Critical value tells us which values are too far o

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Unformatted text preview: alculation of test statistic (z for this case). 2. Set an α. Say, α = 0.05. 3. Figure out the critical value zα such that P (Z ≥ zα ) = α. The z-value with right-tail probability of α. 4. Make a decision about H0 by comparing the test statistic with with the critical value. Critical value tells us which values are too far off from the null hypothesis value. Left-tailed tests: Reject H0 if z < −zα . Right-tailed tests: Reject H0 if z > zα . 5. Choosing an α and finding the critical value creates a rejection region. If the test statistic is in this region, H0 is rejected. Utku Suleymanoglu (UMich) Hypothesis Testing 15 / 37 Testing Hypothesis about the Population Mean: σ known Example Cont. For our example, critical value with α = 0.05 is −1.645 = −zα . Then any test static which is smaller than than −1.645 is in the rejection region. We had z = −1.66, so we reject the the null hypothesis. 0 Utku Suleymanoglu (UMich) Hypothesis Testing z 16 / 37 Testing Hypothesis about the Population Mean: σ known Right-Tailed Test Example Suppose now that you also believe that µ > 1.5. So you want to test the claim µ ≤ 1.5. Let’s do that. First, let’s properly state the hypotheses: H0 :µ ≤ 1.5 H1 :µ > 1.5 This is a right-tailed test. Under the assumption that null hypothesis is true, we need to evaluate the chances of x ≥ 2.5. If small, the null hypothesis is not likely to be true. ¯ Next step: relevant test statistic is z = x −µ0 ¯ √ σ/ n = 2.5−1.5 0.3 = 3.33. Next: Critical value for α = 0.01 is z0.01 = 2.33. Next: Use the decision rule: z > 2.33 so we reject H0 . OR: p-value for z = 3.33 is smaller than 0.001, so we reject the null hypothesis for reasonable α. Utku Suleymanoglu (UMich) Hypothesis Testing 17 / 37 Testing Hypothesis about the Population Mean: σ known Determinants of Tests Results Before we go on and discuss testing with different situations, let’s summarize determinants of the test results. The difference between hypothesized parameter and calculated statistic: x − µ0 . ¯ Generally speaking, if the claim is too far off from the hypothesized value, test statistic would be larger in absolute terms. The effect of this on test depend on the sign of the test statistic and the tail of the test. Precision of the sampling distribution: Standard error of the sample statistic (x ) ¯ determines how much variation we should expect in x from sample to sample. If it is ¯ high, it is more likely to have samples with x that deviates largely from the ¯ hypothesized value. Utku Suleymanoglu (UMich...
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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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