Download The Null Hypothesis

HYPOTHESIS TESTING
Null Hypothesis
and
Research Hypothesis
?
The Null Hypothesis (Ho)
n
n
The null hypothesis
– 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
The Null Hypothesis (Ho)
n
n
Operational Definition:
– MATH KNOWLEDGE
è score obtained on the Stanford Diagnostic Test - Level - Brown
– MATH SKILLS PRACTICE
è 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
The Research Hypothesis (H1)
n
n
The research hypothesis
– 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
The Research Hypothesis (H1)
n
n
Operational Definition:
– MATH KNOWLEDGE
è score obtained on the Stanford Diagnostic Test - Level - Brown
– MATH SKILLS PRACTICE
è number of problems completed on drill-and-practice work sheets
H1
–
Math Knowledge scores will be higher for students that practice
Possible Outcomes in
Hypothesis Testing
True
False
Accept
Correct
Error
Reject
Error
Correct
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“
Possible Outcomes in
Hypothesis Testing
True
Accept
Reject
Correct
Decision
Error
Type I Error
Actuality
False
Error
Type II Error
Correct
Decision
Type I Error: Rejecting a True Hypothesis
Type II Error: Accepting a False Hypothesis
Sampling
50 THROWS
100 THROWS
500 THROWS
1,000 THROWS
5,000 THROWS
10,000 THROWS
Effect of Sampling
Mean and Standard Deviation Comparisons
Mean & Standard Deviation by
Number of Dice Throws
# Throws
MEAN
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
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
1.900
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
Random Samples of Population = 10,000
scores (0 to 99)
SAMPLE
10
20
30
50
100
200
500
1,000
2,000
5,000
10,000
MEAN
s.d.
54.80
58.45
54.63
49.14
47.09
47.05
48.00
50.15
49.45
49.48
49.57
32.19
27.14
30.27
29.59
29.76
28.84
28.73
28.99
29.03
28.96
28.97
Concept of Random Sampling
n
n
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
34%
2%
64.19
14%
66.09
67.99
69.89
71.79
8,547
34.2%
8,532
34.1%
3,419
13.7%
3,361
13.4%
564
2.6%
577
2.3%
64.19
66.09
67.99
69.89
71.79
Contrast Between Random Selection and Random Assignment
Random
Population
Sample
Sample
Measured
Results Generalized
Random Selection
Population
Representativeness
established on
logical basis
Intact Group
Serving as
Sample
Results Generalized
Random Assignment
Experimental
Treatments
Types of Random Sampling
n
Simple Random Sampling
–
all individuals in a population have equal probability
of being in sample
All populations are made up
of many subpopulations:
race, gender, age group,
geographic region, etc.
n
Stratified Random Sampling
–sampling fraction
lratio of sample size to
population size
–sub populations (strata) are
identified
–individuals are randomly chosen
from each strata using: equal,
proportional, or optimal
allocations
Race Black
Race White
Geographic
Geographic Region Region - East
- West
Three Types of Allocation
n
EQUAL
–
n
PROPORTIONAL
–
n
all strata contribute the same number to the sample
Sample allocation is proportional to the strata
population size
OPTIMUM
–
Sample allocation is proportional to the product of
the strata population sizes and variability
Cluster Sampling
n
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
Results
Generalized
Random selection
of classes
Sample of 20
classes (561 students)
All members of these
20 classes are used
as sample
Nonrandom Sampling
n
Systematic Sampling
–
–
n
Convenience Sampling
–
n
every nth individual in the population is selected
sampling interval
a group of individuals available to study
Purposive Sampling
–
selection based on prior knowledge of researcher