Testing of Hypothesis

Introduction

The objective of testing of statistical hypothesis is to determine if an assumption about some characteristic (parameter) of a population is supported by the information obtained from the sample.

The terms hypothesis testing or testing of the hypothesis are used interchangeably. A statistical hypothesis (different from a simple hypothesis) is a statement about a characteristic of one or more populations such as the population mean. This statement may or may not be true. The validity of the statement is checked on the basis of information obtained by sampling from the population.
Testing to Hypothesis refers to the formal procedures used by statisticians to accept or reject statistical hypotheses that include:

i) Formulation of Null and Alternative Hypothesis

Null hypothesis

A hypothesis formulated for the sole purpose of rejecting or nullifying it is called the null hypothesis, usually denoted by H₀. There is usually a “not” or a “no” term in the null hypothesis, meaning that there is “no change”.

For Example, The null hypothesis is that the mean age of M.Sc. students is 20 years. Statistically, it can be written as H₀:μ=20. Generally speaking, the null hypothesis is developed for the purpose of testing.
We should emphasize that, if the null hypothesis is not rejected on the basis of the sample data we cannot say that the null hypothesis is true. In another way, failing to reject the null hypothesis does not prove that the H₀ is true, it means that we have failed to disprove H₀.

For the null hypothesis, we usually state that “there is no significant difference between “A” and “B”. For example, “the mean tensile strength of copper wire is not significantly different from some standard”.

Alternative Hypothesis

Any hypothesis different from the null hypothesis is called an alternative hypothesis denoted by H₁. Or we can say that a statement that is accepted if the sample data provide sufficient evidence that the null hypothesis is false. The alternative hypothesis is also referred to as the research hypothesis.

It is important to remember that no matter how the problem stated, the null hypothesis will always contain the equal sign, and the equal sign will never appear in the alternate hypothesis. It is because the null hypothesis is the statement being tested and we need a specific value to include in our calculations. The alternative hypothesis for the example given in the null hypothesis is H₁:μ≠20.

Simple and Composite Hypothesis

If a statistical hypothesis completely specifies the form of the distribution as well as the value of all parameters, then it is called a simple hypothesis. For example, suppose the age distribution of the first-year college student follows N(16, 25), and the null hypothesis is H₀:μ=16 then this null hypothesis is called a simple hypothesis, and if a statistical hypothesis does not completely specify the form of the distribution, then it is called a composite hypothesis. For example, H₁:μ<16 or H₁:μ>16.

ii) Level of Significance

The level of significance (significance level) is denoted by the Greek letter alpha (α). It is also called the level of risk (as there is the risk you take of rejecting the null hypothesis when it is really true). The level of significance is defined as the probability of making a type-I error. It is the maximum probability with which we would be willing to risk a type-I error. It is usually specified before any sample is drawn so that the results obtained will not influence our choice.

In practice 10% (0.10) 5% (0.05) and 1% (0.01) level of significance is used in testing a given hypothesis. 5% level of significance means that there are about 5 chances out of 100 that we would reject the true hypothesis i.e. we are 95% confident that we have made the right decision. The hypothesis that has been rejected at 0.05 level of significance means that we could be wrong with probability 0.05.

Selection of Level of Significance

The selection of the level of significance depends on the field of study. Traditionally 0.05 level is selected for business science-related problems, 0.01 for quality assurance, and 0.10 for political polling and social sciences.

Type-I and Type-II Errors

Whenever we accept or reject a statistical hypothesis on the basis of sample data, there are always some chances of making incorrect decisions. Accepting a true null hypothesis or rejecting a false null hypothesis leads to a correct decision, and accepting a false hypothesis or rejecting a true hypothesis leads to an incorrect decision. These two types of errors are called type-I error and type-II error.
type-I error: Rejecting the null hypothesis when it is (H₀) true.
type-II error: Accepting the null hypothesis when H₁ is true.

iii) Test Statistics

Procedures that enable us to decide whether to accept or reject the hypothesis or to determine whether observed samples differ significantly from expected results are called tests of hypothesis, tests of significance, or rules of decision. We can also say that test statistics is a value calculated from sample information, used to determine whether to reject the null hypothesis. The test statistics for mean $\mu$ when $\sigma$ is known is $ Z= \frac{\bar{X}-\mu}{\sigma/\sqrt{n}}$, where Z-value is based on the sampling distribution of $\bar{X}$, which follows the normal distribution with mean $\mu_{\bar{X}}$ equal to $\mu$ and standard deviation $\sigma_{\bar{X}}$ which is equal to $\sigma/\sqrt{n}$. Thus we determine whether the difference between $\bar{X}$ and $\mu$ is statistically significant by finding the number of standard deviations $\bar{X}$  from $\mu$ using the Z statistics. Other test statistics are also available such as t, F, $\chi^2$, etc.

iv) Critical Region (Formulating Decision Rule)

It must be decided before the sample is drawn under what conditions (circumstance) the null hypothesis will be rejected. A dividing line must be drawn defining “Probable” and “Improbable” sample values given that the null hypothesis is a true statement. Simply a decision rule must be formulated having specific conditions under which the null hypothesis should be rejected or should not be rejected. This dividing line defines the region or area of rejection of those values which are large or small that the probability of their occurrence under a null hypothesis is rather remote i.e. Dividing line defines the set of possible values of the sample statistic that leads to rejecting the null hypothesis called the critical region.

One-tailed and two-tailed tests of significance

If the rejection region is on the left or right tail of the curve then it is called a one-tailed hypothesis. It happens when the null hypothesis is tested against an alternative hypothesis having a “greater than” or a “less than” type.

and if the rejection region is on the left and right tail (both sides) of the curve then it is called a two-tailed hypothesis. It happens when the null hypothesis is tested against an alternative hypothesis having a “not equal to sign” type.

v) Making a Decision

In this step, the computed value of the test statistic is compared with the critical value. If the sample statistic falls within the rejection region, the null hypothesis will be rejected otherwise accepted. Note that only one of two decisions is possible in hypothesis testing, either accept or reject the null hypothesis. Instead of “accepting” the null hypothesis (H₀), some researchers prefer to phrase the decision as “Do not reject H₀” or “We fail to reject H₀” or “The sample results do not allow us to reject H₀“.