One Sided Tests, Confidence Intervals

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One Sided Tests, Confidence Intervals

II. One Sided Tests III. Confidence Intervals

Lecture number:
Lecture Note
Cornell University
Econ 3120 - Applied Econometrics

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Lecture 7 Outline of Last Lecture I. No Class Outline of Current Lecture II. One Sided Tests III. Confidence Intervals Current Lecture III.2 One-sided tests of the mean Sometimes we are interested in testing against a one-sided alternative. That is, our alternative hypothesis is that the true value is greater than the null: H0 : µ = µ0 HA : µ < µ0 or HA : µ > µ0 If we are conducting a one-sided test, we will only reject the null if the test statistic is on one side of the hypothesized distribution. For example, if our alternative is HA : µ > µ0, then we will only reject if our sample mean Y¯ is very large. In this case, we could also state our null as H0 : µ ≤ µ0. This is equivalent to H0 : µ = µ0 because the we reject when µ > µ0 in either case. For one-sided tests, the critical values will be lower (in absolute value) than the critical values for two-sided tests at the same confidence level. This occurs because we only find evidence against H0 and in favor of HA when our test statistic is on one side of the distribution. For HA : µ > µ0, we reject when our test statistic is large enough such that it only would have been generated 100 ·α% of the time. We therefore reject if t > cα, where cαis chosen such that P(Tn−1 < cα|H0) = 1−α III.3 Testing differences in means (or equality of two different means) We might be interested in whether means of two populations are equal, or if their difference equals some value. We can test differences in means in a similar way that we test single means. In this case, our null hypothesis would be H0 : µx − µy = 4µ0 If 4µ0 = 0, then we are testing equality of means: H0 : µx = µy We saw in earlier that, regardless of the population distributions, t = q X¯ −Y¯ −4µ0 s 2 x/n1 +s 2 y/n2 III.4 has an asymptotic standard normal distribution.3 Therefore, we can test the difference in means in the same manner that we would a single mean. Example: The same poll as above found that 49% of those under 35 and 44% of those over 65 approve of Obama. If 250 of each group were surveyed, test whether the true difference is 0, at the 5% level, against a two-sided alternative. 4.4 P-values The p-value of a hypothesis test gives us the minimum level of α such that H0 is rejected. For a two-sided test, the P-value is computed as p−value = P(|Tn−1| > |t||H0) III.5 For a one-sided test, we want to find the p-value such that p% of the mass is on one side of the distribution. For example, if we are testing against HA : µ > µ0, p−value = P(Tn−1 > t |H0) Thus, p-values for a one-sided test will be lower than ...

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