HYPOTHESIS TESTING Null Hypothesis and Research Hypothesis ?

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

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) 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 in Hypothesis Testing True False Accept Correct Error Error Correct Reject

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 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

Sampling

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Mean and Standard Deviation Comparisons Effect of Sampling Mean and Standard Deviation Comparisons

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

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

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

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

34% 14% 2% 69.89 71.79 64.19 66.09 67.99

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

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

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

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

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 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

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