Unformatted text preview: 39 What determines test results? This time lets start with the Critical Value Approach:
After stating the hypotheses and calculating the z-statistic, the decision rule is going to be:
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 = 2.5−2.6
1/ 25 = 0.1/0.3 = −0.33. Let’s pick α = 0.05, then α/2 = 0.025 and zα/2 = 1.96.
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 24 / 39 What determines test results? Graphical Explanation 0 Utku Suleymanoglu (UMich) Hypothesis Testing z 25 / 39 What determines test results? 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
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 we get a p-value 2 × P (Z > z = 0.33) = 2 × 0.3707 = 0.7414.
This value is bigger than any reasonable α so reach at the same conclusion as before. We fail to
reject the null hypothesis.
Let’s go back one slide and see this on a picture. Utku Suleymanoglu (UMich) Hypothesis Testing 26 / 39 What determines test results? Hypothesis Testing Fundamentals Recap
Before we go on to diﬀerent cases, let’s repeat the general idea of hypothesis testing:
We have an hypothetical value for a population parameter (µ = µ0 ) as a claim and we want
to test this.
We have a sample and a point estimate x = 2, let’s say.
We know the sampling distribution of x assuming the claim is true from chapter 6.
Then we can evaluate the probability of x or a similar draw from this sampling distribution.
To do that we need to transform our normal random variable...
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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.
- Spring '08