1. Testing Group Difference

2. Are These Groups the Same?
Testing group differences
Do Canadians have different attitudes toward Dell than Americans?
Do Fujitsu and Toshiba have different brand images among Pepperdine students?
Does a high income class eat more beef than a lower income class?
Do sales reps taking different training programs in different regions show different performance?

3. Warming Up: Testing One Mean
Average weight = 240? (two-tailed test)
Setting up the null hypothesis (H0: μ = 240; Ha: μ ≠ 240)
Determining the confidence level ( = significance level)
Calculating sample mean
Calculating the standard error of the mean
When population variance (σ2) is known
When population variance (σ2) is unknown
Calculating z-statistic (or t-statistic: d.f.= n-1)
If |statistic| > critical value, then reject the null

4. When Do We Use Normal or t Distribution?
large (>30) sample size?
Yes
Normal Distribution
(Calculating z-statistic) No
Population variance known?
Yes No
t Distribution
(Calculating t-statistic; d.f. = n-1)

5. Example: Testing One Mean
Ho:
Racquet No.
Weight 1
240 2
230 3
220 4 5
250 6
260 7 8 9 10
200
Sum
2350
Confidence interval: 95%
Sample mean = 235
Population variance is unknown
Estimating population variance = 316.7
Standard error of the sample mean = 5.6
t-statistic = -0.89
Critical value = 2.262

6. Example (Cont’d)
Ho: μ = 240
Ha: μ ≠ 240
α = 0.05
0.025
0.025 -t t

7. Decision Rule: When to Reject Ho?
One-Sided (Tailed) (Ha: μ>μo or μ<μo or π>πH or π <πH)
Two-Sided (Tailed) (Ha: μ ≠μo or π≠πH)
Test Statistics
Zobs>Zα or Zobs<-Zα
tobs>tα or tobs<-tα
Zobs>Zα/2 or Zobs<-Zα/2
tobs>tα/2 or tobs<-tα/2
P-value
P-value < α
P-value < α/2
SPSS P-value
(Two-sided)
(Sig.)/2 < α
Sig. < α

8. Comparing Two Independent Means
If two populations are independent
Pop. Mean Pop. Std Dev. Sample Size Sample Mean Sample SD
μA σA nA sA
μB σB nB sB
Setting up the null hypothesis (H0: μA = μB; Ha: μA ≠ μB)
Determining the confidence level ( = significance level)
Calculating sample means

9. Comparing Two Independent Means (Cont’d)
Calculating the standard errors of the means
When population variance (σ2) is known
When population variance (σ2) is unknown
Calculating the standard errors of the “difference in means”
Calculating z-statistic (or t-statistic: d.f.= nA+nB-2)
If |statistic| > critical value, then reject the null

10. Example: Comparing Two Independent Means
Racquet No.
Weight (Machine A)
Weight (Machine B) 1
240 2
230
250 3
220
260 4 5 6 7 8 9 10
200
Sum
2350
2500
Ho:
Confidence interval: 95%
Sample mean = 235 vs. 250
Population variance is unknown
Estimating population variance = vs. 66.7
Standard error of the difference = 6.2
t-statistic = -2.42
Critical value = 2.101

11. Example (Cont’d)
Ho: µA=µB
Ha: µA≠ µB
α = 0.05
0.025
0.025 -t t

12. Comparing Two Related Means
If they are related (e.g., Pretest & Post Test Scores)
Setting up the null hypothesis (H0: D = 0; Ha: D ≠ 0)
Determining the confidence level ( = significance level)
Calculating the difference of each pair (d)
Calculating sample means of the difference
Calculating the standard errors of the mean difference
Calculating t-statistic (d.f. = n – 1)
If |statistic| > critical value, then reject the null

13. Example: Comparing Two Related Means
Racquet No.
Weight (at time 1)
Weight (at time 2) d 1
240 2
230
250
-20 3
220
260
-40 4 5 10 6 7 8 9
-30
200
-50
Sum
2350
2500
-150
Ho:
Confidence interval: 95%
Sample mean of d = -15
Estimating variance of d = 405.6
Standard error of the mean difference = 6.4
t-statistic = -2.36
Critical value = 2.262

14. Example (Cont’d)
Ho: D = 0
Ha: D ≠ 0
α = 0.05
0.025
0.025 -t t

15. Paired T Test: Results

16. Decision Rule: When to Reject Ho?
One-Sided (Tailed) (Ha: μ>μo or μ<μo or π>πH or π <πH)
Two-Sided (Tailed) (Ha: μ ≠μo or π≠πH)
Test Statistics
Zobs>Zα or Zobs<-Zα
tobs>tα or tobs<-tα
Zobs>Zα/2 or Zobs<-Zα/2
tobs>tα/2 or tobs<-tα/2
P-value
P-value < α
P-value < α/2
SPSS P-value
(Two-sided)
(Sig.)/2 < α
Sig. < α

17. Comparing Three or More Means: Analysis of Variance (ANOVA)
Idea: If a significant portion of total variation can be explained by between-group variation, then we can conclude that groups are different
Total variation = Between group variation + Within group variation
Sum of total variation
Sum of between-group variation
Sum of within-group variation

18. Example: Comparing Multiple Groups
No. A B C 1
240
230 2
250 3
220
260 4 5 6 7 8 9 10
200
Sum
2350
2500
1920
Mean
235
Total mean = 6770 / 28 = 241.8
Total variation = 5,410.7
Between-group variation = 1,160.7
Within-group variation = 4,250

19. Calculating Mean Squared Variation
Mean squared variation = sum of squared variation / degrees of freedom
Degrees of freedom:
Total variation: # total observation (n) – 1
Between-group variation: # groups (k) – 1
Within-group variation: # total observation (n) – # group (k)

20. Example: Comparing Multiple Groups
No. A B C 1
240
230 2
250 3
220
260 4 5 6 7 8 9 10
200
Sum
2350
2500
1920
Mean
235
Total mean = 6770 / 28 = 241.8
Total variation = 5,410.7
Between-group variation = 1,160.7
Within-group variation = 4,250
Degrees of freedom for total variation = 27
Degrees of freedom for B variation = 2
Degrees of freedom for W variation = 25
MST = 200.4
MSB = 580.4
MSW = 170.0

21. Then What? Calculating F ratio
This F ratio follows the F-distribution with degrees of freedom of numerator (k-1) and denominator (n-k)
Finding a critical value from the F-distribution table
If the calculated F ratio is greater than the critical value, we reject the null hypothesis that each group mean is the same (i.e., μA = μB = μC)

22. Example: Comparing Multiple Groups
No. A B C 1
240
230 2
250 3
220
260 4 5 6 7 8 9 10
200
Sum
2350
2500
1920
Mean
235
Total mean = 6770 / 28 = 241.8
Total variation = 5,410.7
Between-group variation = 1,160.7
Within-group variation = 4,250
Degrees of freedom for total variation = 27
Degrees of freedom for B variation = 2
Degrees of freedom for W variation = 25
MST = 200.4
MSB = 580.4
MSW = 170.0
F ratio = 3.41
critical value (2, 25) = 3.39

23. ANOVA Procedure Step 1: Calculate each group mean and total mean
Step 2: Calculate total / between-group / within-group variation
Step 3: Calculate degrees of freedom for each variation
Step 4: Calculate mean variation
Step 5: Calculate F ratio
Step 6: Obtain critical value from the F-distribution table depending on significance level and degrees of freedom
Step 7: Reject the null (i.e., conclude that groups are different) if the F ratio is greater than the critical value. Otherwise we fail to reject the null (i.e., conclude that groups are not different)

24. ANOVA Tables
SPSS
Excel

25. ANOVA Tables

26. Check Points for ANOVA What is the null hypothesis?
Do you measure the difference in what?
What is the treatment?
Can you calculate total / between-group / within-group variation?
Can you calculate degrees of freedom for each variation?
Can you calculate the F-ratio and find the critical value from the F-distribution table?

27. Part 2
Cross Tabulations and Chi-square Analysis
Correlation Analysis

28. Are Two Variables Associated?
Categorical variables (i.e., nominal and ordinal scales)
Continuous variables (i.e., interval and ratio scales)

29. Investigating contingent relationships… Are two variables independent?
Cross Tabulation
Investigating contingent relationships…
Are two variables independent?

30. Income vs. Number of Cars
0 or 1 2+
Total
< $37,500 48 6 54
> $37,500 27 19 46 75 25
100
What would be a null hypothesis?

31. Statistical Independence
In general, the probability of two events occurring jointly is
If the two events are independent, then the probability is
Since

32. Example What is the probability of having two aces in a row?
(1) Without replacement
(2) With replacement

33. Expected Numbers Under H0
Income
Number of Cars
0 or 1 2+
Total
< $37,500 54
> $37,500 46 75 25
100 B1 B2 A1 A2
Get the expected probability now……

34. Observed vs. Expected Numbers
Income
Number of Cars
0 or 1 2+
Total
< $37,500 54
> $37,500 46 75 25
100 B1 B2 A1 A2

35. χ2 Test for Statistical Independence
(Step 1) Calculate the test statistic
(Step 2) Find a critical value given the degree of freedom and a significance level (α) from a χ2 table
(Step 3) If the test statistic is greater than the critical value, then reject the null hypothesis (i.e.,Ho: two variables are related each other). Otherwise, fail to reject the null (i.e., Ha: two variables are independent)

36. In Our Example Income Number of Cars 0 or 1 2+ Total < $37,500 54
> $37,500 46 75 25
100

37. χ2 Test for Goodness-of-Fit
H0: The sample represents the population
Observed Expected
Brands oi ei
(oi-ei)2 / ei US 32 38
0.9474
Japanese 27 31
0.5161
European 21 18
0.5000
Korean 9
0.0000
Other 11 4
Total
100

38. Are Two Variables Related?
Correlation Analysis
Measure of a linear association between two interval- or ratio- scaled variables  Correlation Coefficient
Simple Linear Regression
Using an interval- or ratio- scaled variable to predict another interval- or ratio- scaled variable  Simple Linear Regression Model
Multiple Regression Analysis
Introducing multiple predictor variables to predict a focal variable

39. Correlation Analysis Y Y r = 0.8 r = 0 X Y X r = 1.0 Y X Y r = ?

40. Correlation Does Not Mean Causation
High correlation
Rooster’s crow and the rising of the sun
Rooster does not cause the sun to rise.
Ice cream consumption and the virus break outs
Covary because they are both influenced by a third variable

41. Calculating Correlation
Sales (y)
Ad (x)
100 50
160 60
120 55 90 40
150 80
130 35
110 45 65 30
140 70
Means
Standard Deviations
Covariance
Correlation

42. Next Class: Are these variables related?* *
Regression * * * * * * * * * * * * * * * *
* Simple & Multiple Regression *