Econ 3120 1st Edition 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 0 Our null could also be that the that the parameter is less than or greater than a particular value H0 0 H0 0 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 6 0 Our alternative could also be that the average is above or below the hypothesized value HA 0 HA 0 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 0 HA 6 0 This is an example of a two sided test since the alternative is that the 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 true value is larger or smaller than the null Suppose that we have a sample Y1 Yn from a normallydistributed population of unknown mean and variance that is Yi N 2 Intuitively we should reject H0 if our sample mean Y is very large or very small relative to 0 As our test statistic we use the standardized version of Y since it has a distribution that we are familiar with If we know the population variance then we know that Y N 2 n which implies that Y n N 0 1 Usually we don t know the population variance 2 so we have to estimate it We can estimate it using the sample variance s 2 that we defined in the last lecture There is an additional wrinkle however Y s n does not have a standard normal distribution because s is not a known parameter It turns out that s 2 has basically a chi squared distribution since it is a sum of squared normallydistributed random variables so Y s n tn 1 Under the null hypothesis the following statistic called a tstatistic has a t distribution with n 1 degrees of freedom t Y 0 s n tn 1 For our hypothesis test we need to ask the following question Is the estimated value of t likely to be from a tn 1 distribution in which case H0is true or it it more likely to come from another distribution in which case H0 is false If the sample size is large enough then approximately t Y 0 s n N 0 1 This happens for two reasons First as n grows large the t distribution is approximately standard normal You ll see from the table that this occurs when n is above 120 although the numbers look fairly similar above n 30 Second we know that the standardized mean converges to a standard normal regardless of the underlying distribution of the data We ll be using t distributions in the discussion that follows but keep in mind that we would use a standard normal in large samples In our test we can create a rejection rule such that if our computed value of t is above or below that value it is highly unlikely that the value of t would have been generated given that H0 is true For example we can set our rejection rule such that we would have only observed that value of t 5 of the time under the null In carrying out our test there are two types of errors that we can commit Type I errors and Type II errors A Type I error occurs if we reject the null when the null is true We set the probability of a Type I error when we set up our test This probability is defined by and is also called the significance level of the test P Reject H0 H0 is true Now let s go back to our test for the mean We reject if the value of our test statistic is above a certain critical value t c 2where c 2 is chosen such that P Tn 1 c 2 1 A Type II error occurs if we fail to reject the null when the null is false We ll define this as P fail to reject H0 H0 is false In general we don t have direct control over the likelihood of a Type II error We ll see that if we increase and thereby the likelihood of a Type I error then the likelihood of a Type II error decreases A related concept is the power of a test For hypothesis test on the parameter the test s power denoted by is the likelihood that we won t commit a Type II error or 1 P Type II 1 The probability of a Type II error depends on the actual value of the parameter as well as on