. Null stands for zero
hence the symbol. The null hypothesis is also thought of as the
hypothesis of no difference. For example the hypothesis of no
difference between the experimental group and the control group
in an experiment.Lesson 9 will consist of the following topics
For lesson 9, read pages 27-40 in Practical Statistics for Educators,
Third Edition by Ruth Ravid (2005, University Press of America)
or read pages 42-45 & 273-285 in Basic Statistics for Behavioral Science Research
2nd ed by Mary B. Harris (1998, Allyn and Bacon)
or
read pages 49-59 in Practical Statistics for
Educators, 2nd Edition by Ruth Ravid (2000, University Press of America)
or read pages 10-16 in Practical Statistics for Educators
by Ruth Ravid (1994, University Press of America).
We use statistical inference to help us decide if the results of our research are significant, or at least if they are statistically significant. In general we start out with a research problem and end up with a statistical hypothesis which we can test by using the proper inferential statistic. The remaining lessons of this course will be a survey of the most widely used inferential statistics and how to use them properly.
Let's look at some examples of research problems or research questions.
We start out with a research problem or a research questions, such as the examples above, and then we wish to state the research problem in the form of a statistical hypothesis. We actually use two statistical hypotheses, the null hypothesis and the alternative hypothesis. In other words given a research question our next task is to state this question in the form of a null hypothesis and an alternative hypothesis.
The null hypothesis states that a population parameter is
equal to some specific value. The symbol for the null hypothesis
is H sub zero, . Null stands for zero
hence the symbol. The null hypothesis is also thought of as the
hypothesis of no difference. For example the hypothesis of no
difference between the experimental group and the control group
in an experiment.
The alternative hypothesis states that a population parameter
is equal to some value other than that stated by the null hypothesis.
The alterative hypothesis is in the direction we would wish our
experiment to turn out and thus is really a statement of the research
question in the form of a statistical hypothesis. The symbol for the
alternative hypothesis is H sub 1, 
, or
H sub A, 
. In these lessons we will use the
H1 format although our text uses the HA format.
To summarize then, given a research problem, if we wish to test the significance of our results, we must state our research question as a pair of statistical hypotheses. The null hypothesis, H0,states that a population parameter (usually the mean) is equal to some specific value. The alternative hypothesis, H1, states that the population parameter is equal to some value other than that stated by the null hypothesis. Generally the alternative hypothesis has one of three forms.
This does seem like a rather backward process, stating our result as no result (the null hypothesis) and then attempting to reject this hypothesis so that we can accept the alternative hypothesis. In 1935, Sir Ronald Fisher (quoted in Couch, 1987) stated it as follows.
"In relation to any experiment we may speak of this hypothesis as the null hypothesis, and it should be noted that the null hypothesis is never proved or established, but is possibly disproved, in the course of experimentation. Every experiment may be said to exist only in order to give the facts a chance of disproving the null hypothesis."
As we have seen the process of statistical decision making for research involves setting up a null hypothesis and then either rejecting or failing to reject the null hypothesis. If we fail to reject the null hypothesis, that is the end of the process. When we fail to reject the null hypothesis we can say that the results of our experiment are not significant. We could also say that our results are inconclusive. However, if we reject the null hypothesis, we can then accept the alternative hypothesis, and indicate that our results were significant. As we mentioned earlier there are generally three ways we can talk about the significance of the results based on how the alternative hypothesis was stated.
As we will explain in the next section, the last form of the alternative hypothesis is referred to as a two-tailed test that is it can be significant in either of two directions. The other two options are referred to as one-tailed tests. We only look for significance in a single direction.
Now let's go back to the case of rejecting or failing to reject the null hypothesis and consider in each case how we could be correct in our decision or be in error.
First let's take the case in which the null hypothesis,
H0, is true (there truly is no difference between the
two groups). In this case if we reject H0 we are
making an error. This type of error (rejecting H0 when
we shouldn't have) is referred to as a type I error. It is also
referred to as the alpha level or significance level of the experiment.
This type of error can be controlled by the experimenter as he or she
sets the significance level of the experiment. A common level for
alpha is .05 or the 5% level. Another way of thinking of the
alpha level is that it is the probability of making a type I error.
So if we set 
we are saying that
we are willing to make a type I error 5% of the time.
On the other hand, if we fail to reject H0 when it is in fact true, we are making the correct decision. With alpha at .05 we would expect to do so 95% of the time.
Now let's take the case where the true status of the null hypothesis is false. In that case we should reject it. To reject a false H0 is the correct decision. On the other hand if H0 is false and we fail to reject it then we are making an error. This type of error (failing to reject H0 when we should have rejected it) is referred to as a type II error. Beta is used as the symbol for the probability of making a type II error. Beta, the probability of making a type two error, can not be set by the experimenter as can the alpha level, but beta is related to alpha. The higher the alpha level is set (here we mean a less probable setting, .01 is higher than .05, and .001 is higher than .01) the more likely it is that we will make a type II error (or the higher the beta level is). The lower the alpha level (.05 rather than .01) the less likely we are to make a type II error. We are in somewhat of a dilemma here. If we set alpha high then we are less likely to make a type I error but are more likely to make a type II error. On the other hand if we set the alpha level low we are more likely to make a type I error but less likely to make a type II error.I know this is confusing to the potential researcher, but one way of getting around it is just to set your alpha level at .05 (and not at .01 or .001). In this way you are balancing the relationship between type I and type II errors in your decision making process.
The information we have discussed is summarized in the following table.
| True Status of Null Hypothesis | |||
|---|---|---|---|
| H0 is True | H0 is False | ||
| Decision | Reject H0 | Type I Error level |
Correct Decision |
| Fail to Reject H0 |
Correct Decision |
Type II Error level |
|
As a final thought we might also add that although we can not control type II error (beta level) directly except by lowering the alpha level, different statistics, at the same alpha level are more resistant to causing type II error. This characteristic of a statistic is called the power of a statistic. A more powerful statistic is less likely to yield a type II error. The power of a statistic is one minus beta, it is the tendancy of a statistic not to make a type II error.
Power = 1 -
Now that we have looked at the decision making process with regard to rejecting or failing to reject H0 let's look at the entire process of testing statistical hypotheses. This is a process consisting of the following seven steps:
)
Generally this summary rule is either reject H0 or fail to reject H0. In the case of reject H0, you would also indicate the alpha level in the form, p < .05, and indicate if the test is one-tailed or two-tailed. If the decision is fail to reject H0, information on alpha level and number of tails does not need to be given.
Now let's apply this process to an example. For our example we will use the case in which a single score is compared to a population value (the population mean). This is not the typical situation you would use in research (you can not usually base a study on a single observation), but it will illustrate all of the steps. For the rest of the course we will be considering the most widely used inferential statistics and going through this process with each of them.
Sample research problem: Your friend has an IQ of 118. Is this IQ greater than the population mean of 100? Note: the population standard deviation is 15.
The only data we have to collect is the individual's IQ score (X = 118).
The appropriate statistic to compare an individual with the mean is the Z-score test (or the Z-score). We calculate that to be 1.2
reject H0 if 
To determine the Z score value that will cause us to reject H0, we use the table in Appendix A of the Ravid, 2005 text, Z Scores and Percentage of Area under the Normal Curve Between any given Z-Score and the Mean. In the Harris, 1998, text use Table 1, Areas Under the Normal Curve, on pages 508-511. Since our alpha level is .05 we want to find the Z-Score in the table that would have 5% of the scores higher than it. Looking under column 2 of the table (Area from Mean to, it's called Area Between z and Mean in the Harris text) for .45 we find that the associated Z-Score in column 1 is 1.64 (or 1.65 - take your choice).
If our calculated value of Z was greater than (or equal to) 1.64 we would reject H0, however since its value of 1.2 is not greater than or equal to 1.64 we fail to reject H0.
Please send electronic mail to the course instructor if you have any questions about this lesson or other concerns.