1. The future is a vain hope, the past is a distracting thought
The future is a vain hope, the past is a distracting thought. Uphold our loving kindness at this instant, and be committed to our duties and responsibilities right now.

2. Applied Statistics Using SAS and SPSS
Topic: Hypothesis Testing
By Prof Kelly Fan, Cal State Univ, East Bay

3. Hypothesis Testing
A statistical hypothesis is an assertion or conjecture concerning one or more populations.
Agenda:
Types of tests
Types of errors
P-value
Summary of tests
Assumption checking

6. Types of Tests

7. Types of Tests

8. Types of Tests

9. Type II Error, or “ Error” Type I Error, or “ Error”
Types of Errors
H0 true
H0 false
Type II Error, or “ Error”
Good!
(Correct!)
we accept H0
Type I Error, or “ Error”
Good!
(Correct)
we reject H0

10. = Probability of Type I error = P(rej. H0|H0 true)
 = Probability of Type II error = P(acc. H0|H0 false)
We often preset , called significance level. The value of  depends on the specifics of the H1 (and most often in the real world, we don’t know these specifics).

11. Suppose the Critical Value = 141:
EXAMPLE: H0 :  < 100
H1 :  >100
Suppose the Critical Value = 141: X

=100
C=141

12. These are values corresp.to a value of 25 for the Std. Dev. of X
 = P (X < 141/= 150)
= .3594
 = 150
What is ?
141
 = 150
These are values corresp.to a value of 25 for the Std. Dev. of X
 = P (X < 141/= 160)
= .2236
 = 160
141
 = 160
 = 170
 = P (X < 141/= 170)
= .1230
 = 170
141
 = P (X < 141/= 180)
 = 180
= .0594
 = P (X < 141|H0 false)
141
 = 180

13. Note:. Had  been preset at. 025 (instead of
Note: Had  been preset at .025 (instead of .05), C would have been 149 (and  would be larger); had  been preset at .10, C would have been 132 and  would be smaller.
 and  “trade off”.

14. P Value
Definition: the probability that we reject Ho when Ho is true based on the observed data
Idea: the largest “risk” we pay to reject H0
Alternate name: the observed type I error rate / the observed significance level
When will we reject Ho ?
What is the formula to calculate the largest risk?

15. Steps of Hypothesis Tests
Set up Ho and Ha properly
Preset a level (the significant level)
Select an appropriate test
Calculate its p-value
Reject Ho if p-value < or = the significant level; otherwise fail to reject Ho

16. Set Up Hypothesis Properly
Conjecture: The fraction of defective product in a certain process is at most 10%.
Which error is more seriously? Incorrectly claim this conjecture is true? false?
The “=“ sign must be in Ho

17. One Population

18. Two Populations

19. Assumption Checking Tests/graphs for normality
Tests for equal variances

20. Example: Mortar Strength
The tension bond strength of cement mortar is an important characteristic of the product. An engineer is interested in comparing the strength of a modified formulation in which polymer latex emulsions have been added during mixing to the strength of the unmodified mortar. The experimenter has collected 10 observations on strength for the modified formulation and another 10 observations for the unmodified formulation.

21. Example: Mortar Strength
Modified Unmodified

22. SAS/SPSS Data Input SPSS: One variable one column in the work sheet
SAS: One variable one name

23. Normality Tests/Plots
SAS: PROC UNIVARIATE DATA=** NORMAL PLOT;
Tests for Normality
Test Statistic p Value------
Shapiro-Wilk W Pr < W
Kolmogorov-Smirnov D Pr > D >0.1500
Cramer-von Mises W-Sq Pr > W-Sq
Anderson-Darling A-Sq Pr > A-Sq

24. Normality Tests/Plots
SPSS: Analyze >> Descriptive Statistics >> Explore
>> Plots , Normality plots with tests

25. One-sample t tests SAS: PROC UNIVARIATE DATA=** NORMAL PLOT MU0=*;
Tests for Location: Mu0=18
Test
Statistic
p Value
Student's t t
Pr > |t|
0.0003
Sign M
-6.5
Pr >= |M|
0.0044
Signed Rank S
-79.5
Pr >= |S|
0.0005
SPSS: Analyze >> Compare means >> One sample t test , choose Test variable, Test value

26. Two-sample t Tests and Equal-variance Tests
SAS: PROC TTEST DATA=** ;

27. Two-sample Normality Tests/Plots
SAS: PROC UNIVARIATE DATA=** NORMAL PLOT;
BY group-var; VAR testing-var;
SPSS: Analyze >> Descriptive Statistics >> Explore , choose Dependent , Factor lists >> Plots , Normality plots with tests
SPSS output:

28. Two-sample t Tests and Equal-variance Tests
SPSS: see below; choose test and grouping variables

29. Research Question
A researcher claims that a new series of math courses for elementary school is more effective than the current one. Two (1st grade) classes of students are selected to perform an experiment to verify this claim. How would you conduct the experiment to avoid confounding variables as much as possible?

30. Paired Samples
If the same set of sources are used to obtain data representing two populations, the two samples are called paired. The data might be paired:
As a result of the data from certain “before” and “after” studies
From matching two subjects to form “matched pairs”

31. Tests for Paired Samples
Calculate the pair differences
Proceed as in one sample case
Notes:
SAS: all variables must be included in data
SPSS: create/calculate all variables we need

32. Inferences about mean when “beyond the scope”
When population is nonnormal and n is small, how to do inferences about m:
1). Non-parametric tests
2). (Optional) Use Bootstrap methods to simulate the sampling distribution of t test statistic and then the simulated distribution to find an (approximate) C.I. and p-value

33. Non-parametric Tests
Independent samples: Wilcoxon Rank Sum Test (also called Manny-Whitney test)
Assumption: two distributions of the same shape
Paired samples/One sample: Wilcoxon Signed-Rank Test
Assumption: a symmetric distribution (of the differences for paired samples)

34. Introduction to Bootstrap Methods
How to simulate the sampling distribution of a given statistic, say t, based on a given sample of size n:
Pretend the original sample is the entire population
Select a random sample of size n from the original sample (now the population) with replacement ; this is called a bootstrap sample
Calculate the t value of the bootstrap sample, t*
Repeat steps 2, 3 many times, 1000 or more, say B times. Use the obtained t* values to obtain an approximation to the sampling distribution

35. Review: Confidence Interval

37. One Population

38. Two Populations