparing the means of two unrelated samples Comparing the medians of two unrelated samples Old exam question Further study

COMPARING TWO SAMPLES It often happens that we want to compare two situations ▪ do I sell more when there is music in my shop? ▪ is the expensive machine more precise than the cheap one? ▪ are adverisements on TV or internet equally profitable? ▪ do people buy more on Tuesdays than on Wednesday? ▪ in couples, who drinks more: the man or the woman? ▪ etc.

COMPARING TWO SAMPLES In all these questions we compare two populations ▪ Situation 1: two populations (or sub-populations) with similar variable ▪ sales in 105 days without music ▪ sales in 96 days with music ▪ Data matrix: two options SPSS requires this data presentation

COMPARING TWO SAMPLES ▪ Situation 2: one sample with paired observations ▪ drinks of the man in 78 couples ▪ drinks of the woman in the same 78 couples ▪ Data matrix: one option only ▪ Will be discussed in a later lecture

COMPARING TWO UNRELATED SAMPLES Situation 1 ▪ independent samples/unrelated samples ▪ introduce symbols for the two random variables ▪ e.g., using 𝑌 1 en 𝑌 2 ▪ 𝑌 1 with sample 𝑌 1,1 , 𝑌 1,2 , … , 𝑌 1,𝑜 1 and 𝑌 2 with sample 𝑌 2,1 , 𝑌 2,2 , … , 𝑌 2,𝑜 2 ▪ or using 𝑌 and 𝑍 ▪ 𝑌 : 𝑌 1 , 𝑌 2 , … , 𝑌 𝑜 𝑌 and 𝑍 : 𝑍 1 , 𝑍 2 , … , 𝑍 𝑜 𝑍 ▪ sample sizes can be different Or of course using “meaningful” indices: 𝑌 𝐶 and 𝑌 𝐻 for Belgium and Germany. Not 𝐶 and 𝐻 , because we need to stress that it is “about” a variable 𝑌 (like sales)

COMPARING TWO UNRELATED SAMPLES We want to test hypothesis such as ▪ are the means equal? ▪ 𝐼 0 : 𝜈 𝑌 = 𝜈 𝑍 or 𝐼 0 : 𝜈 1 = 𝜈 2 or 𝐼 0 : 𝜈 𝑌 1 = 𝜈 𝑌 2 or ... ▪ are the variances equal? 2 or etc. 2 = 𝜏 𝑍 ▪ 𝐼 0 : 𝜏 𝑌 ▪ are the proportions equal ▪ 𝐼 0 : 𝜌 𝑌 = 𝜌 𝑍 or etc. Also: ▪ inequalities, like 𝐼 0 : 𝜈 𝑌 ≥ 𝜈 𝑍 ▪ and non-zero differences, like 𝐼 0 : 𝜈 𝑌 = 𝜈 𝑍 + 85

COMPARING TWO UNRELATED SAMPLES Context: ▪ sample 𝑌 1 : sales in 𝑜 1 = 105 days without music ▪ sample 𝑌 2 : sales in 𝑜 2 = 96 days with music General idea: 𝑌 1 ~𝑒𝑗𝑡𝑢𝑠𝑗𝑐𝑣𝑢𝑗𝑝𝑜 𝜄 1 ▪ ൠ 𝜄 1 = 𝜄 2 ? 𝑌 2 ~𝑒𝑗𝑡𝑢𝑠𝑗𝑐𝑣𝑢𝑗𝑝𝑜 𝜄 2

COMPARING THE MEANS OF TWO UNRELATED SAMPLES Assumption (for now!): 2 ▪ 𝑌~𝑂 𝜈 𝑌 ; 𝜏 𝑌 2 ▪ 𝑍~𝑂 𝜈 𝑍 ; 𝜏 𝑍 ▪ in words: both samples come from normally distributed populations with known variances Question ▪ are 𝜈 𝑌 and 𝜈 𝑍 different? ▪ can we test this, on the basis of the (limited!) evidence concerning ҧ 𝑦 and ത 𝑧 ? ▪ so, can we reject 𝐼 0 : 𝜈 𝑌 = 𝜈 𝑍 ? So, the sampling To decide distribution of the 2 difference of ▪ use ത 𝑌 − ത 𝑍 ~𝑂 𝜈 ത 𝑍 , 𝜏 ത 𝑌− ത 𝑌− ത 𝑍 means is normal

ҧ COMPARING THE MEANS OF TWO UNRELATED SAMPLES For one sample, we had ത 𝑌 − 𝜈 ത 𝑌 ~𝑂 0,1 𝜏 ത 𝑌 As it turns out, for two samples, we have 𝑌 − ത ത 𝑍 − 𝜈 ത 𝑌 − 𝜈 ത 𝑍 ~𝑂 0,1 𝜏 ത 𝑌− ത 𝑍 𝑍 = 𝜈 𝑌 − 𝜈 𝑍 follows from the null hypothesis ▪ 𝜈 ത 𝑌 − 𝜈 ത ▪ for instance 𝐼 0 : 𝜈 𝑌 = 𝜈 𝑍 or 𝐼 0 : 𝜈 𝑌 − 𝜈 𝑍 = 85 𝑦 and ത 𝑧 are obtained from the data ▪ ▪ but what is 𝜏 ത 𝑍 ? 𝑌−ത

COMPARING THE MEANS OF TWO UNRELATED SAMPLES For one sample, we had 2 2 = 𝜏 𝑌 𝜏 ത 𝑌 𝑜 As it turns out, for two independent samples, we have 2 + 𝜏 ത 2 2 , so 𝜏 ത = 𝜏 ത 𝑌− ത 𝑍 𝑌 𝑍 2 2 𝜏 𝑌 + 𝜏 𝑍 𝜏 ത 𝑍 = 𝑌−ത 𝑜 𝑌 𝑜 𝑍 ▪ recall that variances add up when 𝑌 and 𝑍 are independent 2 but also 𝜏 𝑌−𝑍 2 + 𝜏 𝑍 2 + 𝜏 𝑍 ▪ e.g., 𝜏 𝑌+𝑍 2 2 2 = 𝜏 𝑌 = 𝜏 𝑌

COMPARING THE MEANS OF TWO UNRELATED SAMPLES Example Context: ▪ do I sell more when there is music in my shop? Experiment ▪ on some days the music is turned on, on other days the music is turned off ▪ you keep track of the sales during each day Data: ▪ sample of sales on days with music ( 𝑦 1 , 𝑦 2 , … , 𝑦 105 ) ▪ sample of sales on days without music ( 𝑧 1 , 𝑧 2 , … , 𝑧 96 ) Five step procedure

COMPARING THE MEANS OF TWO UNRELATED SAMPLES ▪ Step 1: ▪ 𝐼 0 : 𝜈 𝑌 = 𝜈 𝑍 ; 𝐼 1 : 𝜈 𝑌 ≠ 𝜈 𝑍 ; 𝛽 = 0.05 ▪ Step 2: sample statistic: ത 𝑌 − ത ▪ 𝑍 reject for “too large” and “too small” values ▪ ▪ Step 3: 𝑌−ത ത 𝑌−ത ത 𝑍 − 𝜈 𝑌 −𝜈 𝑍 𝑍 ▪ null distribution = ~𝑂 0,1 𝜏 ഥ 𝜏 ഥ 𝑌−ഥ 𝑌−ഥ 𝑍 𝑍 ▪ valid because ... in a minute we will supply ▪ Step 4: full details and a worked ▪ example ... 𝑨 𝑑𝑏𝑚𝑑 = ▪ 𝑨 𝑑𝑠𝑗𝑢 = ▪ Step 5: reject or not reject because ... ▪

COMPARING THE MEANS OF TWO UNRELATED SAMPLES ▪ But, wait ... 2 and 𝜏 𝑍 2 are known, while 𝜈 𝑌 ▪ ... isn’t it weird to assume that 𝜏 𝑌 and 𝜈 𝑍 are not known? ▪ In reality the population variances will often be unknown as well! ▪ remember we had the same problem in the one-sample case? ▪ there we decided to estimate the value of 𝜏 2 with the value of 𝑡 2 ▪ and paid a price of using the wider 𝑢 -distribution ▪ here we will do the same: estimate the two 𝜏 2 -values with two 𝑡 2 -values ▪ and pay the same price: use 𝑢 -dsitribution instead of 𝑨 -distribution

ҧ COMPARING THE MEANS OF TWO UNRELATED SAMPLES For one sample, we had ത 𝑌 − 𝜈 ത 𝑌 ~𝑢 df 𝑇 ത 𝑌 As it turns out, for two samples, we have 𝑌 − ത ത 𝑍 − 𝜈 ത 𝑌 − 𝜈 ത 𝑍 ~𝑢 df 𝑇 ത 𝑌− ത 𝑍 𝑍 = 𝜈 𝑌 − 𝜈 𝑍 follows from the null hypothesis ▪ 𝜈 ത 𝑌 − 𝜈 ത 𝑦 and ത 𝑧 are obtained from the data ▪ ▪ but what is 𝑡 ത 𝑍 ? 𝑌−ത ▪ and how to choose df ?

COMPARING THE MEANS OF TWO UNRELATED SAMPLES Two options for 𝑡 ത 𝑍 : 𝑌− ത 2 and 𝜏 𝑍 2 from 𝑡 𝑌 2 and 𝑡 𝑍 2 respectively ▪ 1: estimating 𝜏 𝑌 2 = 𝜏 2 and estimating 𝜏 2 as the 2 = 𝜏 𝑍 ▪ 2: assuming 𝜏 𝑌 weighted average of both sample variances Both options lead to a different value of df

COMPARING THE MEANS OF TWO UNRELATED SAMPLES Option 1: 2 and 𝜏 𝑍 2 from 𝑡 𝑌 2 and 𝑡 𝑍 2 respectively (Welch ’s test) ▪ estimating 𝜏 𝑌 2 2 𝑡 𝑌 + 𝑡 𝑍 𝑡 ത 𝑍 = 𝑌−ത 𝑜 𝑌 𝑜 𝑍 ▪ testing with 𝑢 -distribution with Compare to 2 2 2 𝜏 𝑌 + 𝜏 𝑍 2 2 𝑜 𝑌 + 𝑡 𝑍 𝑡 𝑌 𝜏 ത 𝑍 = 𝑌−ത 𝑜 𝑌 𝑜 𝑍 𝑜 𝑍 df = 2 2 2 2 𝑡 𝑌 𝑡 𝑍 𝑜 𝑌 𝑜 𝑍 𝑜 𝑌 − 1 + 𝑜 𝑍 − 1 quick rule, but bad approximation: 𝑒𝑔 ≈ min 𝑜 𝑌 − 1, 𝑜 𝑍 − 1

COMPARING THE MEANS OF TWO UNRELATED SAMPLES You can read this as Option 2: 2 +df 𝑍 𝑡 𝑍 2 df 𝑌 𝑡 𝑌 ▪ estimating the common 𝜏 2 from both samples df 𝑌 +df 𝑍 2 and 𝑡 𝑍 a “weighted mean” of 𝑡 𝑌 2 , the pooled variance 𝑡 P 2 ▪ 2 + 𝑜 𝑍 − 1 𝑡 𝑍 2 2 = 𝑜 𝑌 − 1 𝑡 𝑌 𝑡 P 𝑜 𝑌 − 1 + 𝑜 𝑍 − 1 ▪ and Compare to 2 2 𝑡 P + 𝑡 P 2 2 𝑡 𝑌 + 𝑡 𝑍 𝑡 ത 𝑍 = 𝑡 ത 𝑍 = 𝑌−ത 𝑌− ത 𝑜 𝑌 𝑜 𝑍 𝑜 𝑌 𝑜 𝑍 ▪ testing with 𝑢 -distribution with You can read this as df = 𝑜 𝑌 − 1 + 𝑜 𝑍 − 1 = 𝑜 𝑌 + 𝑜 𝑍 − 2 df 𝑌 + df 𝑍

COMPARING THE MEANS OF TWO UNRELATED SAMPLES

ҧ ҧ ҧ EXERCISE 1 When to use: 𝑦− ത 𝑧 a. 𝑨 = 2 2 𝜏𝑌 𝜏𝑍 𝑜𝑌 + 𝑜𝑍 𝑦− ത 𝑧 b. 𝑢 = 2 2 𝑡𝑌 𝑡𝑍 𝑜𝑌 + 𝑜𝑍 𝑦− ത 𝑧 c. 𝑢 = 2 2 𝑡𝑄 𝑡𝑄 𝑜𝑌 + 𝑜𝑍

COMPARING THE MEANS OF TWO UNRELATED SAMPLES Use of SPSS a data set on Computer Anxiety Rating split by gender

COMPARING THE MEANS OF TWO UNRELATED SAMPLES Results split by gender Results of 𝑢 -test

COMPARING THE MEANS OF TWO UNRELATED SAMPLES Zoom in 𝑢 -test with pooled 2 = 𝜏 𝑍 2 estimate of 𝜏 𝑌 value of the 𝑢 -statistic 𝑢 -test with separate ( 𝑢 calc ) 𝑞 -value degrees of freedom 2 and 𝜏 𝑍 2 estimates of 𝜏 𝑌 (2-sided)

COMPARING THE MEANS OF TWO UNRELATED SAMPLES And one more thing ... tests of the assumption of equal variance 𝑞 -value for 2 = 𝜏 𝑍 2 ≠ 𝜏 𝑍 2 versus 𝐼 1 : 𝜏 𝑌 2 𝐼 0 : 𝜏 𝑌 this test

COMPARING THE MEANS OF TWO UNRELATED SAMPLES For these two tests, we need both ത 𝑌 and ത 𝑍 to be normally distributed ▪ This means either of the following three requirements holds: ▪ 𝑌 is a normally distributed population ▪ 𝑌 has a symmetric distribution and 𝑜 𝑌 ≥ 15 ▪ 𝑜 𝑌 ≥ 30 ▪ Also for 𝑍 one of these requirements must hold ▪ but not necessarily the same one as for 𝑌