UNIT IV Inferential statistics
What is Hypothesis?
A hypothesis is an assumption that is made based on some evidence. This is the initial point of
any investigation that translates the research questions into predictions. It includes components
like variables, population and the relation between the variables. A research hypothesis is a
hypothesis that is used to test the relationship between two or more variables.
Characteristics of Hypothesis
Following are the characteristics of the hypothesis:




The hypothesis should be clear and precise to consider it to be reliable.
If the hypothesis is a relational hypothesis, then it should be stating the relationship
between variables.
The hypothesis must be specific and should have scope for conducting more tests.
The way of explanation of the hypothesis must be very simple and it should also be
understood that the simplicity of the hypothesis is not related to its significance.
Types of Hypothesis
There are six forms of hypothesis and they are:






Simple hypothesis
Complex hypothesis
Directional hypothesis
Non-directional hypothesis
Null hypothesis
Associative and casual hypothesis
Simple Hypothesis
It shows a relationship between one dependent variable and a single independent variable. For
example – If you eat more vegetables, you will lose weight faster. Here, eating more vegetables
is an independent variable, while losing weight is the dependent variable.
Complex Hypothesis
It shows the relationship between two or more dependent variables and two or more
independent variables. Eating more vegetables and fruits leads to weight loss, glowing skin,
and reduces the risk of many diseases such as heart disease.
Directional Hypothesis
It shows how a researcher is intellectual and committed to a particular outcome. The
relationship between the variables can also predict its nature. For example- children aged four
years eating proper food over a five-year period are having higher IQ levels than children not
having a proper meal. This shows the effect and direction of the effect.
Non-directional Hypothesis
It is used when there is no theory involved. It is a statement that a relationship exists between
two variables, without predicting the exact nature (direction) of the relationship.
Null Hypothesis
It provides a statement which is contrary to the hypothesis. It’s a negative statement, and there
is no relationship between independent and dependent variables. The symbol is denoted by
“HO”.
Associative and Causal Hypothesis
Associative hypothesis occurs when there is a change in one variable resulting in a change in
the other variable. Whereas, the causal hypothesis proposes a cause and effect interaction
between two or more variables.
What Is Hypothesis Testing?
Hypothesis testing is an act in statistics whereby an analyst tests an assumption regarding a
population parameter. The methodology employed by the analyst depends on the nature of the
data used and the reason for the analysis.
Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Such
data may come from a larger population, or from a data-generating process. The word
"population" will be used for both of these cases in the following descriptions.
KEY TAKEAWAYS



Hypothesis testing is used to assess the plausibility of a hypothesis by using sample
data.
The test provides evidence concerning the plausibility of the hypothesis, given the
data.
Statistical analysts test a hypothesis by measuring and examining a random sample of
the population being analyzed.
How Hypothesis Testing Works
In hypothesis testing, an analyst tests a statistical sample, with the goal of providing evidence
on the plausibility of the null hypothesis.
Statistical analysts test a hypothesis by measuring and examining a random sample of the
population being analyzed. All analysts use a random population sample to test two different
hypotheses: the null hypothesis and the alternative hypothesis.
The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a
null hypothesis may state that the population mean return is equal to zero. The alternative
hypothesis is effectively the opposite of a null hypothesis (e.g., the population mean return is
not equal to zero). Thus, they are mutually exclusive, and only one can be true. However, one
of the two hypotheses will always be true.
4 Steps of Hypothesis Testing
All hypotheses are tested using a four-step process:
1. The first step is for the analyst to state the two hypotheses so that only one can be right.
2. The next step is to formulate an analysis plan, which outlines how the data will be
evaluated.
3. The third step is to carry out the plan and physically analyze the sample data.
4. The fourth and final step is to analyze the results and either reject the null hypothesis,
or state that the null hypothesis is plausible, given the data.
Real-World Example of Hypothesis Testing
If, for example, a person wants to test that a penny has exactly a 50% chance of landing on
heads, the null hypothesis would be that 50% is correct, and the alternative hypothesis would
be that 50% is not correct.
Mathematically, the null hypothesis would be represented as Ho: P = 0.5. The alternative
hypothesis would be denoted as "Ha" and be identical to the null hypothesis, except with the
equal sign struck-through, meaning that it does not equal 50%.
A random sample of 100 coin flips is taken, and the null hypothesis is then tested. If it is found
that the 100 coin flips were distributed as 40 heads and 60 tails, the analyst would assume that
a penny does not have a 50% chance of landing on heads and would reject the null hypothesis
and accept the alternative hypothesis.
If, on the other hand, there were 48 heads and 52 tails, then it is plausible that the coin could
be fair and still produce such a result. In cases such as this where the null hypothesis is
"accepted," the analyst states that the difference between the expected results (50 heads and
50 tails) and the observed results (48 heads and 52 tails) is "explainable by chance alone."
Type I & Type II Errors | Differences, Examples, Visualizations
Published on January 18, 2021 by Pritha Bhandari. Revised on May 6, 2022.
In statistics, a Type I error is a false positive conclusion, while a Type II error is a false
negative conclusion.
Making a statistical decision always involves uncertainties, so the risks of making these errors
are unavoidable in hypothesis testing.
The probability of making a Type I error is the significance level, or alpha (α), while the
probability of making a Type II error is beta (β). These risks can be minimized through careful
planning in your study design.
Example: Type I vs Type II errorYou decide to get tested for COVID-19 based on mild
symptoms. There are two errors that could potentially occur:



Type I error (false positive): the test result says you have coronavirus, but you
actually don’t.
Type II error (false negative): the test result says you don’t have coronavirus, but you
actually do.
Example: Type I and Type II errorsA Type I error happens when you get false positive
results: you conclude that the drug intervention improved symptoms when it actually
didn’t. These improvements could have arisen from other random factors or
measurement errors.

A Type II error happens when you get false negative results: you conclude that the drug
intervention didn’t improve symptoms when it actually did. Your study may have
missed key indicators of improvements or attributed any improvements to other factors
instead.


Type I error

A Type I error means rejecting the null hypothesis when it’s actually true. It means
concluding that results are statistically significant when, in reality, they came about
purely by chance or because of unrelated factors.
The risk of committing this error is the significance level (alpha or α) you choose.
That’s a value that you set at the beginning of your study to assess the statistical
probability of obtaining your results (p value).

Type II error
A Type II error means not rejecting the null hypothesis when it’s actually false. This is not
quite the same as “accepting” the null hypothesis, because hypothesis testing can only tell you
whether to reject the null hypothesis.
Instead, a Type II error means failing to conclude there was an effect when there actually was.
In reality, your study may not have had enough statistical power to detect an effect of a certain
size.
Difference Between T-test and Z-test (With Table)
T-test and z-test are terms common when it comes to the statistical testing of hypothesis in the
comparison of two sample means. Notably, the two tests are parametric procedures of
hypothesis testing since they are both their variables are measured on an interval scale.
Comparison Table Between T-Test and Z-Test
Parameter
Comparison
of
T-Test
Type of Distribution Student t-distribution
Z-Test
Normal distribution
Population Variance
Suitable for unknown population
Suitable for known population variance.
variance.
Sample Size
Small sample size.
Key Assumptions
All data points are assumed, not
All data points are assumed to be independent.
dependent.
Large sample size.
Sample values are accurately Distribution of z is assumed to be normal, with a
collected and recorded.
mean of zero and a variance of one.
Use
The sample size is small.
The sample size is large.
For limited sample sizes, not For large sample sizes and known standard
exceeding thirty.
deviation.
What is T-Test?
The t-test is a parameter applied to an identity to identify how the data averages differ from
each other when the variance or standard deviation is not given. The t-test is based on Student
t-statistic, having the mean being known and the variance of the population approximated from
the sample.
The standard deviation of the population is estimated by dividing the standard deviation of the
sample by the square root of the population size.
What is Z-Test?
On the other hand, the z-test is the hypothesis test that ascertains if the averages of two sets of
data differ from each other being given the variance or standard deviation.
The z-test is a univariate test that is based on the standard normal distribution.
Chi-Square Test vs. ANOVA: What’s the Difference?
Chi-Square tests and ANOVA (“Analysis of Variance”) are two commonly used statistical
tests.
In statistics, there are two different types of Chi-Square tests:
1. The Chi-Square Goodness of Fit Test – Used to determine whether or not a categorical
variable follows a hypothesized distribution.
For example:


We want to know if a die is fair, so we roll it 50 times and record the number of times
it lands on each number.
We want to know if an equal number of people come into a shop each day of the week,
so we count the number of people who come in each day during a random week.
2. The Chi-Square Test of Independence – Used to determine whether or not there is a
significant association between two categorical variables.
For example:


We want to know if gender is associated with political party preference so we survey
500 voters and record their gender and political party preference.
We want to know if a person’s favorite color is associated with their favorite sport so
we survey 100 people and ask them about their preferences for both.
Note that both of these tests are only appropriate to use when you’re working with categorical
variables. These are variables that take on names or labels and can fit into categories.
Explanation of ANOVA
In statistics, an ANOVA is used to determine whether or not there is a statistically significant
difference between the means of three or more independent groups.
For example:


We want to know if three different studying techniques lead to different mean exam
scores.
We want to know if four different types of fertilizer lead to different mean crop yields.
Note that it’s appropriate to use an ANOVA when there is at least one categorical variable and
one continuous dependent variable.
When to Use Chi-Square Tests vs. ANOVA
As a basic rule of thumb:












Use Chi-Square Tests when every variable you’re working with is categorical.
Use ANOVA when you have at least one categorical variable and one continuous
dependent variable.
Use the following practice problems to improve your understanding of when to use
Chi-Square Tests vs. ANOVA:
Practice Problem 1
Suppose a researcher want to know if education level and marital status are associated
so she collects data about these two variables on a simple random sample of 50 people.
To test this, she should use a Chi-Square Test of Independence because she is
working with two categorical variables – “education level” and “marital status.”
Practice Problem 2
Suppose an economist wants to determine if the proportion of residents who support a
certain law differ between the three cities.
To test this, he should use a Chi-Square Goodness of Fit Test because he is only
analyzing the distribution of one categorical variable.
Practice Problem 3
Suppose a basketball trainer wants to know if three different training techniques lead
to different mean jump height among his players.
To test this, he should use a one-way ANOVA because he is analyzing one categorical
variable (training technique) and one continuous dependent variable (jump height).
UNIT 5 Report Writing
SIGNIFICANCE OF REPORT WRITING
Research report is considered a major component of the research study for the research task
remains incomplete till the report has been presented and/or written. As a matter of fact
even the most brilliant hypothesis, highly well designed and conducted research study, and
the most striking generalizations and findings are of little value unless they are effectively
communicated to others. The purpose of research is not well served unless the findings are
made known to others. Research results must invariably enter the general store of
knowledge. All this explains the significance of
What is a report?
A report is a document that presents the results of an investigation, project or initiative. It can
also be an in-depth analysis of a particular issue or data set. The purpose of a report is to inform,
educate and present options and recommendations for future action. Reports are an integral
element of dozens of industries, including science, healthcare, criminal justice, business and
academia. Reports typically consist of several key elements, including:






Detailed summaries of events or activities
Analysis of the impact of the event
Evaluations of the facts and data
Predictions for what may happen as a result of an event
Recommendation for next course of action
Conclusion
Many occupations involve writing reports as a primary responsibility. Doctors must write
medical reports that present their analyses of certain patients or cases. Police officers write
reports that outline the details of interrogations and confrontations. Project managers write
regular reports that keep their supervisors updated on how a particular project is developing.
All of these reports must be well-written, accurate and efficient.
Report writing
Report Writing Format
Following are the parts of a report format that is most common.
1. Executive summary – highlights of the main report
2. Table of Contents – index page
3. Introduction – origin, essentials of the main subject
4. Body – main report
5. Conclusion – inferences, measures taken, projections
6. Reference – sources of information
7. Appendix
How to write a report
Knowing how to write a successful report can make you a valuable asset in your current
workplace or an appealing candidate for new employers. Here are some steps to follow when
writing a report:
1.
2.
3.
4.
5.
6.
7.
Decide on terms of reference.
Conduct your research.
Write an outline.
Write a first draft.
Analyze data and record findings.
Recommend a course of action.
Edit and distribute.
1. Decide on terms of reference
Many formal reports include a section that details the document's "terms of reference". These
terms include:




What the report is about
Why it is necessary
When it was written
What its purpose is
Setting these terms helps both the writer and their readers to understand why the report is
important and what it hopes to accomplish. The terms of reference are usually explained in the
first paragraph so that the reader can determine its relevance without having to read the entire
document. Setting concrete terms early on will help you create the report's outline and keep
your discussions on track throughout the writing process.
Related: The Complete Guide to Researching a Company
2. Conduct your research
Most reports will require you to collect a store of data that directly relates to your topic. You
may already have access to this information if, for example, you are a doctor who has copies
of a patient's medical charts. However, if you are tasked with analyzing an issue or
investigating an event, you will likely need to spend some time requesting, finding and
organizing data.
Interpreting data and formatting it in a way that your readers will understand is an important
part of writing a report. You may need to create charts, graphs or timelines that make your raw
information easier to comprehend. You will also need to carefully cite your sources and keep
track of where and how you found your data in order to present it professionally.
3. Write an outline
The next step is to construct your report's outline. This typically looks like a bulleted or
numbered list of all the different sections in the document. Your report's outline might look
similar to this:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Title page
Table of contents
Introduction
Terms of reference
Summary of procedure
Findings
Analysis
Conclusion
References or bibliography
The order of these sections—and whether or not you decide to include them all—will depend
on the specific type of report, how long it is and how formal it needs to be. The most important
thing to do when writing your outline is to include all the necessary sections and eliminate
anything that does not directly contribute to the report's purpose.
4. Write a first draft
Writing a first draft is one of the most important stages of constructing a successful report. The
purpose of the first draft is not to write a perfect document, but rather to get all the main
elements of your information out of your head and onto the page. You will have time to add to
and edit this first attempt later on, so your primary goal is just to organize your data and analysis
into a rough draft that will eventually become a final product.
While writing your first draft, you will likely find gaps in your data or holes in your analysis.
Make note of these, but do not try to address every issue as you write. Instead, finish the draft,
and save the problem-solving for when you begin the editing process.
Related: 10 Resume Writing Tips to Help You Land a Job
5. Analyze data and record findings
The focus of every report is the "findings" section or the part where you present your
interpretation of the data. For an accountant, the findings could involve an explanation as to
why a company's stock drooped during the previous quarter. For an environmental scientist, it
could include a summary of an experiment on biodegradable plastics and how the results could
affect waste management methods.
The findings section of your report should always provide valuable information related to the
topic or issue you are addressing, even if the results are less than ideal. If your final conclusion
is that the data was insufficient or the research method was flawed, you will need to explain
this in a professional and accurate manner.
6. Recommend a course of action
The final section of your report's body is your recommendation. After examining the data and
analyzing any outcomes, you are qualified to present an idea as to what actions should be taken
in response to your findings. After reviewing the number of overtime hours that their team has
been working, a project manager may recommend that an additional employee be added to the
team. A surgeon might recommend that the hospital introduce new sterilization methods into
the operating room after noting an increase in preventable infections in the previous six months.
If you have presented your data well and shown your expertise, your reader is likely to trust
your judgment.
7. Edit and distribute
The final stage of writing a report is editing it thoroughly and distributing it to your audience.
You will need to edit for grammar mistakes, spelling errors and typos. You will also need to
double-check your data, make sure your citations are correct and read over the entire document
to make sure it presents a cohesive narrative. If the report is going to be read by a wide
audience, you may decide to ask someone else to proofread it or give you their opinion on the
readability of the content.
Distributing the report can take different forms depending on your particular occupation. You
might email it to your supervisor, present it verbally during a staff meeting or publish it in a
professional journal. Regardless of how or where it is read, your goal is always to create a
concise, informative and effective document that will contribute to increased productivity in
your workplace.
writing research report. There are people who do not consider writing of report as an integral part
of the research process. But the general opinion is in favour of treating the presentation of
research results or the writing of report as part and parcel of the research project. Writing of
report is the last step in a research study and requires a set of skills somewhat different from
those called for in respect of the earlier stages of research. This task should be accomplished
by the researcher with utmost care; he may seek the assistance and guidance of experts for the
purpose.
What is a Bibliography?
A bibliography is a listing of the books, magazines, and Internet sources that you use in
designing, carrying out, and understanding your science fair project.
How to write a bibliography
A bibliography is not just “works cited.” It is all the relevant material you drew upon to write
the paper the reader holds.
Do I need a bibliography?
If you read any articles or books in preparing your paper, you need a bibliography or footnotes.

If you cite the arguments of “critics” and “supporters,” even if you don’t name them or
quote them directly, you are likely referring to information you read in books or articles
as opposed to information you’ve gathered firsthand, like a news reporter, and so you
need a bibliography.

If you quote sources and put some of the reference information in the text, you still need
a bibliography, so that readers can track down the source material for themselves.

If you use footnotes to identify the source of your material or the authors of every quote,
you DO NOT need a bibliography, UNLESS there are materials to which you do not
refer directly (or if you refer to additional sections of the materials you already
referenced) that also helped you reach your conclusions. In any event, your footnotes
need to follow the formatting guidelines below.
How to write a bibliography
These guidelines follow those of the American Psychological Association and may be slightly
different than what you’re used to, but we will stick with them for the sake of consistency.
Notice the use of punctuation. Publication titles may be either italicized or underlined, but not
both.
Books
Books are the bibliography format with which you’re probably most familiar. Books follow
this pattern:
12
Author Last Name, Author First Name. (Publication Year) Title. Publisher’s City:
Publisher. Page numbers.
Alexander, Carol. (2001) Market Models: A Guide to Financial Data Analysis. New
York, NY: John Wiley & Sons. pp. 200-220.
Periodicals
Periodicals remove the publisher city and name and add the title of the article and the volume
or issue number of the periodical. Notice article titles are put in quotation marks and only the
publication title is italicized or underlined.
Author Last Name, Author First Name. (Publication Date—could be more than a year)
“Article Title.”Publication Title, Vol. #. (Issue #), Page numbers.
Salman, William A. (July-August 1997) “How to Write a Great Business Plan.” Harvard
Business Review 74. pp. 98-108.
Research Ethics and Integrity
What are research ethics and research integrity and why are they important?
Research ethics and integrity practices make sure that research is conducted according to the
highest standards of practice, and with the minimal risk of adverse or harmful outcomes or
consequences.
The research community and a wider public will have confidence in the outcomes of your
research and the quality of your research output will be enhanced.
 Research is conducted honestly
 Provides confidence that conclusions drawn from research can be relied upon to be accurate
 Minimises potential risks to researchers and participants of research, protecting the
vulnerable and ensuring their safety and wellbeing
 Safeguards data collected during the course of research, particularly sensitive data,
respecting confidentiality
 Avoids unfair allegations of misconduct, whilst ensuring that genuine concerns are
appropriately investigated
 Prevents people being drawn into terrorism
 Ensures conflicts of interest are identified and avoided
What do I need to consider?
If your project involves any of the following, you will need to think about how to address
these.
 Are there potential risks to researchers and participants involved in the research?
13
This could be either the activity to be undertaken or the location the research is to be conducted.
For example:  Are you investigating illegal behaviours or activities?
 Could the dissemination of your findings adversely affect participants?
 Will your research be carried out in a hazardous area or in an area not recommended for
travel?
 Does your research concern groups which are legally construed as terrorist or extremist?
 Will the research expose either researcher or participants to situations or circumstances they
might find distressing?
 Will any of the participants be classed as vulnerable?
For example
:  Are any of them under 16 years of age?
 Unable to communicate in the language in which the research is conducted?
 Members of a stigmatised or marginalised social group?
 Have a relationship with the researcher (either personally or professionally)?
 How will you show that any participants have agreed to take part?
 Will they be able to give informed consent individually?
 Will consent need to be obtained from parents or guardians?
 Will the purpose of your research be concealed from participants at the outset?
 Are you collecting personal data, either face to face or online? If so,
 How will you obtain the consent of participants?
 How will this be securely stored and maintained?
 How will this data be used?
 With whom will it be shared?
 How and when will it be disposed of?
Misconduct in Research and consequences of misconduct:
Research report is a channel of communicating the research findings to the readers of the
report. A good research report is one which does this task efficiently and effectively. As
such it must be prepared keeping the following precautions in view:
1. While determining the length of the report (since research reports vary greatly in
length), one should keep in view the fact that it should be long enough to cover the
14
subject but short enough to maintain interest. In fact, report-writing should not be a
means to learning more and more about less and less.
2. A research report should not, if this can be avoided, be dull; it should be such as to
sustain reader’s interest.
3. Abstract terminology and technical jargon should be avoided in a research report.
The report should be able to convey the matter as simply as possible. This, in other
words, means that report should be written in an objective style in simple
language, avoiding expressions such as “it seems,” “there may be” and the like.
4. Readers are often interested in acquiring a quick knowledge of the main findings
and as such the report must provide a ready availability of the findings. For this
purpose, charts,
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
graphs and the statistical tables may be used for the various results in the main
report in addition to the summary of important findings.
The layout of the report should be well thought out and must be appropriate and in
accordance with the objective of the research problem.
The reports should be free from grammatical mistakes and must be prepared
strictly in accordance with the techniques of composition of report-writing such as the
use of quotations, footnotes, documentation, proper punctuation and use of
abbreviations in footnotes and the like.
The report must present the logical analysis of the subject matter. It must reflect a
structure wherein the different pieces of analysis relating to the research problem
fit well.
A research report should show originality and should necessarily be an attempt to
solve some intellectual problem. It must contribute to the solution of a problem and
must add to the store of knowledge.
Towards the end, the report must also state the policy implications relating to the
problem under consideration. It is usually considered desirable if the report makes
a forecast of the probable future of the subject concerned and indicates the kinds
of research still needs to be done in that particular field.
Appendices should be enlisted in respect of all the technical data in the report.
Bibliography of sources consulted is a must for a good report and must necessarily
be given.
Index is also considered an essential part of a good report and as such must be
prepared and appended at the end.
Report must be attractive in appearance, neat and clean, whether typed or printed.
Calculated confidence limits must be mentioned and the various constraints
experienced in conducting the research study may also be stated in the report.
Objective of the study, the nature of the problem, the methods employed and the
analysis techniques adopted must all be clearly stated in the beginning of the report
in the form of introduction.
15