Lecture 12Outline of Last Lecture I. Exploratory Data AnalysisII. ProbabilityOutline of Current Lecture III. Formal HypothesesIV. Type 1 and Type 2 ErrorsCurrent Lecture- Inferential statistics always starts with a claim- A null hypothesis is the negation of a claim/research hypothesis; H0 is the skeptical choice- The research hypothesis is a formal statement of the claim; H1 is assertive, positive- Positive result: We assume that H0 is true when we analyze our data; we reject H0 if it is very unlikely to result in what we observed; in research, we are very conservative and skeptical. We do not reject H0 unless we really have to- Rejecting H0 is a positive result- Not rejecting H0 is a negative result- A test statistic is a value that comes from your sample data; It is used to test the null hypothesis; it describes how extreme the data is- Scientific Method:o Assume that H0 is trueo Select an appropriate sampleo Perform experiment (or make observations)o Collect Datao Given that H0 is true, is it likely that you would end up with the data that you got? YES: Fail to reject H0 ; NO: Reject H0 - The evidence is conclusive- The significance level (denoted by alpha) is the probability representing how rare or unusual (or extreme) must a test statistic be in order to reject the null hypothesis- A critical value is a value of the test statistic that is used to determine the result of the hypothesis test; If the test statistic has a smaller probability than the critical value, the null hypothesis will be rejected- Reject H0 if the test statistic falls within the critical region- Fail to reject H0 if the test statistic does not fall within the critical region- The p-value is the probability of getting a value more extreme than the test statistic by chance, assuming that the null hypothesis is actually true- If the p-value is less than the level of significance, we reject the null hypothesis- Reject H0 if the P-value is less than or equal to alpha (where alpha is the significance level, such as 0.05)- Fail to reject H0 if the P-value is greater than alphaKIN 4310- Another option: Instead of using a significance level such as 0.05, simply identify the P-value and leave the decision to the reader- A Type I error is the mistake of rejecting the null hypothesis when it is true; The symbol (alpha) is used to represent the probability of a type I error- A Type II error is the mistake of failing to reject the null hypothesis when it is false; The symbol (beta) is used to represent the probability of a type II error- For any fixed alpha, an increase in the sample size n will cause a decrease in beta; for anyfixed sample size n, a decrease in alpha will cause an increase in beta. Conversely, an increase in alpha will cause a decrease in beta; To decrease both alpha and beta, increase the sample sizeThese notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used asa supplement to your own notes, not as a