Utku suleymanoglu umich hypothesis testing 21 37

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Unformatted text preview: othesis tests: H 0 : µ = µ0 H 1 : µ = µ0 These type of tests judge the claim that unknown population parameter is exactly equal to some number. In economics, two-tailed tests are performed a lot to test things like: Whether a production technology have constant returns to scale. (population parameter=1) Whether a job training program has any effect on wages whatsoever. (population parameter=0) We will come back to the latter one again when we do regression analysis. Utku Suleymanoglu (UMich) Hypothesis Testing 21 / 37 Testing Hypothesis about the Population Mean: σ known Example Test procedure is very similar to the one-tailed tests with a few but important differences. Suppose with your lightbulb sample (remember x = 2.5, n = 25 and σ = 1.5.) Now ¯ suppose that there is a claim that says the mean life expectancy of lightbulbs is 2.6 years: H0 :µ = 2.6 H1 :µ = 2.6 The test statistic is going to be identical with one-tailed tests: z= x − µ0 ¯ 2.5 − 2.6 √= = −0.33 0.3 σ/ n The test statistic calculates the relative position of 2.5 with respect to hypothesized value for µ: 2.6. You can see it is fairly close as measured by z . Given that normal ¯ distribution is bell-shaped, we know x = 2.5 draw from the distribution of X is quite ¯ probable if µ = 2.6, so we should not reject the H0 . Key thing: Because of the equality in the null, what we consider unlikely if the null hypothesis is true can be on either tail. We will build our rejection regions on both tails. Utku Suleymanoglu (UMich) Hypothesis Testing 22 / 37 Testing Hypothesis about the Population Mean: σ known This time let’s start with the Critical Value Approach: After stating the hypotheses and calculating the z-statistic, the decision rule is going to be: x −µ ¯ Reject H0 if | σ/√0 | > zα/2 n In other words, reject the null if the test statistic is outside the interval (−zα/2 , zα/2 ) where critical value zα/2 is the z-value for upper tail probability α/2. For our example, we have z = α/2 = 0.025 and zα/2 = 1.96. 2.5−2.6 √ 1/ 25 = 0.1/0.3 = −0.33. Let’s pick α = 0.05, then Because z = 0.33 lies inside the interval (−1.96, 1.96), we do not reject the null hypothesis. We don’t have enough evidence to assert that µ = 2.6 is not the case. Let’s see what we are doing on a picture. Utku Suleymanoglu (UMich) Hypothesis Testing 23 / 37 Testing Hypothesis about the Population Mean: σ known Graphical Explanation 0 Utku Suleymanoglu (UMich) Hypothesis Testing z 24 / 37 Testing Hypothesis about the Population Mean: σ known p-value Approach for Two-Tailed Tests We can also calculate a p-value for the test statistic z , to compare with different α’s to get a conclusion. The p-value can be calculated as the area outside the interval (−z , z ), or simply (due to symmetricity) as 2 × P (Z > z = 0.33) In our example w...
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This note was uploaded on 03/17/2014 for the course ECON 404 taught by Professor Staff during the Spring '08 term at University of Michigan.

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