Econ 3120 1st Edition Lecture 7Outline of Last Lecture I. No ClassOutline of Current Lecture II. One Sided TestsIII. Confidence IntervalsCurrent LectureIII.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 valueswill 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−αIII.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/n2III.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) These notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.Thus, p-values for a one-sided test will be lower than those of two-sided tests. Example: Findthe p-value of the test that Obama’s approval is the same for older and younger adults.Confidence Intervals 5.1 Computation and Interpretation Once we have an estimator for the population mean, µˆ , we’d like to say something about the range of potential values that could have generated our data. A confidence interval gives us such a range. In general, we can construct a 95% confidence interval such that in 95% of random samples, ourinterval contains the true parameter µ. In otherwords, we’d like to find bounds a and b such that P(a ≤ µ ≤ b) = 0.95 Note that while 95% is a commonly used convention, a confidence interval can defined at other levels, say 90% or 99%. We can generalize the confidence level as 1 − α. So a 95% confidence level has α = 0.05. Supposethat our population is distributed Normal(µ,σ 2 ), and we have an independent random sample Y1,...,Yn with sample mean Y¯ and sample variance s 2 . Let’s first examine the distribution for Y¯.To do this, we start with the fact that Y¯−µ s/ √ n ∼ tn−1. For a large enough sample, this implies that P(−1.96 ≤ Y¯ − µ s/ √ n ≤ 1.96) = 0.95 After some rearranging, we can write this as P(Y¯ −1.96 ·s/ √ n ≤ µ ≤ Y¯ +1.96 ·s/ √ n) = 0.95 The way to think about this equation is that 95% of the time, the random interval [Y¯ − 1.96 · s/ √ n,Y¯ +1.96 ·s/ √ n] contains µ. Some people express the confidence interval as “the probability that µ lies within the interval is 95%.” This is technically incorrect, since probability is associated only with estimates (in this case, the bounds), and not the true parameter µ. More generally, our 100(1−α)% confidence interval is given by [y¯−cα/2 ·s/√ n, y¯+cα/2 ·s/ √ n] where cα/2 is the 100∗α% critical value of the tn−1 distribution (or the normal distribution in large samples)5.2 Relationship between hypothesis testing and confidence intervals Confidence intervals and hypothesis tests are closely related. A 95% confidence interval contains all of the possible values of µ for which the sample mean would have been generated 95% of the time. It follows that a confidence interval represents all possible values of the null hypothesis µ0 which are not rejected in favor of the two-sided alternative µ 6= µ0. Therefore, once we have constructed an 95% confidence interval, we can easily test any value of H0 against a two-sided alternative at the95%