Introductory Statistics for Economists
ECON 2500 – Winter 2011 – Xianghong Li
Chapter 6 – Introduction to Inference – Mar 15
Confidence Interval – Caveats
- The data must be a random sample from the population (for the formula we
o Random: sampling errors come from chance variation, and they die down
as the sample size increases.
o Non-random sample: systematic errors could dominate sampling errors,
and they do not fade away as the sample size increases.
- Outliers can have a large effect on the CI.
- If n is small then you need X to be normal. If n > 15, then underlying distribution
can often be different from normal and the sample mean will still be close to
- 95% confidence interval does say:
o Among all possible samples, the interval will cover the population mean
for 95% of the time.
o Note that this interval varies from sample to sample.
- 95% confidence interval does not say:
o For 95% of probability, the CI computed from a specific sample has
contained the population parameter.
6.2 Significance Testing
What is Hypothesis?
- A tentative explanation for a phenomenon, used as a basis for further
- It could be either right or wrong.
The Logic of Court Trial
- The defendant has been accused of certain crime (e.g. robbery), guilty?
- Under the British common law and those systems derived from it, the defendant
is assumed to be innocent before the trial.
- The prosecutor collects and presents evidence. This evidence takes the form that
strives to contradict the presumption of innocence.
- The jury considers the evidence in light of the presumption of innocence and
judges whether the evidence against him would be plausible if the defendant
where in fact innocent. It leaves to the jury to decide the degree to which the
evidence contradicts the presumption of innocence.
- Definition of ‘strong’ evidence against the presumption of innocence: the
evidence is too unlikely under the assumption that the person is innocent
(beyond reasonable doubt).
- Given strong evidence, conclude ‘guilty’. - Not strong evidence, conclude ‘not guilty’.
- What is considered to be ‘beyond reasonable doubt’, a decision of the jury.
Procedures for Significance Testing
- State the null hypothesis and the alternative hypothesis.
- Calculate the value of the test statistic, e.g. sample mean.
- Given that the null hypothesis is true, calculate the probability of observing the
realization of the test statistic to be this or more unusual.
- If the probability is small, reject the null hypothesis.
One-Sided vs. Two-Sided Alternative
- One-sided alternative: the parameter differs from its null value in a specific
direction. ‘Unusual’ evidence means the test statistic is too far in the specific