Page 206 - Computational Statistics Handbook with MATLAB

Chapter 6: Monte Carlo Methods for Inferential Statistics 193

There must be an alternative hypothesis such that we would decide in
, then this
favor of one or the other, and this is denoted by H 1 . If we reject H 0
. Returning to the engineering example, the
leads to the acceptance of H 1
alternative hypothesis might be that there is a difference in the instruments
or that one is more accurate than the other. When we perform a statistical
hypothesis test, we can never know with certainty what hypothesis is true.
For ease of exposition, we will use the terms accept the null hypothesis and
reject the null hypothesis for our decisions resulting from statistical hypoth-
esis testing.
To clarify these ideas, let’s look at the example of the transportation official
who wants to determine whether the average travel time to work has
increased from the time it took in 1995. The mean travel time to work for
northern Virginia residents in 1995 was 45 minutes. Since he wants to deter-
mine whether the mean travel time has increased, the statistical hypotheses
are given by:

H 0 : µ = 45 minutes
H 1 : µ > 45 minutes.

The logic behind statistical hypothesis testing is summarized below, with
details and definitions given after.

STEPS OF HYPOTHESIS TESTING
1. Determine the null and alternative hypotheses, using mathematical
expressions if applicable. Usually, this is an expression that in-
volves a characteristic or descriptive measure of a population.
2. Take a random sample from the population of interest.
3. Calculate a statistic from the sample that provides information
about the null hypothesis. We use this to make our decision.
4. If the value of the statistic is consistent with the null hypothesis,
.
then do not reject H 0
5. If the value of the statistic is not consistent with the null hypothesis,
and accept the alternative hypothesis.
then reject H 0
The problem then becomes one of determining when a statistic is consistent
with the null hypothesis. Recall from Chapter 3 that a statistic is itself a ran-
dom variable and has a probability distribution associated with it. So, in
order to decide whether or not an observed value of the statistic is consistent
with the null hypothesis, we must know the distribution of the statistic when
the null hypothesis is true. The statistic used in step 3 is called a test statistic.
Let’s return to the example of the travel time to work for northern Virginia
residents. To perform the analysis, the transportation official takes a random
sample of 100 residents in northern Virginia and measures the time it takes

© 2002 by Chapman & Hall/CRC