Chapter 9: Hypothesis Tests Based on a Single Sample

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Chapter 9: Hypothesis Tests Based on a Single Sample http://www.rmower.com/statistics/Stat_HW/0801HW_sol.htm

Assumptions for Inference We have an SRS from the population of interest. The variable we measure has a Normal distribution (or approximately normal distribution) with mean  and standard deviation σ. We don’t know  but we do know σ (Section 9.3) We do not know σ (Section 9.5)

9.1: The Parts of a Hypothesis Test - Goals State the steps that are required to perform a hypothesis test. Be able to state the null and alternative hypothesis.

Hypothesis In statistics, a hypothesis is a declaration, or claim, in the form of a mathematical statement, about the value of a specific population parameter (or about the values of several population characteristics). A Hypothesis Test is a formal procedure for comparing observed data with a claim (also called a hypothesis) whose truth we want to assess.

Parts of a Hypothesis Tests A. The claim assumed to be true. B. Alternative claim. C. How to test the claim. D. What to use to make the decision.

Statistical Hypotheses A. Null Hypothesis: H0: Initially assumed to be true. B. Alternative Hypothesis: Ha Contradictory to H0

Parts of a Hypothesis Tests A. The claim assumed to be true. B. Alternative claim. C. How to test the claim. A test statistic, TS calculated from the sample data measures how far the data diverge from what we would expect if the null hypothesis H0 were true. D. What to use to make the decision.

Parts of a Hypothesis Tests A. The claim assumed to be true. B. Alternative claim. C. How to test the claim. D. What to use to make the decision. The p-value for a hypothesis test is the smallest significance level for which the null hypothesis, H0, can be rejected.

9.2: Hypothesis Test Errors and Power - Goals Describe the two types of possible errors and the relationship between them. Define the power of a test and what affects it.

Error Probabilities Decision Reject H0 Fail to reject H0 Truth H0 is true H0 is false (Ha is true) If we reject H0 when H0 is true, we have committed a Type I error. P(Type I error) =  If we fail to reject H0 when H0 is false, we have committed a Type II error. P(Type II error) = , Power = 1 - 

Types of Error http://www.rmower.com/statistics/Stat_HW/0801HW_sol.htm

Type I vs. Type II errors (4)

Type I vs. Type II errors (5)

Errors  measures the strength of the sample evidence against H0. The power measures the sensitivity (true negative) of the test.

Increase the power  a  n

Type I vs. Type II errors (4)    a    n

Extra Discussion Question Why is the US Judicial system a good example for hypothesis tests? What is H0? What is Ha?