MOI UNIVERSITY

RESEARCH METHODS

HYPOTHESIS TESTING

Ong'anya

3/16/2010

Presented to Dr Harry O. Ododah

Hypothesis testing

A hypothesis is a proposition of a set of propositions that guide a researcher in finding facts about a phenomena being investigated. The hypothesis may be a mere statement of a probable result, or may be a statement backed by empirical proof. Whichever way, hypothesis needs to be tested to ascertain their validity within the framework of scientific investigation.

Characteristics of hypothesis

Must be testable and allow for other provable deductions to be made from it. It must provide for the ability to prove or disprove these deductions.

The hypothesis must use a clear and simple language.
It must be testable within a reasonable time.
Must be concisely stated for ease of testing
It must be factually based, consistent with existing knowledge.
In case of a relational hypothesis, the relationship between variables must be clear.
The statement must be simple, devoid of jargons for ease of understanding.
It must explain what it claims to explain, with empirical data. In other words, the need for investigation must be clearly elaborated.

Basic concepts in testing hypothesis.

Null versus Alternative Hypothesis

A null hypothesis is the statement about the population parameter and the basis of investigation, which should be stated precisely. It is put forward either because it is believed to be true, or it is to be used as a basis of argument, but is yet to be proved. The opposite is alternative hypothesis which is accepted whenever the null is rejected. Null hypothesis is when two variables are assumed to be equal in terms of quality. In case one is stated to be more superior to the other, then we are stating an alternative hypothesis. The final outcome is always stated in terms of the null hypothesis, which may be its rejection or acceptance. However, by its rejection it does not mean that it is wrong. It basically means that no sufficient evidence was found supporting it. The converse is true.

The null hypothesis dominates our interest. It is concisely stated so that whenever adequate proof is found as to lead to its rejection, then it is rejected in favour of the alternative hypothesis. It should be clear that the aim of testing is not to reject the null, but rather to test it on the basis of data available.

Whatever the sample result, if it does not support the null hypothesis then it is concluded that something else other than the null hypothesis is true. The alternative hypothesis H₁ is then accepted at the expense of H₀. The alternative hypothesis H₁ is a statement which a statistical hypothesis test is set up to establish. As stated earlier, it is more complicated as compared to a null hypothesis.

Significance level

It is the level of proof of a null hypothesis thus the chance of making a Type I error. It fixes the probability of wrongly rejecting the null hypothesis. It is usually small so as to prevent an investigator from inadvertently making false claims to disprove the null hypothesis. Supposing we have a significance of 5%, this would mean that the probability that the null hypothesis would be rejected is 0.05.

Decision rule/test of hypothesis

This is simply a decision made concerning the use/acceptance of a null or alternative hypothesis. We may decide that given 100 items we may accept H₀ if up to 10 will be favourable, otherwise we get with H₁. This basis is the decision rule.

Type I and Type II error

These are errors credited to the null hypothesis. Type I error (error of a first kind) is where a null hypothesis that could have otherwise been accepted is rejected. Type II error (error of the second kind) is the reverse of this. Type I error is denoted by alpha α while type II by beta β. In a scientific research, care should be taken as to balance these errors noting that increasing the probability of one has a counter effect of reducing the probability of another. For instance, a medic is researching on certain chemicals. He is confronted with a situation where he has to make a solution that requires time and extra material resources. For economic purposes he would work on reducing costs. On the other polar end, these chemicals may cost a patient's life. There is therefore a need to increase the probability that the chemicals be safe, while decreasing the aspect of cost effectiveness.

On the basis of this analogy, situation 1 represents Type I error while situation 2 Type II error. If a sample size is too small then the chances of a Type II error is more likely. It depends on the kind of research to determine the balance of these two errors.

Test Statistic

A test statistic is a quantity calculated from our sample of data. Its value is used to decide whether or not the null hypothesis should be rejected in our hypothesis test.

The choice of a test statistic will depend on the assumed probability model and the hypotheses under question.

Two-tailed and one-tailed tests

A two-tailed (two-sided) test rejects the null hypothesis if the sample mean significantly differ with the hypothesised value of the mean population. It is a statistical hypothesis test in which the values for which we can reject the null hypothesis, H₀ are located in both tails of the probability distribution. The critical region for a two-sided test is the set of values less than a first critical value of the test and the set of values greater than a second critical value of the test.

In such a case we may have the alternative hypothesis either greater than or smaller than the hypothesised mean but not equal. Represented in a curve, the rejection region would be found on either side.

A one-tailed (one-sided) test tests the one sidedness of the hypothesis. It tests whether the population mean is either lower than or higher than the hypothesised value.

A one-sided test is a statistical hypothesis test in which the values for which we can reject the null hypothesis, H₀ are located entirely in one tail of the probability distribution. In other words, the critical region for a one-sided test is the set of values less than the critical value of the test, or the set of values greater than the critical value of the test. A one-sided test is also referred to as a one-tailed test of significance.

The choice between a one-sided and a two-sided test is determined by the purpose of the investigation or prior reasons for using a one-sided test.

Procedure for hypothesis testing

Testing hypothesis means telling whether or not the hypothesis is valid on the basis of data collected. It will determine whether or not to accept the null hypothesis. Various steps are at play in the testing process:

Making a formal statement

Statements of a hypothesis must be clear on the basis of the research problem. The formulation of a hypothesis would determine whether to use one tailed or two tailed test as it would determine whether we use ≠ or </>.

For instance, in determining the number of students in an undergraduate class, the class should not be less than five students. The hypothesis would be stated as:

H₀: µ=5

H₁: µ>5

This represents a one tailed test. Illustration II is where a national popularity of a politician is 60%. In evaluating this popularity from one region of the country it is found to be 40%. The pollsters want to know if there is a significant deviation of this regional opinion vis-a-vis the national poll. The hypothesis may be stated as:

H₀: µ=60%

H₁: µ≠40%.

This is an example of a two tailed test.

Selecting a significance level

Usually, hypotheses are tested on pre determined level of significance. Often a range of 5% level or 1% is adopted. Determinants of choice of level vary. They include:

Magnitude of the difference between sample means
Size of sample population: the smaller the sample the higher the risk of type II error
Variability of measurements within samples
Whether the hypothesis is directional or non directional. A directional hypothesis is that which predicts the direction of the difference between measures involved
Deciding the distribution to use

Normal and t-distribution are the generally used sampling distribution. One needs to select the method best suited for his research.

Selection of a random sample and computing an appropriate value.

This involves coming up with a sample to aid in the research. The sample must be as representative as appropriately possible. This is done using a relevant distribution.

Calculation of the probability

In case the null hypothesis was true, what is the probability that the sample result would diverge as expressed in the expectations?

Comparing the probability

Here a comparison is made between the calculated probabilities with a specific value of significance level. If it turns out that the calculated probability equals or smaller than the significance level, then for a one tailed test the null hypothesis is rejected. For a two tailed test the null hypothesis is rejected when half the significance levels equals or less than the value of significance. By rejecting H₀ we run the risk of committing Type I error up to the significance. However, by accepting H₀ we run the risk of committing Type II error, the extent of which cannot be specified as long as we have a vague null hypothesis.

Diagrammatically, the above procedure may be shown as:

State H₀as well as H₁
Specify the level of significance
Decide the correct sampling distribution
Sample a random sample (s) and work out an appropriate value from sample data
Calculate the probability that sample result would diverge as widely as it has from expectations if H₀ were true
Is this probability equal to or smaller than α value in case of one tailed test and α/2 in case of two tailed test
Yes Reject H₀	No Accept H₀
Run the risk of committing Type I error	Run some risk of committing Type II error