HYPOTHESIS TESTING

The Nature of Statistical Hypotheses

A statistical hypothesis Is a statement or assumption about an unknown population parameter. For example, we may hypothesize about a population mean about a population proportion, about the difference between two population means or the difference between two population proportions. Such assumptions might be true or false. A test performed in order to verify whether a hypothesis is true or false is called a test of hypothesis. The hypothesis about any population parameter is tested by using information obtained from a sample drawn from the population in question. If the result obtained from the sample, (in form of calculated sample quantities), is Inconsistent with the hypothesis being tested, we have to reject the hypothesis; on the other hand, sample evidence supporting the hypothesis leads to its acceptance. Consequently, a statistical hypothesis is formulated for the purpose of rejecting it or accepting it.

The Null and Alternative Hypotheses

The hypothesis that is being tested is called the null hypothesis and its denoted by H_0. The hypothesis that we are willing to accept if we reject the null hypothesis is called the alternative hypothesis and is denoted by  H₁. Let us illustrate these two hypotheses with the following example.

Suppose that during the first semester, a class of 100 students was taught mathematics by the use of a certain method. At the end of the semester, the class recorded an average score of 65% in this subject with a standard deviation of 5%. During the second semester, a new mathematics lecturer was employed to handle the course. The new lecturer claims that he has developed a new method for teaching the course, which according to him, is more effective than the first one. In order to test this claim, a sample of 30 students taught by the new method was examined at the end of the second semester. Their average score was found to be 68%. The question arises: Is this new method really more effective than the old one? In words, should a decision be taken to adopt the new method of teaching mathematics? In order to answer this question, we need to carry out a test of hypothesis. Such a test will enable us verify the following pos

1. whether the observed higher sample average of 68% was actually due to the fact that the second method is better than the first one that is, whether there is a significant difference between the two methods, or
2. whether the observed sample average occurred by chance for example, it might have happened that the sample was made up of the very brilliant students. If that is the case, it would be wrong to attribute the higher average score observed for the 30 students to the superiority of the. second method,

Suppose the sample of 30 students so selected recorded an average score of say, 63%, which is lower than the average scored by the entire class taught by the first method during the first semester. Here, again, a test of hypothesis will enable us verify:

1. whether the observed low sample score of 63% has occurred because the new method is by no means more effective than the old one; or
2. whether the observed low sample average score occurred because the sample happened to be made up of less brilliant students; and not because the new method is less effective than the old one.

The null hypothesis is usually specified in terms of the population parameter of interest It is the hypothesis of no difference, and consequently, it is stated with the equality sign. Throughout the process of analysis, the null hypothesis is assumed true. Evidence from the sample that is inconsistent with the null hypothesis leads to its rejection. On the other hand, evidence supporting the hypothesis leads to its acceptance.

In the example on teaching methods, we shall test the hypothesis that there is no difference in the two teaching methods. In other words, we shall assume that the average scores of students taught by the two methods are equal. This will be our null hypothesis (H0). The alternative hypothesis (H­₁) will be that the two teaching methods are not the same. In a similar pay, if we want to show that vaccine A is better than vaccine B in the prevention of a certain disease, our null hypothesis will be that there is no difference between the two vaccines.

Returning to our example on the two teaching methods, we shall state hypothesis in terms of the mean (average) score of students taught by the two methods. For the case where the observed sample average was 68%, our null and alternative hypotheses could be stated as follows:

In case (1), we are assuming that the average score of students is equal to 65%. Rejection of H₀ implies the acceptance of the alternative hypothesis, H₁ that the average score is not equal to 65%. In case (2) we are making the same assumption as in case (1). But here, rejection of H₀ leads to the acceptance of the alternative that the average score of students is greater than 65%.

If the observed sample average was, say 63%, instead of 68%. Then our alternative hypothesis could be stated as: in case (1) above. i.e. H₁: or as H1: The null hypothesis remains the same, i.e. H₀ :  In other words, we have case (3) as follows:

In case (3), a rejection of H0 implies the acceptance of the alternative that the mean score is less than 65%.

For all these three cases, the aim is to test the significance of the observed difference on the hypothesis of no difference, if0. This is done by selecting a test statistic* whose sampling (probability) distribution is known under the assumption that H0 is true. Using the probability distribution of this test statistic, we work out the probability of the occurrence of the observed difference to know whether it is significant.

Two-tailed Test and One-tailed Test

A test could be two-tailed or one-tailed. A two-tailed test is a test in which the alternative hypothesis is non-directional (i.e. it is two-sided), e.g. H₁ : . Case (1) above is a two-tailed test. For a two-tailed test, when the null hypothesis is rejected, the alternative hypothesis does not indicate whether the true mean is greater or less than that specified in the hypothesis. A two-tailed test has two critical values. We shall explain the meaning of critical values later in this section.

A one-tailed test, sometimes called a one-sided test, is one where the alternative hypothesis is one-sided (directional) as follows:

Cases (2) and (3) above are one-sided tests. In these two cases, when the null hypothesis is rejected, we conclude that the true mean is as specified by the alternative hypothesis. A one tailed test has one critical value – to the right of the distribution for case (2) and to the left of the distribution, for case (3).

We can also test hypothesis about a population proportion. For example, we may wish to test that a population proportion, P – is equal to a specified value P₀, against the alternative that it is not. We state our hypotheses as follows:

Two Types of Errors

In testing statistical hypothesis we can commit two types of errors, – the type I error and the type II error.

A type I error has been committed if we accept the null hypothesis, H₀ when in fact it is true. This error is sometimes called the  – error (alpha error).

A type II error has been committed if we accept the null hypothesis, H₀ when in fact it is false. This error is sometimes called the  – error (beta error). In other words, a type II error is made when H₀ is erroneously accepted; i.e. when H₀ is accepted even though it is false. It is pertinent to mention here that these two types of errors arise because the truth or falsity of H₀ is unknown; even after it is accepted or rejected. Consequently, that accept H₀ does not necessarily mean it is true. In the same manner, that reject H₀ does not mean it is false.

Level of Significance

The level of significance of a test is the probability of committing the type I error; that is, the probability of rejecting H₀ when in fact it should be accepted. It represents the highest probability with which we are willing to risk a type I error. The level of significance is denoted by and its magnitude is usually specified before samples are drawn for the t most frequently used levels of significance in hypothesis testing are (i.e. 5% level of significance), and 0.05 (i.e1% level of significance). Other values of a, say  =0.00l,  = 0.10, etc., could also beused. Sui test is performed at a 5% level of significance (i.e. 0.05), it means that there are 5 chances in a hundred that a true null hypothesis would be rejected. A test is said to be significant if the null hypothesis is rejected at the 5% level. A test at a 1% level of significance means that there is only one chance in a hundred that a true null hypothesis would be rejected. A test is said to be highly significant if the null hypothesis is rejected at the 1% level.