Null Hypothesis andResearch Hypothesis

HYPOTHESIS TESTING

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The Null Hypothesis (Ho) 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) 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

The Research Hypothesis (H1) 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) 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

Possible Outcomes inHypothesis Testing

True False

Accept

Reject

Correct

CorrectError

Error

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 inHypothesis Testing

True False

Accept

Reject

CorrectDecision

CorrectDecisionError

Error

Type I Error

Type II Error

Type I Error: Rejecting a True HypothesisType II Error: Accepting a False Hypothesis

Actuality

Sampling

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

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

Effect of Sampling on Mean & Standard DeviationHeight: U.S. Women

Random Samples of Population = 10,000 scores (0 to 99)

SAMPLE MEAN s.d.

10 54.80 32.1920 58.45 27.1430 54.63 30.2750 49.14 29.59

100 47.09 29.76200 47.05 28.84500 48.00 28.73

1,000 50.15 28.992,000 49.45 29.035,000 49.48 28.96

10,000 49.57 28.97

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

Concept of Random Sampling

34%

14%2%

67.99 69.89 71.7966.0964.19

67.99 69.89 71.7966.0964.19

5772.3%

3,36113.4%

8,53234.1%

8,54734.2%

3,41913.7%

5642.6%

PopulationIntact GroupServing asSample

Representativenessestablished on logical basis

Random Assignment

ExperimentalTreatments

Results Generalized

Population SampleRandom SampleMeasured

Results GeneralizedRandom Selection

Contrast Between Random Selection and Random Assignment

Types of Random Sampling 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.

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

Race - Black Geographic

Region - West

Geographic Region - East

Race - White

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

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 4thgrade classes:83 classes in 33 schools

Random selectionof classes

Sample of 20classes (561 students)

All members of these20 classes are usedas sampleResults

Generalized

Nonrandom Sampling Systematic 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