The Testing of Hypothesis

 

Introduction

Generally, the decision-maker has to draw a conclusion about the population parameter on the basis of the sample statistic. The statement does have two meanings viz.: (i) we can estimate the population parameter by taking into consideration of sample statistic which we discussed in the previous chapter (Estimation). That is, if we have to estimate the average score of MBS students in statistical methods we may estimate the average score of whole students of MBS students in Tribhuvan University by taking the sample statistics (ii) another meaning is that by examining the gap between a population parameter and sample statistic we will have a decision on population parameter as right or wrong. The population parameter will be correct if there is no significant difference between a population parameter and sample statistic and won't be correct if there is a significant difference. This is the basic concept of testing of hypothesis. We will try to make it clear by following the example.

Let's take the can of certain paint. The company's specification shows that each can contains one-liter paint on average. Suppose, the decision-maker takes some sample (say 20 cans) from the store and gets the weights of the sample. He examines the difference between the weights of the samples and one liter, stated in the specifications; he will have two types of decisions.

(1) He will find no significant difference between the weights of the sample and stated weight in the company's specifications or (1) He will find the significant difference as he gets the weights of the Samples too little or too high compared to the stated weight.

In the first case, the decision-maker will decide to bring the products in the market while in the second case he will decide oppositely.

Hence hypothesis is a quantitative statement about the population parameter. Then testing of hypothesis is to test the reliability of the hypothesis assumption or Presumption about population parameters) by me simple statistics. In other words, it is an assumption about the population parameter from a sample statistic and its validity is tested.

Before discussing hypothesis testing methodology, we should be able to get used to using the following terminology.

The Null and Alternative Hypothesis

We already come to know that hypothesis is merely an assumption or presumptions about the population parameter. There are two types of hypotheses.

(i) Null hypothesis

(ii) Alternative hypothesis

Null Hypothesis

The assumptions or presumptions about the population the parameter is called the null hypothesis and is denoted by Ho. Hypothesis testing always starts with it.

According to a professor. R.A. Fisher "Null the hypothesis is that the hypothesis that is tested for potential rejection under the belief that it's true."

Generally, a Null hypothesis is a hypothesis of no difference which means there is no significant difference between the sample statistic and the population parameter, in case of difference, is seen, that is merely due to fluctuations in sampling.

For example, Let's consider the paint problem which states that the average weight of particular paint is 1 liter. This can be written as

H_o:µ=1

Alternative Hypothesis

If the decision-maker rejects the null hypothesis on the basis of sample information, he/she should accept another hypothesis that is complementary to the null hypothesis and known as the alternative hypothesis donated by H_1.  Simply, an alternative hypothesis is the opposite of the null hypothesis.

The idea of an alternative hypothesis was propounded by J. Neyman. If H₀ is accepted H₁ is rejected and vice versa.

If we consider the problem of paint, the alternative hypothesis is stated as

H₁:µ ≠ 1

The meaning is that the average weight of paint isn't 1 liter. Either it is too low or high on the basis of sample information. Hence, the hypothesis has two sides. In this condition, the test is known as a two-tailed test. Depending on the questions, the alternative hypothesis can be one side.

1. Do we have sufficient evidence to indicate that the average weight of the paint is greater than 1 liter. Then,

H₁:µ > 1

In this condition, the test is known as right tailed test.

2. Do we have sufficient evidence to indicate that average weight of the point is less than 1 liter. Then,

H₁:µ < 1

In this condition, the test is known as left tailed test.

The following rules are helpful in stating Null and Alternative Hypothesis:

1.      The conclusion expected as a result of the test should be placed in the alternative hypothesis.

2.    The null hypothesis should contain a statement of equality either s? or = but while conducting the test we use the only equal sign in the null hypothesis.

       3.   The null hypothesis is the hypothesis that is tested

       4.  The null and alternative hypothesis is complementary.

Errors in Hypothesis Testing

There are four possible ways in decision making in the testing of hypothesis.

(i)    Event of accepting the null hypothesis when the null hypothesis was true. This is the correct decision.

(ii)     Event of rejecting the null hypothesis when the null hypothesis was true. This is an incorrect decision and known to have errors in decision making. This error is called a type I error in hypothesis testing. The probability of having a type I error is denoted by  (alpha) and referred to our level of significance. The significance level is always specified by the decision-maker himself/herself. In most of the hypothesis testing problems, the level of significance is fixed at 5%. Hence, the risk of committing a type I error is always under the control of the decision-maker. This risk is also known as the producer's risk. The complement of , 1-  is called the confidence coefficient. Hence, the confidence coefficient (1 - ) is the probability of accepting H₀ when H₀ is true. The discussion about and 1 -  reveals the size of the rejection and size of the acceptance under the null hypothesis.

(iii)  Event of failing to reject the null hypothesis when the null hypothesis was false. This is an incorrect decision and known to have errors in decision making. This error is called type II error in hypothesis testing. The probability of having type II error is denoted by  (beta) and is referred to as the consumer's risk. The probability  depends upon the gap between a sample statistic and population parameter. If the gap is too low, then  is high and vice–versa. The complement of, 1 -  is called the power of a statistical test which is the probability of rejecting the null hypothesis when it is false. 

(iv)    Event of rejecting the null hypothesis when the null hypothesis was false. This is the correct decision.

 

State of Nature   Decision Alternatives	H0 is true	H0 is false/H1 is true
Accept H0	Correct decision confidence coefficient = 1-	Type II error probability =
Reject H0/Accept H1	Type I error significance level probability =	Correct decision power of test probability = 1-

If we consider the consequence of both types of errors, we find type II error is more serious than type I error. To make it clear, let's consider the example. Suppose a drug is administered to a few patients to cure a particular disease and the drug is curing the disease but if it is discontinued by claiming the drug has an adverse effect. It is a type II error. In contrary to this, the drug has, in fact, adverse effect but continue to administer to patients claiming that the drugs have a good effect. This is a type II error.

It should be noted that both errors can't be minimized at some time. Hence, it is general practice to assign abound to type I error thereby minimizing type II error while assigning abound to the type of error, that is choosing a level of significance, the decision-maker should consider the power of a test against various alternatives. In case, the power of a test is too low, the decision-maker should choose a higher value of the level of significance (  ) say 0.1, 0.2, 0.5 etc.