Chapter 5: Hypothesis Testing and Statistical Inference The null hypothesis is typically a statement of the values that the researcher does not expect. Null hypothesis (the values you do not expect) The alternative hypothesis is typically a statement of the values that the research expects. Alternative hypothesis (the values you expect) Two-sided test in which the alternative hypothesis has values on both sides of the null hypothesis. Type I and Type II Errors Type I error = sending an innocent defendant to jail Type II error = freeing a guilty defendant A decision rule is a method of deciding whether to reject a null hypothesis. A critical value is a value that divides the “acceptance” region from the rejection region when testing a null hypothesis. Assumption of normality of the error term: the error term should follow a normal distribution. This assumption is optional but is commonly invoked for hypothesis testing purposes. When the error term is normally distributed, it facilitates the use of t-tests and F-tests, which rely on the normality of the sampling distribution of the estimators. This assumption is especially relevant in small samples, where the central limit theorem may not apply sufficiently to approximate a normal distribution. The t-Test The t-test is used to test hypotheses about individual regression slope coefficient. The larger the absolute value this t-value is, the greater the likelihood that the estimated regression coefficient is different from zero. A critical t-value is the value that distinguishes the “acceptance” region from the rejection region. You reject the null hypothesis if the calculated t-value is greater in absolute value than the critical t-value and if the calculated t-value has the sign implied by . Reject also has the sign implied by / do not reject otherwise. The level of significance indicates the probability of observing an estimated t-value greater than the critical t-value if the null hypothesis were correct. The most common use of the one-sided t-test is to determine whether a regression coefficient is significantly different from zero in the direction predicted by theory. The four steps to use when working with the t-test are: 1. Set up the null and alternative hypotheses. 2. Choose a level of significance and therefore a critical t-value. 3. Run the regression and obtain an estimated t-value (or t-score). 4. Apply the decision rule by comparing the calculated t-value with the critical t-value in order to reject or not reject the null hypothesis. Week 2 15 The P-Values A p-value, or marginal significance level, for a t-score, is the probability of observing a t-score that size or larger (in absolute value) if the null hypothesis were true. A p-value is a probability, so it runs from 0 to 1. A small p-value casts doubt on the null hypothesis, so to reject a null hypothesis we need a low p-value. Reject 𝐻0"𝑖𝑓"𝑝" −" 𝑣𝑎𝑙𝑢𝑒𝑘"<"the level of significance and if 𝛽J" has the sign implied by 𝐻𝐴. Do not reject 𝐻0"otherwise.