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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, twotailed 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 onetailed 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 onetailed 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 bellshaped, 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 zstatistic, 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 zvalue 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 pvalue Approach for TwoTailed Tests We can also calculate a pvalue for the test statistic z , to compare with diﬀerent α’s to
get a conclusion.
The pvalue 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.
 Spring '08
 STAFF

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