In **section** **3b**,
we discovered that an important component of scientific
method was the testing of **hypotheses** either
through **experiments** or **predictive** forms
of analysis. A hypothesis can be defined as a tentative
assumption that is made for the purpose of empirical
scientific testing. A hypothesis becomes a theory
of science when repeated testing produces the same
conclusion.

In most cases, hypothesis testing involves
the following structured sequence of steps. The first
step is the formulation of a **null
hypothesis**. The null hypothesis is the assumption
that will be maintained by the researcher unless
the analysis of data provides significant evidence
to disprove it. The null hypothesis is denoted symbolically
as **H0**.
For example, here is a formulated null hypothesis
related to the investigation of precipitation patterns
over adjacent rural and urban land-use types:

H0: There is no difference in precipitation levels between urban and adjacent rural areas.

The second step of hypothesis testing
is to state the **alternative
hypothesis** (**H1**).
Researchers should structure their tests so that
all outcomes are anticipated before the tests and
that results can be clearly interpreted. Some tests
may require the formulation of multiple alternative
hypotheses. However, interpretation is most clear
cut when the hypothesis is set up with only one alternative
outcome. For the example dealing with precipitation
patterns over adjacent rural and urban land-use types,
the alternative might be:

H1: There is an increase in precipitation levels in urban areas relative to adjacent rural areas because of the heating differences of the two surface types (the urban area heats up more and has increased convective uplift).

Step three involves the collection
of data for hypothesis testing. It is assumed that
this data is gathered in an unbiased manner. For
some forms of analysis that use **inferential **statistical
tests the data must be collected randomly, data observations
should be independent of each other, and the variables
should be normally distributed.

The fourth step involves testing the null hypothesis through predictive analysis or via experiments. The results of the test are then interpreted (acceptance or rejection of the null hypothesis) and a decision may be made about future investigations to better understand the system under study. In the example used here, future investigations may involve trying to determine the mechanism responsible for differences in precipitation between rural and urban land-use types.

**Inferential Statistics and Significance
Levels **

Statisticians have developed a number
of mathematical procedures for the purpose of testing
hypotheses. This group of techniques is commonly
known as **inferential
statistics** (see **sections** **3g** and **3h** for
examples). Inferential statistics are available both
for predictive and experimental hypothesis testing.
This group of statistical procedures allow researchers
to test assumptions about collected data based on
the laws of probability. Tests are carried out by
comparing calculated values of the test statistic
to assigned critical values.

For a given null hypothesis, the calculated value of the test statistic is compared with tables of critical values at specified significance levels based on probability. For example, if a calculated test statistic exceeds the critical value for a significance level of 0.05 then this means that values of the test statistic as large as, or larger than calculated from the data would occur by chance less than 5 times in 100 if the null hypothesis was indeed correct. In other words, if we were to reject the null hypothesis based on this probability value of the test statistic, we would run a risk of less than 5% of acting falsely.

**One-tailed and Two-tailed Tests **

When using some types of inferential
statistics the alternative hypothesis may be directional
or non-directional. A directional hypothesis (or
one-sided hypothesis) is used when either only positive
or negative differences are of interest in an experimental
study. For example, when an alternative hypothesis
predicts that the mean of one sample would be greater
(but not less) than another, then a directional alternative
would be used. This type of statistical procedure
is known as a **one-tailed
test**. A non-directional (or two-sided) hypothesis
would be used when both positive and negative differences
are of equal importance in providing evidence with
which to test the null hypothesis. We call this type
of test **two-tailed**.