1. Inferential Statistics: SPSS

2. Testing Population Mean
One Sample T-Test
Independent Samples T-Test
Paired Sample T-Test
SPSS
Analyze -> Compare Mean -> …

3. One Sample T-Test Test mean of one sample against a test value
Variable is either interval or ratio
EX: Test if average total score is more than 55
H0 : μ <= 55
H1 : μ > 55
If the hypothesis is true then we should reject H0 and accept H1
Calculate statistic (use SPSS)

4. SPSS Analysis Result
t: the calculated T value
df: degree of freedom
SPSS uses “Sig.(2-tailed)” or p-value to show test result
SPSS only does Two-tailed
Divide this p-value by 2 to get one-tailed
If p-value is less than α (e.g. 0.05) then the test is significant
Reject H0, accept H1
Thus the average total score is more than 55 at significance level 0.05

5. Independent Samples T-Test
Test mean of one sample against another
Assumptions
Independent variable consists of two independent groups.
Dependent variable is either interval or ratio
Dependent variable is approximately normally distributed
Similar variances between the two groups (homogeneity of variances)

6. Independent Samples T-Test
EX: Test if male students get lower total score than female students
H0 : μm >= μf
H1 : μm < μf
If the hypothesis is true then we should reject H0 and accept H1
Calculate statistic (use SPSS)

7. Levene’s Test for Equality of Variances
For independent samples T-Test, the calculation for T value is different when:
Both samples have the same variance (σ12 = σ22) AND
The variances are difference (σ12 != σ22)
Use variance test to determine this
See Levene’s Test for Equality of Variances in the table
If the value of Sig. is >= α (e.g. 0.05) then the two variance is equal – use the first row of the result
If the value of Sig. is >= α (e.g. 0.05) then the two variance is NOT equal – use the second row of the result

8. Result
According to Levene’s Test, use the first row (Sig. = > α)
The p-value (2-tailed) is < α, thus the average score of male and female students are different
The p-value (1-tailed) is 0.033/2 = < α, thus the result is significant
Check the Group Statistics, female group has higher mean, thus reject H0 and accept H1 - the research hypothesis is true

9. Paired Sample T-Test Test means of paired samples against each other
Same sample group (or two dependent samples)
Assumptions
Dependent variable is interval or ratio
The differences in the dependent variable between the two related groups are approximately normally distributed.
Independent variable consists of two related groups or "matched-pairs".
No outliers in the differences between the two related groups.

10. Paired Sample T-Test H0 : μD = 0 H1 : μD != 0
EX: Test if final score is not different from midterm score of the same group of student
H0 : μD = 0
H1 : μD != 0
If the hypothesis is true then we should accept H0 and reject H1
Calculate statistic (use SPSS)

11. Result
The p-value (2-tailed) is < α, thus the result is significant
Thus reject H0 and accept H1 - the research hypothesis is false

12. Testing Categorical Data or Proportion
One variable – binomial proportion
One variable – multiple groups proportion (Goodness of Fit Test)
Two variables – Chi-square Test of Independence
Two variables – Test of Homogeneity

13. Binomial
Determining the proportion of people in one of two categories is different from a specified amount
H0 : pD = p0
H1 : pD != p0
SPSS assumes numerical data
Recode data into number e.g. M,F -> 1,2
Analyze->Nonparametric Tests->Legacy Dialogs->Binomial
E.g. the proportion of male student is 0.5
H0 : pD = 0.5
H1 : pD != 0.5

14. Result
Careful about the “Test Prop.” SPSS considers the first observation (row) as first group
Exact Sig. is 0.04 < α, the result is significant, thus reject H0 and accept H1 - proportion of male students is not 0.5
If tested at 0.6

15. Multiple Groups Goodness of Fit Test
Determining the proportion of groups is different from a specified ratio
O: Observed
E: Expected
Analyze -> Nonparametric Tests -> Legacy Dialogs ->
Chi-Square
E.g. the proportion of sections is 1:2:1:2:1

16. Result
The values in the “expected values” ratio correspond to groups in order of appearance in the observation row.
Asymp.Sig. = < α, the result is significant, thus reject H0 and accept H1 - the proportion is not 1:2:1:2:1
Frequency less than 5 might make the analysis not meaningful

17. Chi-square Test of Homogeneity
Used to determine whether the proportion of one variable is similar when grouped by another variable
two or more groups in each variable
H0: p1 = p2 = p3 = … = pn
H1: p1 ≠ p2 ≠ p3 ≠ … ≠ pn
Data -> Weight Cases -> Weight cases by ->
Do not weight cases – SPSS uses proportion of total population
Select frequency variable to test dependency
Analyze -> Descriptive Statistics -> Crosstabs
Statistics -> Tick “Chi-square”
Cells -> Tick “Expected” (optional)

18. Result
E.g. The proportion of selected of major is similar in both genders of student?
H0: pm = pf
H1: pm ≠ pf
Pearson Chi-Square: Asymp.Sig < α
Reject H0 and accept H1 - the proportion is not similar in each gender

19. Chi-square Test of Independence
Used to determine whether the effects of one variable depend on the value of another variable (2 variables)
H0: Variable x and variable y are independent of each other
H1: Variable x and variable y are dependent of each other
H0: ΣΣ(O - E)2 = 0
H1: ΣΣ(O - E)2  0
Data -> Weight Cases -> Weight cases by ->
Do not weight cases – SPSS uses proportion of total population
Select frequency variable to test dependency
Analyze -> Descriptive Statistics -> Crosstabs
Statistics -> Tick “Chi-square”
Cells -> Tick “Expected” (optional)

20. Result
E.g. Determine if gender and major are independent based on total score
H0: gender and major are independent of each other
H1: gender and major are dependent of each other
Pearson Chi-Square: Asymp.Sig < α
Reject H0 and accept H1 - the two variables are dependent of each other based on total score

21. One-way ANOVA: SPSS Analyze -> Compare Means -> One-way ANOVA
Option -> Tick…
Homogeneity of variance test
Descriptive (optional)
Welch
Post Hoc - used when the result is significant (at least one of the means is different) to find the group with the different mean

22. At least one pair is different
Example
Determine if the means of total score are different in the 5 Sections
H0 : μ1 = μ2 = μ3 = μ4 = μ5
H1 : μ1 != μ2 != μ3 != μ4 != μ5
At least one pair is different

23. Result: Descriptives and Variances
Check Levene test
“Sig.” > = 0.05, thus variances are equal in all groups
If not, need to refer to the Robust Tests of Equality of Means Table (Welch) instead of the ANOVA Table

24. Result: ANOVA Table
Sig. = < α, thus at least one of the group has different means
Use Post-Hoc tests To find the pair with different mean

25. Result: Post Hoc Tests The pair that Sig. < α has different mean
Section 1 and 4
Section 2 and 4
Section 2 and 5
Section 3 and 4
Section 4 and 5

26. Two-way ANOVA: SPSS
Analyze -> General Linear Model -> Univariate
Multivariate is MANOVA
Add dependent variable and two or more factors (independent variables)
Option -> tick “Homogeneity tests” (optional “Descriptive”)
Plot -> add one factor (containing more groups) to “Horizontal Axis” and other to “Separate Lines” then click “Add”
To obtain profile plot
Post Hoc to find pair that has different means (similar to One-way ANOVA, optional)

27. Example Determine the effect of major and gender on the total score

28. Result Compare Error to Corrected Total
Error should be less than 20% of corrected total
Error is very large compared to corrected total
Total score is effected by other external factors
Gender row Sig. = < α, gender has effect on total score
Major row Sig. = > α, major has no effect on total score
Major*Gender row Sig. = > α, the interaction between two factors has no effect on total score

29. Result: Profile Plot

30. Example
Determine the effect of section and gender on the total score