The document discusses hypothesis testing and introduces key concepts such as: 1) The null and alternative hypotheses for testing differences in means. 2) How to calculate a p-value and use it to reject the null hypothesis based on a significance level. 3) How to calculate a t-statistic and use t-tables to test hypotheses when sample sizes are small. 4) How confidence intervals are related to hypothesis tests and contain the true population parameter a certain percentage of the time.

1. Undergraduate Econometrics
Wei-Chih Chen
SIBA, SHUFE
Fall 2010
Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 1 / 10

2. Hypothesis Testing
The hypothesis testing problem (for the mean): make a provisional
decision, based on the evidence at hand, whether a null hypothesis is
true, or instead that some alternative hypothesis is true.
That is, test
H0 : E (Y ) = µY ,0 vs. H1 : E (Y ) > µY ,0 (1-sided, >)
H0 : E (Y ) = µY ,0 vs. H1 : E (Y ) < µY ,0 (1-sided, <)
H0 : E (Y ) = µY ,0 vs. H1 : E (Y ) 6= µY ,0 (2-sided)
Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 2 / 10

3. Hypothesis Testing
Some terminology for testing statistical hypotheses:
p-value = probability of drawing a statistic (e.g. Y ) at least as adverse
to the null as the value actually computed with your data, assuming
that the null hypothesis is true.
The signi…cance level of a test is a pre-speci…ed probability of
incorrectly rejecting the null, when the null is true.
Calculating the p-value based on Y :
act act
p value = PrH 0 [jY µY 0 j > jY µY 0 j], where Y is the value
of Y actually observed (nonrandom)
Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 3 / 10

4. The p-value
In practice, σY is unknown - it must be estimated
Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 4 / 10

5. Estimation of the Variance
n 1 Σi =1 (Yi
2 1 n
The sample variance: sY = Y )2
q
The sample stadard deviation: sY = 2
sY
sY
The standard error of Y : SE(Y )= pn
The p-value when σY is unknown: see note
Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 5 / 10

6. The t Statistic
Y µY 0
The t-statistic or t-ratio: t = SE (Y )
When n is large, the distribution of the t-statistic is approximated by
the standard normal distribution (CLT)
Suppose the signi…cance level is 5%. We reject H0 if jt act j > 1.96
When n is small and Y is normally distributed, the t-statistic has a
Student t distribution with n 1 degrees of freedom
Suppose the signi…cance level is 5% and n = 10. We reject H0 if
jt act j > 2.26 (check the table)
Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 6 / 10

7. The Link between the p-value and the Signi…cance Level
The signi…cance level is prespeci…ed. For example, if the prespeci…ed
signi…cance level is 5%,
you reject the null hypothesis if jt j > 1.96
equivalently, you reject if p 0.05.
Often, it is better to communicate the p-value than simply whether a
test rejects or not – the p-value contains more information than the
“yes/no” statement about whether the test rejects.
Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 7 / 10

8. Con…dence Intervals
A 95% con…dence interval for µY is an interval that contains the true
value of µY in 95% of repeated samples.
A 95% con…dence interval can always be constructed as the set of
values of µY not rejected by a hypothesis test with a 5% signi…cance
level.
95% con…dence interval for µY = fY 1.96 SE (Y )g (see note)
What is random here? The values of Y1 , . . . , Yn and thus any
functions of them, including the con…dence interval. The con…dence
interval it will di¤er from one sample to the next.
The population parameter, µY , is not random, we just don’ know it.
t
Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 8 / 10

9. Di¤erences-of-Means Estimation
Suppose we want to test the hypothesis that two means are equal:
µm µf = 0
Y Yf Ym Yf
The t-statistic: t= r m2 = SE (Y m Y f )
, where sm and sf2 are
2
sm s2
nm + nf
f
sample variances
If both nm and nf are large, the t-statistic has a standard normal
distribution
If nm and nf are small, the t-statistic doesn’ have a Student t
t
distribution, even if the population distribution of Y in the two groups
is normal!
There is a statistic testing this hypothesis that has a normal
2
distribution, the “pooled variance” t-statistic – spooled
However the pooled variance t-statistic is only valid if the variances of
the normal distributions are the same in the two groups. Would you
expect this to be true, say, for men’ v. women’ wages?
s s
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10. Summary
From the two assumptions of:
1 simple random sampling of a population, that is, {Yi , i = 1, . . . , n} are
i.i.d.
2 0 < E (Y 4 ) < ∞
We developed, for large samples (large n):
Theory of estimation (sampling distribution of Y )
Theory of hypothesis testing (large-n distribution of t-statistic and
computation of the p-value)
Theory of con…dence intervals (constructed by inverting test statistic)
Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 10 / 10