Comprehensive Essay on Hypothesis: Formulation, Types and Testing

In hypothesis testing, we must state the hypothesized value of the population parameter before we begin sampling. The hypothesis we wish to test is called Null Hypothesis and is denoted as H₀. Example: If we want to test the hypothesis that the population mean is equal to 600, we can write it as follows: H₀: p = 600 and read, “The null hypothesis is that the population mean is equal to 600.”

Alternative Hypothesis

Whenever we reject the null hypothesis, the conclusion that does accept is called Alternative Hypothesis and is symbolised by H, for the null hypothesis:

H₀: p. = 600 (Read as “The null hypothesis is that the population mean is equal to 600.”) Three possible alternative hypotheses may be as follows:

H,: p * 600-The alternative hypothesis is that the population mean is not equal to 600. H,: p > 600-The alternative hypothesis is that the population mean is greater than 600. H,: p < 600-The alternative hypothesis is that the population mean is less than 600.

Significance Level

The purpose of hypothesis testing is not to question the computed value of the sample statistic but to make an estimate about the difference between that sample statistic and a hypothesized population parameter. Now, the next step is to decide what criterion should be used for deciding whether to accept or reject the null hypothesis.

If we test a hypothesis at the 5 per cent level of significance, this means that we will reject the null hypothesis if the difference between the sample statistics and the hypothesized population parameter is very large. i

Assuming the hypothesis to be correct, then the significance level indicates the percentage of sample means that is outside certain limits.

Figure 8.7 is interpreting the significance level. In each tail, 2.5 per cent of the area is located. We can see that 95 per cent of all the area under the curve is included in an interval extending 1.96 s_x on either side of the hypothesized mean. Thus, there is no significant difference between the sample statistics and the hypothesized population parameter, which is 95 per cent of the area. In the remaining 5 per cent, a significant difference does exist.

In Fig. 8.7, 95 per cent of the area under the curve is the one where we accept the null hypothesis. The two small parts under the curve, representing a total of 5 per cent of the area, are those where we reject the null hypothesis.

Two-Tailed and one-Tailed Tests

1. Sometimes, we use one-tail test. Consider the aforementioned example once again. The wholesaler buys light bulbs in large lots and does not accept a lot of bulbs unless their mean life is 800 hours. At the arrival of each consignment, the wholesaler tests a sample to decide whether it should accept the consignment.

The buyer company would reject the consignment only if he feels that the mean life is less than 800 hours. If he feels that the bulbs have more life spans than expected (with a mean life above 800 hours), he would not reject the consignment, He would certainly accept better items at no extra cost. Therefore, the wholesaler’s hypotheses are: H₀: p = 800 hours and H, : p < 800 hours. He rejects H₀ only if the mean life of the sampled bulbs is significantly below 800 hours. This situation has been depicted.

2. A two tailed test of a hypothesis would reject the null hypothesis if the sample mean is significantly higher or lower than the hypothesized population mean. Hence, in a two- tailed test, there are two rejection regions.

A two tailed test is applied when the null hypothesis is p = m_H0 (which is a specified value) and the alternative hypothesis is p * p_H0. For example, a manufacturer of light bulbs plans to manufacture bulbs with a mean life of p = p_H0 = 800 hours. If the life time is shorter, he would lose customers to his competitors.

If the lifetime is longer, he has to bear a very high manufacturing cost because the filaments would be excessively thick. In order to check, if the manufacturing process is working well, he takes a sample of the output to test the hypothesis H₀: p = 800. Because, he does not want to deviate significantly from 800 hours in either direction, he will use a two-tailed test alternative hypothesis is Hj : p = 800, this means, he would reject the null hypothesis if the mean life of bulbs in the sample is either too far above 800 hours or too far below 800 hours.

Hypothesis Testing Test of Means

1. By One-Tailed Test: Take an example of a drug, which is frequently used by a hospital. The individual dose of this drug is 125 cc. There is no harm when body takes excessive does of this drug. But on the other hand, insufficient doses do not assist doctors in the necessary medical treatment. The hospital has been purchasing the same drug from the same manufacturer for many years and the population’s standard deviation is 4 cc. The hospital inspects 50 doses of this drug at random from a very large consignment and calculates the mean of these doses to be 99.5 cc. The data in this case are:

p_H0 =125 (hypothesised value of the population mean) cr = 4 (population standard deviation) n = 50 (sample size) x = 99.5 (sample mean)

The hospital sets a 0.10 significance level. We have to find out “whether the dosages in this consignment are too small.”

In order to find the answer, we can state the problem as follows: H₀: p = 125 (null hypothesis) H,:p <125 (alternative hypothesis)

a = 0.10 -level of significance for testing this hypothesis

Here, we would calculate the standard error or the mean, (the population size is assumed to be infinite).

The hospital wants to know whether the actual dosage is 125 cc, or the dosage is too small. The hospital must see that the dosage should be more than a certain amount otherwise it should reject the consignment. This is one-tailed test and the shaded portion is representing the 0.10 significance level. In Fig. 8.9, the acceptance region includes 40 per cent of the area on the left side of the mean and 50 per cent of the area on the right side of the mean. The non-acceptance region has an area of 10 per cent. It has been shown by shaded portion.

+As we know the population standard deviation and n is larger than 30, we can use the nor distribution. The appropriate z value for 40 per cent of the area under the curve is 1.28. Using information, we can calculate the acceptance region’s lower limit: p_H0 – 1.28 o_xen = 125 – 1.28 (0.5658) = 125 – 0.7242 = 124.276 cc (lower limit)

As a result, the hospital should accept the null hypothesis, because there is no significant difference between our hypothesized mean of 125 cc and the observed mean of the sample i.e., 99.5. On the basis of this sample of 50 doses, the hospital should accept the consignment.

2. By Two-Tailed Test: An engineering firm supplies water pumps to a hotel. These pumps must have a pumping capacity of 40,000 gallon per minute each. If the pumping capacity has to be very exact, then production costs rise (because of the use of extremely sturdy impellers and other testing procedures). Researchers have concluded that the standard deviation of the pumps is 2000 gallons per minute. The manufacturer selects a sample of 40 pumps from the fresh production lot. These pumps are subjected to tests. It is found that the mean pumping capacity of the sample of pumps is 39,500 gallons per minute. If we write these data in a symbolic notation, we get:

p_H0 (hypothesised value of the population mean) = 40000 gpm o population’s standard deviation) = 2000 gpm n (sample size) = 40 pumps x (sample mean) = 39,500

It the supplier uses a significance level (a) of 0.05 in testing, we can state the problem as:

H₀: p = 40000 (null hypothesis: the true mean is 40,000) H,: p * 40000 (alternative hypothesis is not 40,000) a = 0.05 <- level of significance for testing this hypothesis

We know that standard deviation and size of the population are large enough to be treated as infinity. So, we may use normal distribution in our testing. First of all, we calculate standard error of the mean by using the following equation:

2000

= 316.225

6.3246

= 316.225 (standard error of the mean) Figure 8.10 shows the significance level of 0.05 (the two shaded regions contain 0.025 part of the area). The 0.95 acceptance region contains two equal areas of 0.475 each. The appropriate value for 0.475 part of the area under the curve is 1.96. Now, we can find out the limits of the acceptance regions:

p_H0 + 1.96 = 40000 + 1.96 (316.2255) = 40000 + 619.80

= 40619.80 (upper limit)

Pho ” 196 ct_x = 40000 – 1.96 (316.225) = 40000-619.80 = 39380.20 (lower limit)

Now, there are two limits of the acceptance region-40619.80 and 39380.20. It is quite obvious that sample mean, 39,500, lies within the acceptance region. The supplier should accept the null hypothesis because there is no significant difference between the hypothesis mean of40, 000 and the observed mean of the sample.

Research Report

Formally, a research report usually consists of the following parts:

1. Title pages

2. Abstract: If the researcher feels that his report could also have come applications outside the scientific readership, he can, in addition to the abstract, also prepare an application account for this publication.

3. Table of contents (can be at the back of the book as well)

4. Preface

5. The text itself

6. List of sources and of literature

7. Possible indexes of people or a general index

8. Appendices

Title Pages

1. The title pages are the first pages of the book.

2. The uppermost one only indicates the series of publications and the title of the book.

3. The title is the name of the publication.

4. A pithy name will help future researchers working on a list of literature about the same subject.

5. It is advisable to give the book a title in which the first noun expresses the most important topic that was studied: for this reason, it is not wise to start the title with the word ‘Research.’

6. Author(s) has/have no honorary titles.

7. Published at’…’ means the location of the publishing house.

8. The next page is the actual title page.

9. The actual title page contains the series of publications, if any, and the number of the publications in this series.

10. The title page also indicates, possibly on its back side; printing house, printing place, data and ISBN (International Standard Book Number), provided by the publisher.

Abstract

1. The abstract clarifies the title and is an account of no more than 250 words dealing with what has been studied and how.

2. The gives an outline of the results in such a way that the reader understands the subject matter without having to read the report itself.

3. Outlines of the report is best acquired by the researcher writing his own report.

4. The purpose of the abstract is to make subsequent search for information easier.

5. It helps subsequent researchers to decide if it is worth their while to seek that particular report.

6. The abstract is written in the same language as the report itself, and often also in English.

7. The researcher can also indicate the UDC number describing the topics of the study. These topics should preferably be picked in the thesaurus of the field, if there is one. In this matter, it is advisable for the researcher to turn to the library systems analyst.

8. If researcher prepares an application account, it can also serve as an announcement to various professional periodicals.

Table of Contents

It can be at the back of the book.

Preface

1. The preface is addressed to its readers.

2. The preface tells about the origins of the research and about the different parties that have contributed to it.

3. It is customary to thank here the sponsors and the people having promoted the work.

The Text Itself

1. The text itself presents the purpose of the research, the problem or the aim.

2. It contains the nature of the data analysed and the based for selecting data.

3. It tells how the results were acquired.

4. The text should also provide the reader with a possibility to estimate the reliability of the results.

5. The text should contain method of research, results obtained and conclusions.

6. It makes an estimate of the reliability of the results and of how generally they can possibly be applied.

List of Sources and of Literature

It is given in the end of each of each chapter or in the end of book.

Index and Appendices

Possible indexes of people (or a general index) and appendices are, generally, given in the end of the book.

Steps in Writing (Direction to the Writer)

In report writing a writer should take following steps:

Step I: Start the writing by outlining the questions, the results and then the logical conclusions which combine these two on a sheet of paper.

Step II: Select the most suitable order of presentation of these three. This is not necessarily the same order in which the sections will be written: it is often better to start at those passages that feel the easiest, like descriptions of research methods or of the new findings.

Step III: Modify a research many times. These revisions should not be allowed to endanger the finished passages.

Final Report

1. After the report is printed, the researcher has still something to do; he should try to prevent that the report only gathers dust on a bookshelf. He should try to find out which are those people who would want to apply the new knowledge, and inform these.

2. If the report is published in a scientific series, it will automatically be distributed within the series to libraries of the field. Information about the publication and its abstract will thus go to the databases of all the scientific libraries.

3. It is nevertheless worthwhile considering if there would be other parties who might need the information but do not know where to look for it. Usual channels for spreading research are articles to professional journals and lectures at the seminars of the people working in the pertinent business.