TWO μs OR MEDIANS: COMPARISONS. Business Statistics

Transcription

1 TWO μs OR MEDIANS: COMPARISONS Business Statistics

2 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

3 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.

4 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

5 COMPARING TWO SAMPLES Situation : 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

6 COMPARING TWO UNRELATED SAMPLES Situation 1 independent samples/unrelated samples introduce symbols for the two random variables e.g., using X 1 en X X 1 with sample X 1,1, X 1,,, X 1,n1 X,1, X,,, X,n or using X and Y X: X 1, X,, X nx sample sizes can be different and Y: Y 1, Y,, Y ny and X with sample Or of course using meaningful indices: X B and X G for Belgium and Germany. Not B and G, because we need to stress that it is about a variable X (like sales)

7 COMPARING TWO UNRELATED SAMPLES We want to test hypothesis such as are the means equal? H 0 : μ X = μ Y or H 0 : μ 1 = μ or H 0 : μ X1 = μ X or... are the variances equal? H 0 : σ X = σ Y or etc. are the proportions equal H 0 : π X = π Y or etc. Also: inequalities, like H 0 : μ X μ Y and non-zero differences, like H 0 : μ X = μ Y + 85

8 COMPARING TWO UNRELATED SAMPLES Context: sample X 1 : sales in n 1 = 105 days without music sample X : sales in n = 96 days with music General idea: X 1 ~distribution θ 1 ൠ θ X ~distribution θ 1 = θ?

9 COMPARING THE MEANS OF TWO UNRELATED SAMPLES Assumption (for now!): X~N μ X ; σ X Y~N μ Y ; σ Y in words: both samples come from normally distributed populations with known variances Question are μ X and μ Y different? can we test this, on the basis of the (limited!) evidence concerning xҧ and തy? so, can we reject H 0 : μ X = μ Y? To decide use തX തY ~N μx ത തY, σx ത തY So, the sampling distribution of the difference of means is normal

10 COMPARING THE MEANS OF TWO UNRELATED SAMPLES For one sample, we had തX μ ത X σ ത X ~N 0,1 As it turns out, for two samples, we have തX തY μx ത μy ത ~N 0,1 σ ത X തY μx ത μy ത = μ X μ Y follows from the null hypothesis for instance H 0 : μ X = μ Y or H 0 : μ X μ Y = 85 xҧ and തy are obtained from the data but what is σx ത തY?

11 COMPARING THE MEANS OF TWO UNRELATED SAMPLES For one sample, we had σx ത = σ X n As it turns out, for two independent samples, we have = σx ത + σy ത, so σ ത X തY σx ത തY = σ X + σ Y n X n Y recall that variances add up when X and Y are independent e.g., σ X+Y = σ X + σ Y but also σ X Y = σ X + σ Y

12 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 (x 1, x,, x 105 ) sample of sales on days without music (y 1, y,, y 96 ) Five step procedure

13 COMPARING THE MEANS OF TWO UNRELATED SAMPLES Step 1: H 0 : μ X = μ Y ; H 1 : μ X μ Y ; α = 0.05 Step : sample statistic: തX തY reject for too large and too small values Step 3: null distribution ത X തY μ X μ Y σ ഥ X ഥY valid because... Step 4: z calc = z crit = Step 5: reject or not reject because... X = ത തY ~N 0,1 σ X ഥY ഥ in a minute we will supply full details and a worked example...

14 COMPARING THE MEANS OF TWO UNRELATED SAMPLES But, wait isn t it weird to assume that σ X and σ Y are known, while μ X and μ Y 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 σ with the value of s and paid a price of using the wider t-distribution here we will do the same: estimate the two σ -values with two s -values and pay the same price: use t-dsitribution instead of z-distribution

15 COMPARING THE MEANS OF TWO UNRELATED SAMPLES For one sample, we had തX μ ത X S ത X ~t df As it turns out, for two samples, we have തX തY μ ത X μ ത Y S ത X തY ~t df μx ത μy ത = μ X μ Y follows from the null hypothesis xҧ and തy are obtained from the data but what is sx ത തY? and how to choose df?

16 COMPARING THE MEANS OF TWO UNRELATED SAMPLES Two options for s ത X തY: 1: estimating σ X and σ Y from s X and s Y respectively : assuming σ X = σ Y = σ and estimating σ as the weighted average of both sample variances Both options lead to a different value of df

17 COMPARING THE MEANS OF TWO UNRELATED SAMPLES Option 1: estimating σ X and σ Y from s X and s Y respectively (Welch s test) testing with t-distribution with quick rule, but bad approximation: df min n X 1, n Y 1 sx ത തY = s X + s Y n X df = s X n X s X n X + s Y n X 1 + n Y n Y s Y n Y n Y 1 Compare to σ തX തY = σ X + σ Y n X n Y

18 COMPARING THE MEANS OF TWO UNRELATED SAMPLES Option : estimating the common σ from both samples a weighted mean of s X and s Y, the pooled variance s P and s P = n X 1 s X + n Y 1 s Y n X 1 + n Y 1 sx ത തY = s P + s P n X n Y testing with t-distribution with df = n X 1 + n Y 1 = n X + n Y You can read this as df X s X +df Y s Y df X +df Y Compare to s തX തY = s X + s Y n X n Y You can read this as df X + df Y

19 COMPARING THE MEANS OF TWO UNRELATED SAMPLES

20 EXERCISE 1 When to use: x a. z = ҧ തy σ X + σ Y n X n Y x b. t = ҧ തy s X + s Y n X n Y x c. t = ҧ തy s P + s P n X n Y

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

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

23 COMPARING THE MEANS OF TWO UNRELATED SAMPLES Zoom in t-test with pooled estimate of σ X = σ Y t-test with separate estimates of σ X and σ Y value of the t-statistic (t calc ) degrees of freedom p-value (-sided)

24 COMPARING THE MEANS OF TWO UNRELATED SAMPLES And one more thing... tests of the assumption of equal variance H 0 : σ X = σ Y versus H 1 : σ X σ Y p-value for this test

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

26 EXERCISE a. Suppose X and Y are two distributions. We sample with n X = 10 and n Y = 0 and we want to test H 0 : μ X = μ Y. Do we need to make any assumption? b. Same as a, but now we know that X and Y are symmetrically distributed. c. Same as a, but now we know that X and Y are normally distributed

27 COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES Recall the non-parametric one-sample test for the median the Wilcoxon signed ranks test replacing the values by ranks and testing the sum of the positive ranks Can we also develop a non-parametric (rank-order) order test for two unrelated samples? Yes we can: Wilcoxon-Mann-Whitney test named after Frank Wilcoxon, Henry Mann, and Donald Whitney also named Wilcoxon (Mann-Whitney) test, Mann-Whitney test, etc. requirement: similar shape of distribution

28 COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES Computational steps of the Wilcoxon-Mann-Whitney test combine both samples (X and Y) assign ranks to the combined sample ties get an average rank sum the ranks of both samples separately (T X and T Y ) compare the test statistic T X (or T Y ) to a critical value from the table

29 COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES Example (same as before) Sample data are collected on the capacity rates (in %) for two factories factory A, the rates are 71, 8, 77, 94, 88 factory B, the rates are 85, 8, 9, 97 Are the median operating rates for two factories the same (at a significance level α = 0.05)?

30 COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES Example data A: x i (n X = 5) data B: y i (n Y = 4) one case of ties (8) T Y = 4.5 a tie: observations 3 and 4 are 8, so assign rank 3.5 to facilitate the discussion, we focus on the sample with the smallest sample size Capacity Rank Factory A Factory B Factory A Factory B Rank sums:

31 COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES Testing the Wilcoxon-Mann-Whitney T statistic using a table of critical values included in tables at exam using a normal approximation valid for large samples when Wilcoxon-Mann-Whitney table of critical values is not sufficient

32 COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES Table of critical values of Wilcoxon statistic for n x = n 1 = 4 and n y = n = 5 at α = 0.05: T lower = 11, T upper = 9 R crit = 0,11 [9,45] Why 0 and 45? 0 because the sum of the ranks cannot be smaller than because the sum of the ranks here cannot be larger than 45.

33 COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES Conclusion from small sample Wilcoxon-Mann-Whitney test T Y = 4.5 is between T lower = 11 and T upper = 9 Therefore, do not reject the null hypothesis (H 0 : M X = M Y ) at the 5% level There is not enough evidence to conclude that the medians are different

34 COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES Large sample approximation Under H 0, it can be shown that E T Y = n Y n X +n Y +1 var T Y = n Xn Y n X +n Y +1 1 Further, when n X 10 or n Y 10, we use a normal approximation: T Y ~N n X n X +n Y +1 Z = T Y n Y n X+n Y+1 n X n Y n X +n Y +1 1, n Xn Y n X +n Y +1 1 ~N 0,1

35 COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES Large sample approximation (continued) so you can compute z calc = T Y,calc n Y n X+n Y+1 n X n Y n X +n Y +1 1 and compare it to z crit (e.g., ±1.96)

36 COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES Use of SPSS T = 345 z-score with normal approximation p-value (-sided)

37 OLD EXAM QUESTION 1 May 015, Qa

38 FURTHER STUDY Doane & Seward 5/E , 16.4 extra document Wilcoxon Mann-Whitney test Tutorial exercises week 3 z-test (known variance) t-test (pooled variance) t-test (separate variance estimates) Wilcoxon Mann-Whitney test