13-116-4S-Ch-10 - 115-Review-2 One Sample Statistical...

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One Sample Statistical Inference (Chapters 8 10) 1 115-Review-2 How sample data can be used to draw conclusions about the entire population .
Hypothesis Tests Regarding a Parameter Basic Topics: 1. The Language of Hypothesis Testing (the elements of hypothesis testing). 2. Hypothesis Tests for a Population Proportion 3. Hypothesis Tests for a Population Mean 4. Hypothesis Tests for a Population Standard Deviation and Variance. 2
3 Statistical Inference Problem: Given a population with an unknown parameter θ , use a sample (SRS) of size n to make Statistical Inferences about unknown parameter θ . In Hypothesis Testing : an explicit statement or hypothesis is generated about a specific value θ 0 of a population unknown parameter θ . a sample statistic is used to support or reject the stated hypothesis. a measure of the reliability of the test. Hypothesis Testing Problem
4 We test these types of statements using sample data because it is usually impossible or impractical to gain access to the entire population. If population data are available, there is no need for inferential statistics. Hypothesis testing is a procedure, based on sample evidence and probability, used to test statements regarding a parameter of one or more populations. Hypothesis Testing Problem
Hypothesis Testing Problem 1. The Elements of Hypothesis Testing: 1. Specification of Hypotheses: Null Hypothesis Alternative Hypothesis 2. Test Statistic and Rejection (Critical) Region 3. The Logic of Hypotheses Testing 4. Type I and Type II Errors 5. Significance level and Power of a test. 5
Elements of Hypothesis Testing 6 1.1. Specification of hypotheses. Definition: A statistical hypothesis is a statement or a claim about the population unknown parameter . HT problem involves two competing hypotheses: 1. Null or basic hypothesis, denoted by H 0 2. Alternative or research hypothesis, denoted by H 1 (or H a ) .
7 1. The Null or Basic Hypothesis - H 0 Is a statement to be tested, indicating no change, no effect , or no difference . Provides an ”information” about the unknown parameter values that we would like to reject . Is assumed true until evidence indicates otherwise. Refers to the status quo . Always contains “ = ” sign. May or may not be rejected. HT Problem: Specification of hypotheses
8 2. The Alternative or Research Hypothesis H 1 : Is a statement to be tested, indicating change, effect , or difference . Is the logical alternative or opposite of H 0 . Provides an ”idea” about the unknown parameter that we would like to support. Challenges the status quo. Never contains the “ = ” sign. May or may not be rejected. HT Problem: Specification of hypotheses
The null hypothesis H 0 we always specify in the form of single value : H 0 : parameter = some value: H 0 : θ = θ 0. Depending on alternative hypothesis H 1 , we distinguish three possible cases : 9 HT Problem: Specification of hypotheses θ = θ 0 θ = θ 0 θ = θ 0 θ θ 0 θ < θ 0 θ > θ 0 θ 0
Test Statistic: A Test Statistic is a function of the data (sample), whose value is used to decide whether to reject H 0 .

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