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Unformatted text preview: to standard normal, this gives
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:
H0 :µ ≥ µ0
H1 :µ < µ0
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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- Spring '08