21 Hypothesis Testing: Developing null and alternative hypotheses

Learning Objectives:

After the completion of this module the student will understand:
Concept of the null and alternative hypotheses
How to develop null and alternative hypothesis about the population parameter.
Rationale for hypothesis testing
Various formats of hypothesis

1 Introduction

There are three techniques of inferential statistics namely,

(i) a point estimate,

(ii) confidence interval and

(iii) degree of confidence associated with a parameter value which lies within an interval values, help decision makers in determining an interval estimate of a population parameters value with the help of sample statistic along with certain level of confidence of the interval containing those values.

Another method to determine the true value of population parameters is to validate the claim ( belief, assumption or statement), also called hypothesis, made about this true value by using sample statistics.

2 Hypothesis and Hypothesis Testing

A statistical hypothesis (assertion, statement, belief or assumption) about an unknown population parameter value is tested and analyzed based on sample data. Accordingly, the hypothesized value of the population parameter is either not rejected or rejected. The method of testing the validity of one’s claim by evaluating the difference between the value of sample statistic and the corresponding hypothesized population parameter value is known as hypothesis testing. Few examples of hypothesis are.

A judge assumes that a person charged with a crime is innocent and subject this assumption (hypothesis) to verification by reviewing the evidence and hearing testimony before reaching to a verdict.

A medicine manufacturing organization claims that 96 per cent of all persons suffering from the said disease get cured by using its medicine.

An investment company claims that the average return across all its investments is 20 per cent, and so on.

3 Formats of Hypothesis

As stated earlier, a hypothesis is a statement which uses sample statistics to test about the true value of population parameters. To examine whether any significant difference exists or not between true value of population parameters and sample statistics, any hypothesis can be written in the form of ‘If–then‘ statement.

Consider, for example, the following statements:-

If inflation rate has decreased, then wholesale price index will also decrease. If employees are unhealthy, then they may take sick leave more frequently.

Hypothesis which involves terms such as ‘positive’, ‘negative’, “more than’ or ‘less than’ is called directional hypothesis and indicates the direction of the relationship between two or more populations under the study with respect to rrrrrrwa‘parameter value as illustrated on next page.

Side effects of particular medicine were experienced by less than 20 per cent of people. Greater stress causes lower job satisfaction to employees in any organization.

The non-directional hypothesis indicates the relationship but does not indicate the direction of relationship. In other words, there may be a significant relationship between two populations with respect to a parameter, even than nothing can be said whether the relationship would be positive or negative. Similarly, even if two populations differ with respect to a parameter, nothing can be said whether parameter value of any population will be more or less. The following examples illustrate non-directional hypotheses: –

The relationship between age and sick leaves.

The difference between average heart beats of men and women.

4 Rational for Hypothesis Testing

If a hypothesis claim or assumption is made about the specific value of population parameter, then it is expected that the corresponding sample statistic is close to the hypothesized parameter value. It is possible only when hypothesized parameter value is reasonably close to true value of parameter and the sample statistic turns out to be the good estimator of the parameter value. This approach to test a hypothesis is called a test statistic.

Since sample statistics are random variables therefore their sampling distributions show the tendency of variation. Consequently, sample statistic value is not expected to be’ exactly equal to the hypothesized parameter value. The difference, if any, is due to chance and/or sampling error. But if the value of the sample statistic differs significantly from the hypothesized parameter value, then hypothesized parameter value may not be correct and increases the doubt about the correctness of the hypothesis.

In statistical analysis, difference between the value of sample statistic and hypothesized parameter is specified in terms of the probability whether the particular level of difference is significant or not. The probability that a particular level of difference exists by chance can be calculated from the known sampling distribution of the test statistic.

The probability of concluding that observed difference between the test statistic value and hypothesized parameter value cannot be due to chance is called the level of significance of the test.

5 A null hypothesis is a statement that tells that there is no real difference in the sample and the population with respect to the issue under consideration. It maintains the status quo. If the null hypothesis is not rejected, no changes will be made. It is denoted as H0. It represents the claim or statement made about the value or range of values of the population parameters The capital letter H stands for hypothesis and the subscript‘ zero’ implies‘ no difference’ between sample statistic and the parameter value. Thus, hypothesis testing requires that the null hypothesis be considered true (status quo or no difference) until it is proved false on the basis of results derived from the sample data. The null hypothesis is expressed using mathematical sign (≤, =, ≥) to make a claim about specific value of the population parameter as follows: –

H0 : µ(≤, =, ≥)µ0

Where µ is population mean and µ0 represents a hypothesized value of µ. Only one sign out of ≤ = and ≥ will appear at a time when stating the null hypothesis.

6 An alternative hypothesis is one in which some difference is expected. It specifies those values that are believed to hold true and which is expected to hold the truth. Accepting the alternative hypothesis will lead to changes in opinions or actions. Thus, the alternative hypothesis is the opposite of the null. It is denoted as HA.. It is the counter claim or statement made against the value or range of values of the population parameter. Thus, an alternative hypothesis represents the claim that specific value or range of values of the population parameters is not equal to the value claimed in the null hypothesis and is written as

HA : µ ≠ µ0

HA : µ < µ0 or H1 : µ > µ0

Consequently

Each of the following statements is an example of a null hypothesis and alternative hypothesis: –

H0 : µ = µ0 HA : µ ≠ µ0 H0 : µ ≤ µ0 HA : µ > µ0

H0 : µ ≥ µ0 HA : µ < µ0

As a general principle, whatever that needs to be established is placed in the alternative hypothesis whereas the null hypothesis includes the status quo. The null hypothesis regarding the population parameter of interest is specified with either of the symbols like =, ≥, ≤ and the corresponding alternative hypothesis is then specified with either of the signs like ≠, <, >.

It is important to note that it is always the null hypothesis that is tested. The null hypothesis refers to a specified value of the population parameter (e.g. µ, σ, π) and not of a sample statistic .Secondly, a null hypothesis can either be rejected or not rejected, but it can never be accepted based on a single test. Thus, a statistical test can have one of the two outputs i.e. the null hypothesis is rejected and the alternative hypothesis accepted, or that the null hypothesis is not rejected based on the evidence and under such condition it would not be correct to conclude that since the null hypothesis is not rejected, alternative hypothesis can be accepted as valid. Thirdly, in marketing research, the null hypothesis is formulated in such a way that its rejection leads to the acceptance of the desired conclusion. The alternative hypothesis represents the conclusion for which evidence is sought.

7 Summary-

This module provides an opportunity to the students to develop an understanding about the basic concepts related to the hypothesis. It starts by explaining the meaning of hypothesis that it is an assumption, pre-conceived notion which needs to test. Two type of hypotheses are developed, i.e., null hypothesis and alternative hypothesis. Null hypotheis is a statement that tells that there is no real difference in the sample and the population with respect to the issue under consideration. It maintains the status quo. An alternative hypothesis is one in which some difference is expected. It specifies those values that are believed to hold true and which is expected to hold the truth. The null hypothesis regarding the population parameter of interest is specified with either of the symbols like =, ≥, ≤ and the corresponding alternative hypothesis is then specified with either of the signs like ≠, <, >.

8 Self-Check Exercise with solutions:

1- A promotional activity carried out by an ayurvedic pharma company propagates that its new medicine if consumed regularly for fifteen days will result in average weight loss of more than 5 kilograms. If you want to find out whether the claim made by the company is valid or not, following hypothesis shall be tested.

H0:µ ≤ 5

HA :µ> 5

Where, µ is the population mean.

Q-2: A gardeners claims that by using his particular method, a plant of a particular variety may be grown to the an average height of 50.00cm. To check if the claim made by the farmer about his technique is correct or not, nine hundred plants were introduced to the field and then the heights of some poppies were measured. Formulate the hypothesis.

Sol. H0: µ = 50 against the alternative hypothesis that they have changed HA: µ≠ 50.

Q-3: A micro biologist wants to find out whether a particular micro organism will grow at a different rate at four degree centigrade than at twenty five degree centigrade. Given that the average number of micro organism that may grow at twenty five degree centigrade is 3.

Sol.

Null hypothesis will be H0: µ = 3

Alternative hypothesis will be HA µ: ≠3

Q-4: A trade group predicts that back-to-school spending will average Rs 1000/-per family. A different economic model is needed if the prediction is wrong. Develop the null and alternative hypotheses to determine if the alternative model may be required.

Sol.

H0:µ =1000

HA :µ≠1000

Q-5: A television research analyst wishes to test a claim that more than fifty percent of the house-holds will tune in for a TV episode. Specify the null and the alternative hypotheses to test the claim. Sol.

H0:p ≤ 50

HA: p> 50

p signifies population proportion.

Q-6: An advertisement for a popular weight loss clinic suggests that participants in its new diet program lose, on average, more than 10 kg. A consumer activist wants to determine if the advertisement’s claim. Specify the null and alternative hypotheses to validate the advertisement’s claim.

Sol.	H0:µ≤ 10
	HA: µ >10

Q-7-It is generally believed that at least .60 of the residents in a small town in a town are happy with their lives. A sociologist is concerned about the lingering economic crisis and wants to determine whether the crisis has adversely affected the happiness level in this town. Specify the null and alternative hypotheses to determine if the sociologist’s concern is valid.

Sol.

H0:p≥ 50

HA: p < 50

Learn More:

https://www.youtube.com/watch?v=R2hxisYFKxM
https://www.youtube.com/watch?v=spwEHuHOMuk
Sharma, J K (2014), Business Statistics, S Chand & Company, N Delhi.
Bajpai, N (2010) Business Statistics, Pearson, N Delhi.
Trevor Hastie, Robert Tibshirani, Jerome Friedman (2009), The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd Edition, Springer.
Darrell Huff (2010), How to Lie with Statistics, W. W. Norton, California.
K.R. Gupta (2012), Practical Statistics, Atlantic Publishers & Distributors (P) Ltd., N. Delhi.