Unformatted text preview: t consists of 90
people aged 65. They are measured to have a weekly labor supply (hours worked) of 30 hours.
The standard deviation is calculated to be s = 10. You believe that in the population mean hours
worked per week is less than 32 hours/week. Perform a test to support your idea.
Step 1: The null and alternative hypotheses are:
H0 :µ ≥ 32
H1 :µ < 32 Utku Suleymanoglu (UMich) Hypothesis Testing 30 / 39 σ not Known Step 2: Calculate the relevant test statistic:
t= −2
x − µ0
¯
30 − 32
=
= −1.897
√
√=
s/ n
1.054
10/ 90 Step 3: Find the critical value: Let’s pick α = 0.05. Then −tα,n−1 = t0.05,89 ≈ −1.662
Step 4: Use the decision rule for lefttailed: Reject H0 if t < −tα,n−1 . t = −1.897 < −1.662, so
reject the null.
Finally, state your conclusion: We have enough evidence to reject the claim that µ ≥ 32. So our
claim that it is less than 32 is supported by the data.
Notice that with a lower α choice, the result could have been diﬀerent. Utku Suleymanoglu (UMich) Hypothesis Testing 31 / 39 σ not Known TwoTailed Tests: σ Unknown We will again use tdistribution instead of standard normal. First state the null and alternative
hypotheses.
H0 :µ = µ0
H1 :µ = µ0
We reject the null the hypothesis if test statistic (same as onetailed tests):
t= x − µ0
¯
√
s/ n is such that
t  > tα/2,n−1
where tα/2 is the critical value with probability α/2 in the upper tail.
So if test statistic falls outside the interval (−tα/2 , tα/2 ), the null is rejected. Utku Suleymanoglu (UMich) Hypothesis Testing 32 / 39 σ not Known Example Now let’s say the data set has 900 people aged 65. They are measured to have a weekly labor
supply (hours worked) of 30 hours. The standard deviation is calculated to be s = 10. Now,
somebody claims that mean hours worked is exactly 29.5. Test this claim.
Step 1: The null and alternative hypotheses are:
H0 :µ = 29.5
H1 :µ = 29.5 Utku Suleymanoglu (UMich) Hypothesis Testing 33 / 39 σ not Known Step 2: Calcu...
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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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