Hypothesis testing
Introduction
An important component of the scientific method is 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 H_{0}. For example, here is a formulated null hypothesis related to the investigation of precipitation patterns over adjacent rural and urban land-use types:
- H_{0}: 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 (H_{1}). 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:
- H_{1}: 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). </dd>
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. 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.
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