Hypothesis Testing

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Hypothesis Testing

IV. Hypothesis Testing V. Two Sided Hypothesis Testing VI. Type II Errors

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Lecture Note
Cornell University
Econ 3120 - Applied Econometrics

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Lecture 6 Outline of Last Lecture I. Asymptotic Normality II. Central Limit Theorem III. Distribution of Difference in Means Outline of Current Lecture IV. Hypothesis Testing V. Two Sided Hypothesis Testing VI. Type II Errors Current Lecture 4 Hypothesis Testing: Introduction Oftentimes we want to test whether the data are likely to be generated by a specific value or values for the true mean. If we only have a random sample, we cannot use sample averages (i.e., estimates) to tell us definitively whether these hypotheses are true. However, we can use hypothesis testing to inform whether the estimates could have been generated by particular true values. We start with a null hypothesis. A null hypothesis is a hypothesized value for the parameter. Our null hypothesis might be that the parameter takes on a particular value. This is written as: H0 : θ = θ0 Our null could also be that the that the parameter is less than or greater than a particular value: H0 : θ < θ0 H0 : θ > θ0 We then define an alternative hypothesis as hypothesized values of the parameter outside of the null. This generally takes two forms. First, our alternative hypothesis can be any value of the parameter outside of the null: HA : θ 6= θ0 Our alternative could also be that the average is above or below the hypothesized value: HA : θ > θ0 HA : θ < θ0 In a hypothesis test, we can either reject or fail to reject the null hypothesis. If we reject the null hypothesis, we are essentially saying that it is highly unlikely that the estimate would have been generated if the null were true. Essentially, we need evidence against the null in order to reject. If we fail to reject, we are finding evidence consistent with the null hypothesis. Note that failing to reject the null is not the same thing as accepting the null. The most we can say is that the evidence is consistent with the null hypothesis, but we cannot say with any certainty that the null is true. Thus, we never accept the null. 4.1 Two-sided hypothesis tests for the mean Suppose that we are interested in testing whether the mean of a population takes on a particular value. Our alternative is that the mean is not equal to that value. That is, H0 : µ = µ0 HA : µ 6= µ0 This is an example of a two-sided test, since the alternative is that the true value is larger or smaller than the null. Suppose that we have a sample {Y1,...,Yn} from a normally- distributed population of unknown mean and variance; that is, Yi ∼ N(µ,σ 2 ). Intuitively, we should reject H0 if our sample mean Y¯ is very large or very small relative to µ0. As our test statistic, we use the Econ 3120 1st Edition

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