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Unformatted text preview: UMich) Hypothesis Testing 33 / 37 σ not Known Case 3: σ not Known, Population Not Normal The ttest relies on the assumption that population values are distributed normally.
If not, we can still test hypothesis thanks to CLT, but these are going to be
approximations which are close to true values in large samples.
To make the procudure work: you switch back to standard normal distribution even
though σ is not known.
So the test statistic is z = x −µ0
¯
√,
s/ n and ztest rules apply. PLEASE BE CAREFUL: We have the same formula for test statistic for cases 2 and 3.
The diﬀerence is
Case 2: Population values are normally distributed, so test statistic has
tdistribution. Hence called “t”. And you use the ttable.
Case 3: Population values are from unknown distribution, so test statistic is
approximately normal. So it is called “z” and ztable is the relevant one.
If one knows that population has normal distribution, “ttest” must be performed
(especially with small n). This yields more accurate test results. Utku Suleymanoglu (UMich) Hypothesis Testing 34 / 37 Hypothesis about Population Proportions Hypothesis Tests about Population Proportion We have learned
Sampling distribution of p
¯
Building conﬁdence intervals for p using p
¯
These worked only in large samples (CLT). Using the same ideas, we can perform
hypothesis testing for population proportions.
The test statistic you will need will be a zstatistic:
z= p − p0
¯
p0 (1−p0 )
n where p0 is the value in the null hypothesis. Note that denominator uses p0 , not p .
¯
And critical values looked up from the standard normal table. Utku Suleymanoglu (UMich) Hypothesis Testing 35 / 37 Hypothesis about Population Proportions Example Suppose you work at a polling company and you are interested the proportion of
Americans who support Obamacare. In particular, you are interested in whether the
majority of people support it. Your company interviewed 10, 000 people, 5, 200 of which
said they support Obamacare. Perform a test to support the Democratic claim that the
majority of Americans support Obamacare with 0.01 signiﬁcance level. [this is ﬁctitious
data]
We have a large enough sample, so we can do hypothesis testing.
The null and the alternative:
H0 :p ≤ 0.50
H1 :p > 0.50 Utku Suleymanoglu (UMich) Hypothesis Testing 36 / 37 Hypothesis about Population Proportions p = 0.52 so the test statistic is:
¯
z= 0.52 − 0.50
0.5(1−0.5)
10000 = 0.02
= 40
0.005 Critical value zα = z0.01 = F (0.99) = 2.33
Because this is righttailed test, decision rule says “reject H0 if z > zα ”.
So we reject the null hypothesis that says a minority supports Obamacare. The ﬁctitious
data supports the Democratic claim. Utku Suleymanoglu (UMich) Hypothesis Testing 37 / 37...
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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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