1. HYPOTHESIS TESTING
Null Hypothesis and Research Hypothesis ?

2. The Null Hypothesis (Ho)
relates to a statistical method of interpreting conclusions about population characteristics that are inferred from observations made with a sample
asserts that observed differences or relationships merely result from chance errors inherent in the sampling process
If the researcher rejects the null hypothesis
she accepts the research hypothesis
concluding that the magnitude of difference between observed and anticipated is too great to attribute to sampling error

3. The Null Hypothesis (Ho)
Operational Definition:
MATH KNOWLEDGE
score obtained on the Stanford Diagnostic Test - Level - Brown
MATH SKILLS PRACTICE
number of problems completed on drill-and-practice work sheets H0
There will be no difference in Math Knowledge scores for students who practice and students that do not practice

4. The Research Hypothesis (H1)
is a formal affirmative statement predicting a single research outcome
a tentative explanation of the relationship between two or more variables
is directional
In behavioral sciences
the variables may be abstractions that cannot be directly observed
these variables must be defined operationally by describing some sample of actual behaviors that are concrete enough to be observed directly

5. The Research Hypothesis (H1)
Operational Definition:
MATH KNOWLEDGE
score obtained on the Stanford Diagnostic Test - Level - Brown
MATH SKILLS PRACTICE
number of problems completed on drill-and-practice work sheets H1
Math Knowledge scores will be higher for students that practice

6. Possible Outcomes in Hypothesis Testing
True
False
Accept
Correct
Error
Error
Correct
Reject

7. Errors: Type I and Type II
Type I error
A type I error, also known as an error of the first kind, occurs when the null hypothesis (H0) is true, but is rejected. It is asserting something that is absent, a false hit. A type I error may be compared with a so called false positive (a result that indicates that a given condition is present when it actually is not present) in tests where a single condition is tested for. Type I errors are philosophically a focus of skepticism and Occam's razor. A Type I error occurs when we believe a falsehood.[1] In terms of folk tales, an investigator may be "crying wolf" without a wolf in sight (raising a false alarm) (H0: no wolf).
The rate of the type I error is called the size of the test and denoted by the Greek letter (alpha). It usually equals the significance level of a test. In the case of a simple null hypothesis is the probability of a type I error.
"convicting an innocent person"
NASA throw out suspected electric circuit
Type II error
A type II error, also known as an error of the second kind, occurs when the null hypothesis is false, but it is erroneously accepted as true. It is failing to assert what is present, a miss. A type II error may be compared with a so-called false negative (where an actual 'hit' was disregarded by the test and seen as a 'miss') in a test checking for a single condition with a definitive result of true or false. A Type II error is committed when we fail to believe a truth.[1] In terms of folk tales, an investigator may fail to see the wolf ("failing to raise an alarm"; see Aesop's story of The Boy Who Cried Wolf). Again, H0: no wolf.
The rate of the type II error is denoted by the Greek letter (beta) and related to the power of a test (which equals ).
What we actually call type I or type II error depends directly on the null hypothesis. Negation of the null hypothesis causes type I and type II errors to switch roles.
The goal of the test is to determine if the null hypothesis can be rejected. A statistical test can either reject (prove false) or fail to reject (fail to prove false) a null hypothesis, but never prove it true (i.e., failing to reject a null hypothesis does not prove it true).
"letting a guilty person go free“

8. Possible Outcomes in Hypothesis Testing
Actuality
True
False
Correct
Decision
Accept
Error
Type II Error
Correct
Decision
Error
Reject
Type I Error
Type I Error: Rejecting a True Hypothesis
Type II Error: Accepting a False Hypothesis

9. Sampling

10. Mean & Standard Deviation by Number of Dice Throws
S.D.
10,000
7.0000
2.4157
5,000
7.0072
2.4360
1,000
7.0460
2.4300
500
7.0480
2.3040
100
7.6400
2.4599 50
6.6400
2.1453 25
6.9200
2.5807

11. Effect of Sampling on Mean & Standard Deviation Height: U.S. Women
Sample
MEAN
S.D.
25,000
67.993
1.902
20,000
67.984
1.900
15,000
67.997
1.903
10,000
67.986
5,000
67.965
1.884
1,000
67.998
1.915
500
67.922
1.907
100
68.182
1.653 50
68.037
2.010 25
67.633
1.965

12. Random Samples of Population = 10,000 scores (0 to 99)
MEAN
s.d. 10
54.80
32.19 20
58.45
27.14 30
54.63
30.27 50
49.14
29.59
100
47.09
29.76
200
47.05
28.84
500
48.00
28.73
1,000
50.15
28.99
2,000
49.45
29.03
5,000
49.48
28.96
10,000
49.57
28.97

13. Concept of Random Sampling
Probability Sampling: every member of the population has a nonzero probability of being selected for the sample
Random Selection and Random Assignment: used to obtain representativeness and eliminate possible bias

15. 34%
14% 2%
69.89
71.79
64.19
66.09
67.99

16. 8,547
34.2%
8,532
34.1%
3,419
13.7%
3,361
13.4%
564
2.6%
577
2.3%
69.89
71.79
64.19
66.09
67.99

17. Contrast Between Random Selection and Random Assignment
Population
Sample
Random
Measured
Results Generalized
Random Selection
Intact Group
Serving as
Sample
Representativeness
established on
logical basis
Random Assignment
Experimental
Treatments
Results Generalized
Population

18. Types of Random Sampling
Simple Random Sampling
all individuals in a population have equal probability of being in sample

19. Stratified Random Sampling sampling fraction
All populations are made up of many subpopulations: race, gender, age group, geographic region, etc.
Stratified Random Sampling
sampling fraction
ratio of sample size to population size
sub populations (strata) are identified
individuals are randomly chosen from each strata using: equal, proportional, or optimal allocations
Geographic Region - East
Race - Black
Geographic Region - West
Race - White

20. Three Types of Allocation
EQUAL
all strata contribute the same number to the sample
PROPORTIONAL
Sample allocation is proportional to the strata population size
OPTIMUM
Sample allocation is proportional to the product of the strata population sizes and variability

21. Cluster Sampling
When the selection of individuals of the population is impractical:
a procedure of selection in which the unit of selection (cluster) contains two or more population members
Population of 4th
grade classes:
83 classes in 33 schools
Random selection
of classes
Sample of 20
classes (561 students)
All members of these
20 classes are used
as sample
Results
Generalized

22. Nonrandom Sampling Systematic Sampling Convenience Sampling
every nth individual in the population is selected
sampling interval
Convenience Sampling
a group of individuals available to study
Purposive Sampling
selection based on prior knowledge of researcher