Econ 3120 1st Edition 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 0 HA 0 or HA 0 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 0 then we will only reject if our sample mean Y is very large In this case we could also state our null as H0 0 This is equivalent to H0 0 because the we reject when 0 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 0 we reject when our test statistic is large enough such that it only would have been generated 100 of the time We therefore reject if t c where c is chosen such that P Tn 1 c H0 1 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 x y 4 0 If 4 0 0 then we are testing equality of means H0 x y We saw in earlier that regardless of the population distributions t q X Y 4 0 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 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 0 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 Find the 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 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 In other words we d like to find bounds a and b such that P a 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 So a 95 confidence level has 0 05 Suppose that our population is distributed Normal 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 s n tn 1 For a large enough sample this implies that P 1 96 Y s n 1 96 0 95 After some rearranging we can write this as P Y 1 96 s n 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 Some people express the confidence interval as the probability that 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 More generally our 100 1 confidence interval is given by y c 2 s n y c 2 s n where c 2 is the 100 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 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 0 which are not rejected in favor of the two sided alternative 6 0 Therefore once we have constructed an 95 confidence interval we can easily test any value of H0 against a two sided alternative at the 95 level