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Econ 3120 1st Edition Lecture 6Outline of Last Lecture I. Asymptotic NormalityII. Central Limit TheoremIII. Distribution of Difference in MeansOutline of Current Lecture IV. Hypothesis TestingV. Two Sided Hypothesis TestingVI. Type II ErrorsCurrent Lecture4 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 havebeen 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 meanof 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 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 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 standardized version of Y¯, since it has a distribution that we are familiar with. If we know the populationvariance σ, then we know that Y¯ ∼N(µ, σ 2 n ), which implies that Y¯ − µ σ/ √ n ∼ N(0,1)Usually we don’t know the population variance σ 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¯−µ 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¯ − µ s/ √ n ∼ tn−1 Under the null hypothesis, the following statistic (called a t-statistic) has a t distribution with n−1 degrees of freedom: t = Y¯ − µ0 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¯ − µ0 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, butkeep 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 α and is also called the “significance level” of the test: α = 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| > cα/2where cα/2 is chosen such that P(|Tn−1| < cα/2 ) = 1−αA Type II error occurs if we fail to reject the null when the null is false. We’ll define this as β: β = 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 α (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 θ, the test’s power, denoted by π(θ), is the likelihood that we won’t commit a Type II error, or π(θ) = 1−P(Type II|θ) = 1−β The probability of a Type II error depends on the actual value of the parameter as well as on


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