But it is good to enough the reasoning and the

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Unformatted text preview: to standard normal, this gives us z-statistic. Then we can either calculate the probability associated with the z-statistic and see if it is small or big (p-value approach) compare it to some critical z-value so that we can assess how far off it is from the claimed value. (critical value approach) Either way, based on the assumption the claim is true, we assess the correctness of the claim by comparing it to what we observe in the data. If two are “different enough”, we say the claim is (probably) not correct. Notice ALL these boils down to a simple step-by-step procedure, so that when you need to do a test, you don’t have to think about and explain what you are doing, but just follow the procedure and get a result. But it is good to enough the reasoning and the mathematics behind the procedure. Utku Suleymanoglu (UMich) Hypothesis Testing 27 / 39 σ not Known Case 2: σ not known, population normal When σ is not known, we can use s , sample standard deviation, instead. Just like we did before. . . for CI’s. But remember, we need a modification to make this work. We need to use t-distribution instead of standard normal distribution. p-value approach is hard to perform with t-distribution without the help of a computer, so we will just use the critical value approach in the notes and exam. In the section and problem sets, you might use p-value approach. Let’s do a one tailed example first . . . Utku Suleymanoglu (UMich) Hypothesis Testing 28 / 39 σ not Known One-Tailed t-Tests For one tailed tests involving hypotheses: Left tailed: H0 :µ ≥ µ0 H1 :µ < µ0 Right tailed: H0 :µ ≤ µ0 H1 :µ > µ0 We reject the null the hypothesis if test statistic: t= x − µ0 ¯ √ s/ n is such that t < −tα,n−1 for left-tailed tests t > tα,n−1 for right-tailed tests where tα is the t value with probability α in the upper tail. You look this up via the t-table. Utku Suleymanoglu (UMich) Hypothesis Testing 29 / 39 σ not Known Example Suppose you are interested in the labor supply of elderly. You have a data set tha...
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