Econ 209
Handout Hypothesis Testing
Statistical reasoning is a refinement of every day logical reasoning. Consider the sta tement, “ If
A , then B”. Lets suppose that this is a true statement. And let’s suppose that we wanted to investigate
whether A was true. The knowledge that B is true does not provide us with evidence that A is true but
the knowledge that B is false does provide us with evidence that A is false. If B is false we know that A
is false but if B is true we don’t know anything about the truth or falsity of A.
Consider the following example. Suppose that the statement “all criminals have blonde hair”
were true. We can rewrite this statement in the form “ If X is a criminal, then X has blonde hair” where
X is a person. Clearly if X does not have blonde hair we know that X is not a criminal, but if X does
have blonde hair we do not know whether or not X is a criminal. If we think of the statement “X is a
criminal” as a hypothesis, then if X does not have blonde hair we know that the hypothesis is false and
we say that we “reject the hypothesis”. If X has blonde hair however we don’t know that X is a criminal
so we do not say that we “accept the hypothesis” that X is a criminal. We say instead that we “fail to
reject the hypothesis” that X is a criminal. The same kind of reasoning applies to statistical hypothesis
testing.
Example I. Suppose we want to test the hypothesis that a coin is a fair coin. (A fair coin is one
for which the probability of getting a head is one-half.) In order to test this hypothesis we might choose
to flip the coin 100 times. Our “if A, then B” statement becomes the following: If the coin is fair, then
the number of heads which occur when we flip the coin 100 times is a normally distributed random
variable with mean 50 and variance 25. Suppose that we flip the coin 100 times. (If the coin is fair this
is equivalent to a random draw from a normal population with mean 50 and variance 25.) Suppose we
get 70 heads. It would be extraordinarily unlikely that we would get this result if the population is
normal with mean 50 and variance 25. So we reject the hypothesis that the coin is fair. If on the other
hand we got 49 heads this is not an unusual outcome from a normal population with mean 50 and
variance 25 we would fail to reject the hypothesis that the coin is fair.
Now you will notice a difference between this example and the former example. In the former
example we assumed that we can say with certainty that “X does not have blonde hair.” In this example
we can only say that it would be extraordinarily unlikely that we would get this result from a population
normally distributed with mean 50 and variance 25.
Briefly, hypothesis testing consists of the following. A hypothesis is formulated. We call the
hypothesis to be tested the “null” hypothesis. A experiment is run or some data is collected and a test
statistic is calculated. Then the following question is asked.: Assuming that the hypothesis is true is the
test statistic which we calculated unusual? If the test statistic is sufficiently unusual we conclude that the
hypothesis is false and we “reject the hypothesis”. If the test statistic is not sufficiently unusual we “fail
to reject the hypothesis.
Example II. Suppose that we have a population with unknown mean and standard deviation 10.
Suppose that we want to test the hypothesis that the population mean is 100. We take a sample of size n
= 25 and find a sample mean of 110. If the null hypothesis were true then we know
that and If then z = 5. This is a very unusual
result for a standard normal random variable. The probability of getting a result that differs from 0 by
