Download Slides for Session #19

Statistics for Social
and Behavioral Sciences
Session #19:
Estimation and Hypothesis Testing,
Wrap-up & p-value
(Agresti and Finlay, from Chapter 5 to Chapter 6)
Prof. Amine Ouazad
Outline
1. True or False?
1. The p-value
2. One sided t tests
3. Type II error
Next time: Mid term covering Chapters 1—6 and 9 of A&F
True or False ??
• For the test of a null hypothesis, the confidence interval
method can only be applied when sample size N is large.
• For the test of a null hypothesis, the t test method can
only be applied when the sample size is small.
• The sampling distribution of a t statistic is left-skewed, or
right skewed, depending on the number of degrees of
freedom.
• When testing a null hypothesis with 95% confidence
level, the probability of a Type I error (i.e. when the null
hypothesis is true) is 5%.
Outline
1. True or False?
1. The p-value
2. One sided t tests
3. Type II error
Next time: Mid term covering Chapters 1—6 and 9 of A&F
The p-value
is the probability (in other potential samples,
unobserved) that the absolute value of the t
statistic is greater than the observed t statistic in
our sample.
• A low p value means…..
– that larger absolute values of the t test are unlikely.
In practice:
• Reject the null hypothesis H0 at 95% if the pvalue is lower than 0.05.
• Similarly for 90% and 99%! (0.10 and 0.01 resp.)
Back to Amartya Sen:
Are there really fewer women per
man in India?
• “More than 100 million women are
missing”
• Dataset collected by Simple
Random Sampling.
• Population: Indian residents.
• Sample: 2,878,380 respondents.
• Little response bias.
• Non response bias is possible.
Thomas W. Lamont University
Professor, and Professor of
Economics and Philosophy, at
Harvard University
Back to Amartya Sen:
Are there really fewer women per
man in India?
Obtaining Table 5.1’s t scores values in Stata: type `display invttail(df,0.025)’ for 95%.
Replace df by the degrees of freedom (N-1)
Back to Amartya Sen:
Are there really fewer women per
men in India?
• Either with the confidence interval method,
• Or with the t test method,
• We reject the null hypothesis:
“H0: the fraction of women in India is 50%”
• At 99% (and thus at 95%, 90%).
• Draw the p-value on a graph of the
distribution.
Outline
1. True or False?
1. The p-value
2. One sided t tests
3. Type II error
Next time: Mid term covering Chapters 1—6 and 9 of A&F
Back to Cory Gardner
• With 1850 respondents, the polling company PPP
found that 48% of respondents would vote for
Gardner.
• Pollsters think that
Gardner wins if the vote share is > 46%.
• Can you test the null hypothesis that the true vote
share is = 46%, with Ha: vote share > 46% ?
One sided test of H0: m=v, Ha: m>=v
• When testing m=v vs m>=v, use the t statistic
method.
– avoid the confidence interval method.
Assume that m=v and build t = (m-v)/SE.
☞ Reject the null hypothesis at 95%
if the t statistic is greater than +t0.05.
☞ Do not reject the null if t < 0.
• Similarly for 90% and 99% (t0.10 and t0.01).
Back to Cory Gardner
• Can we reject the null hypothesis at 95% with
a one sided test?
• Can we reject the null hypothesis at 95% with
a two sided test?
☞ Beware of the dangers of a one-sided test !!
Why it is objectionable
to use one-sided tests….
• Assuming that the parameter is higher than a certain
value….
– Lowers the threshold t necessary to reject the null hypothesis.
– For a one sided test, we use t0.05 for a test at 95%
– For a two sided test, we use t0.025 for a test at 95%.
• Ask yourself: Can I assume that the m will be
necessarily higher than v?
But here, not such a big issue:
Are there really fewer women per man in India?
Obtaining one sided t scores values in Stata: type `display invttail(df,0.05)’ for 95%.
Replace df by the degrees of freedom (N-1)
Outline
1. True or False?
1. The p-value
2. One sided t tests
3. Type II error
Next time: Mid term covering Chapters 1—6 and 9 of A&F
Type II error
• Type I error is well-known.
• But what is the level of type II error?
Wrap up
Confidence interval method for the test of H0 : m = v. Ha: m ≠ v.
CI = [ m – t * SE ; m + t * SE ]
– Reject the H0 with significance level 1% if the 99% confidence interval for the
sample mean m does not include v.
– Reject the H0 with significance level 5% if the 95% confidence interval for the
sample mean m does not include v.
– Reject the H0 with significance level 10% if the 90% confidence interval for the
sample mean m does not include v.
t test method for the test of H0 : m = v. Ha: m ≠ v.
– Build the t statistic (m-v)/SE
– Reject the H0 with significance level 1%
if the t statistic is outside the range [-t0.005 , t0.005]
– Reject the H0 with significance level 5%
if the t statistic is outside the range [-t0.025 , t0.025]
– Reject the H0 with significance level 10%
if the t statistic is outside the range [-t0.05 , t0.05]
I will not ask for the one sided t-test, but good to know for your future life.
Coming up:
Readings:
• Mid term on Tuesday, November 25.
– Coverage: up to Chapter 6 inclusive.
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No online quiz this week.
Make sure you come to sessions and recitations.
For help:
• Amine Ouazad
Office 1135, Social Science building
[email protected]
Office hour: Tuesday from 5 to 6.30pm.
• GAF: Irene Paneda
[email protected]
Sunday recitations.
At the Academic Resource Center, Monday from 2 to 4pm.