statistics report topic hypothesis testing

The difference between the sample means of 60mm Hg is shown to the right together with the 95% confidence interval from 11 to 10-9 23 TABLE OF TABLES Table 1 Hypotheses to be tested 13

An Ngọc Khánh 11205585

Nguyễn Hằng Nga 11206248 Nguyễn Thanh Thảo 11206962 Nguyễn Hoàng Đức Anh 11204380

Hanoi, November 2022

Table of Contents

1 INTRODUCTION: Hypothesis testing

1.1 WHAT IS HYPOTHESIS TESTING?

1.2 ITS NECESSITY AND SOME APPLICATION

1.2.1 The necessity of hypothesis testing

1.2.2 Understanding Hypothesis Testing

2.3 OTHER RESEARCHES AND TECHNIQUES

2.3.1 RELIABILITY ANALYSIS ON THE LIFE OF THE LIGHT BULBS

2.3.2 CONFIDENCE INTERVALS RATHER THAN P VALUES: ESTIMATION RATHER THAN HYPOTHESIS TESTING

TABLE OF FIGURES

Figure 1 (Figure 9.7) 19 Figure 2 (Figure 9.8) 20 Figure 3 (Figure 9.9) 21 Figure 4 Systolic blood pressures (BP) in 100 diabetics and 100 non-diabetics with mean levels of 1464 and 1404

mm Hg respectively The difference between the sample means of 60mm Hg is shown to the right together with the 95% confidence interval from 11 to 10-9 23

TABLE OF TABLES

Table 1 Hypotheses to be tested 13 Table 2 t test (Control Group) 14 Table 3 t test (Experimental Group) 14 Table 4 Hypothesis status 15 Table 5 Critical Points for Different Levels of Significance 18

1 INTRODUCTION: Hypothesis testing

1.1 WHAT IS HYPOTHESIS TESTING?

Hypothesis testing is an assessment method that allows researchers to determine the plausibility of a hypothesis It involves testing an assumption about a specific population parameter to know whether it’s true or false These population parameters include

variance, standard deviation, and median

Typically, hypothesis testing starts with developing a null hypothesis and then

performing several tests that support or reject the null hypothesis The researcher uses test statistics to compare the association or relationship between two or more variables Researchers also use hypothesis testing to calculate the coefficient of variation and determine if the regression relationship and the correlation coefficient are statistically significant

1.2 ITS NECESSITY AND SOME APPLICATION

1.2.1 The necessity of hypothesis testing

The most significant benefit of hypothesis testing is it allows you to evaluate the strength

of your claim or assumption before implementing it in your data set Also, hypothesis testing is the only valid method to prove that something “is or is not” Other benefits

4 Hypothesis testing is one of the most important processes for measuring the validity and reliability of outcomes in any systematic investigation

5 It helps to provide links to the underlying theory and specific research questions 1.2.2 Understanding Hypothesis Testing

The goal of hypothesis testing is to determine the likelihood that a population parameter, such as the mean, is likely to be true In this section, we describe the four steps of hypothesis testing:

Step 1: State the hypotheses

Step 2: Set the criteria for a decision

Step 3: Compute the test statistic

Step 4: Make a decision

Step 1: State the hypotheses We begin by assuming that the hypothesis or claim we are testing is true This is stated in the null hypothesis The basis of the decision is to determine whether this assumption is likely to be true

The null hypothesis (Ho), stated as the null, is a statement about a population parameter, such as the population mean, that is assumed to be true We will test whether the value stated in the null hypothesis is likely to be true

An alternative hypothesis (H,) is a statement that directly contradicts a null hypothesis by stating that the actual value of a population parameter is less than, greater than, or not equal to the value stated in the null hypothesis

The alternative hypothesis states what we think is wrong about the null hypothesis, which

is needed for Step 2

Step 2: Set the criteria for a decision To set the criteria for a decision, we state the level of significance for a test

Level of significance, or significance level, refers to a criterion of judgment upon which a decision is made regarding the value stated in a null hypothesis The criterion is based on the probability of obtaining a statistic measured in a sample if the value stated in the null hypothesis were true In behavioral science, the criterion or level of significance is typically set at 5% When the probability of obtaining a sample mean is less than 5% if the null hypothesis were true, then we reject the value stated in the null hypothesis Step 3: Compute the test statistic The test statistic is a mathematical formula that allows researchers to determine the likelihood of obtaining sample outcomes if the null hypothesis were true The value of the test statistic is used to make a decision regarding the null hypothesis (discussed in Step 4)

Step 4: Make a decision We use the value of the test statistic to make a decision about the null hypothesis The probability of obtaining a sample mean, given that the value stated in the null hypothesis is true, is stated by the P value

The p value is a probability: It varies between O and 1 and can never be negative In Step

2, we stated the criterion or probability of obtaining a sample mean at which point we will decide to reject the value stated in the null hypothesis, which is typically set at 5% in behavioral research To make a decision, we compare the p value to the criterion we set

in Step 2

A P-value is the probability of obtaining a sample outcome, given that the value stated in the null hypothesis is true The p value for obtaining a sample outcome is compared to the level of significance

In sum, there are two decisions a researcher can make:

1 When the p value is less than 5% (p < 05), we reject the null hypothesis

2 When the p value is greater than 5% (p > 05), we retain the null hypothesis

They may perform a hypothesis test using the following hypotheses:

¢ Null Hypothesis (Ho): wafter = before (the mean sales is the same before and after spending more on advertising)

- Alternative Hypothesis (4:): uafter > tbefore (the mean sales increased after spending more on advertising)

If the p-value of the test is less than some significance level (e.g a = 05), then the company can reject the null hypothesis and conclude that increased digital advertising

leads to increased sales

2) In Manufacturing

Hypothesis tests are also used often in manufacturing plants to determine if some new process, technique, method, etc causes a change in the number of defective products produced

For example, suppose a certain manufacturing plant wants to test whether or not some new method changes the number of defective widgets produced per month To test this, they may measure the mean number of defective widgets produced before and after using

the new method for one month

They can then perform a hypothesis test using the following hypotheses:

¢ Null Hypothesis (Ho):(the mean number of defective widgets is the same before and after using the new method)

« Alternative Hypothesis (H;): (the mean number of defective widgets produced is different before and after using the new method)

If the p-value of the test is less than some significance level (e.g a = 05), then the plant can reject the null hypothesis and conclude that the new method leads to a change in the number of defective widgets produced per month

3) In Real Estate

Take another example of a claim that the housing price depends upon the average income

of people staying in the locality This is the claim which is required to be proved or otherwise The alternate hypothesis will be formulated first as the statement that “housing price depends upon the average income of people staying in the locality” Hence, the null hypothesis will be formulated as the statement that housing price does NOT depend upon the average income of people staying in the locality

¢ Null Hypothesis (Ho):The housing price does not depend upon the average income of people staying in the locality

¢ Alternative Hypothesis (H,):The housing price depends upon the average income

of people staying in the locality

Hypothesis tester in Real estate’s industry can take advantage of this to focus on approaching customers in more aspects such as public opinions, taste, Feng Shei, rather than only looking at investors’ income

4) Biology

Hypothesis tests are often used in biology to determine whether some new treatment,

fertilizer, pesticide, chemical, etc causes increased growth, stamina, immunity, etc in

plants or animals

For example, suppose a biologist believes that a certain fertilizer will cause plants to grow more during a one-month period than they normally do, which is currently

15 inches To test this, she applies the fertilizer to each of the plants in her laboratory for one month

She then performs a hypothesis test using the following hypotheses:

¢ Null Hypothesis (Ho): = 15 inches (the fertilizer will have no effect on the mean plant growth)

¢ Alternative Hypothesis (H,): 1 > 15 inches (the fertilizer will cause mean plant growth to increase)

If the p-value of the test is less than some significance level (e.g a = 05), then she can reject the null hypothesis and conclude that the fertilizer leads to increased plant growth 5) Courtroom trial

A statistical test procedure is comparable to a criminal trial; a defendant is considered not guilty as long as his or her guilt is not proven The prosecutor tries to prove the guilt of the defendant Only when there is enough evidence for the prosecution is the defendant convicted

¢ Null Hypothesis (4): the defendant is not guilty

¢ Alternative Hypothesis (H,): the defendant is guilty

The hypothesis of innocence is rejected only when an error is very unlikely, because one doesn't want to convict an innocent defendant Such an error is called error the first kind

(.e., the conviction of an innocent person), and the occurrence of this error is controlled

to be rare As a consequence of this asymmetric behavior, an error of the second kind (acquitting a person who committed the crime), is more common

2 ARTICLE AND SOURCES ANALYSIS

2.1 ARTICLE SUMMARY

STRESS MANAGEMENT THROUGH REGULATION OF BLOOD PRESSURE AMONG COLLEGE STUDENTS

Anurag Joshia,b,*, Ravi Kirana, Harish Kumar Singlac and Ash Narayan Saha School of

Humanities and Social Sciences, Thapar University, Patiala, Punjab, India bElectrical

Engineering, Thapar Polytechnic College, Patiala, Punjab, India cNational Institute of

Construction Management and Research, Indore, India

The case

Stress is a whole-body illness, a mental instability affecting the way a person sleeps, works and more generally, lives When people stressed out, they malfunction Many industrial researchers have indicated that two dominant of the many reasons contributing

to stress is job- and academic-related; however, there seems to be a lack of study investigating the first aid (or instant cure) to reduce stress

This paper introduces a concept of Deep Breathing Technique (DBT) and its applications

as one of the means towards stress management through regulation of blood pressure among Indian College Engineering students in comparison to the Ordinary Breathing Technique (OBT)

The purpose

The aim of this article is to find out whether deep breathing techniques are able to control blood pressure, and in turn, the level of stress After identifying the influence of breathing technique, the article made recommendations that relied on the results

The method

For the target of the research, a total of 123 students from Engineering colleges from Punjab (India) were selected Sample students are filtered and selected via an initial

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screening (a questionnaire on academic stress) and the ones reported high mental stress during the interview were chosen for the main drills The total data set was divided into two groups named as control group and experimental group In the control group, the first readings were recorded as “before the drill readings” for Systolic Blood Pressure (SBP) and Diastolic Blood Pressure (DBP) The second reading was recorded as “after Ordinary Breathing Technique (OBT)”

Student's tests are used for the purpose of hypothesis testing The applied method of decision rule is P-Value for a two-tailed t test (paired two samples for means)

1) Define P-value method:

P-value is a statistical metric that represents the probability of an extreme result occurring This result is at least as extreme as an observed result in a statistical hypothesis test by random chance, assuming the null hypothesis is correct

The null and alternative hypotheses offer competing answers to the research question For example, when the research question asks, “Does the independent variable affect the dependent variable?”:

« The null hypothesis (Ho) answers “No, there’s no effect in the population.”

« The alternative hypothesis (H2) answers “Yes, there is an effect in the population.” Also, there is another term called Significance — In hypothesis testing, significance refers

to when a result is very unlikely to have occurred if the null hypothesis is correct

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2) How to calculate P-value:

The P-Value is calculated using the sampling distribution of

the test statistic under the null hypothesis, the sample data, and RIGHT

the type of test being done (one-tailed or two-tailed test) If Ha

is “>” or “<”, use the One-tailed (left-tailed or right-tailed) test #1 ÄÌLx dang

(QESECT

and if Ha is “#”, use the Two-tailed test Formulas for P-value

in different situations:

« H,1s “< > left-tailed test: P-value = P(Z < Tstats)

e H,is “>” > right-tailed test: P-value = P(Z > Tstats)

« Hạis “£” > two-tailed test: P-value = P(Z < Tsas) +

P(Z > Tsuu)

P: Probability of an event

where: | Tstats = test statistic = - Tk (*) _ oa: ECB

(*): sample mean subtracts hypothesized mean, all together divided by the standard error (SE) of the mean

And to evaluate P-value then come to conclusion: if p-value < a (0.05) > reject Ho and vice versa if p-value > o (0.05) > do not reject Ho

3) Steps to by hand test the hypothesis with P-value method:

e Step 1: State null and alternative hypothesis

e Step 2: Calculate test statistics using the following formula

e Step 3: Use a t-test and its formula (Tsag)

e Step 4: Use a t-distribution table to find the associated P-value

e Step 5: Formulate the Decision Rule using P-value

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hypothesis

The application of this method in context

Step 6: Reach the conclusion regarding whether to accept the null or alternative

Although the calculation prepared in the article was ran by computer, by using the mentioned formulas and following the criterion procedure discussed above, an analysis of the whole data pack can be manually executed The initiatory step is to exhibit pair(s) of hypothesis The entire data set is arranged and the hypothesis to be tested are shown in Table 1, using students ¢ test at 1% level of significance (a = 0.001)

Table 1 Hypotheses to be tested

Table | Hypotheses to be tested

S No Null Hypothesis Ho; Ho2 Ho3 Hos Alternative Hypothesis and Explanation H; H2 H3 He

1 There is no effect of Ordinary Breathing Technique on There is a significant effect of OBT on DBP Diastolic Blood Pressure (Control Group) The readings are taken before and after the event i.e OBT

Two tail f test is used to test the above hypothesis at 1% significance level

2 There is no effect of Ordinary Breathing Technique on There is a significant effect of OBT on SBP

Systolic Blood Pressure (Control Group) The readings are taken before and after the event i.e OBT

Two tail f test is used to test the above hypothesis at 1% significance level

3 There is no effect of Deep Breathing Technique on Diastolic There is a positive effect of DBT on DBP

Blood Pressure (Experimental Group) The readings are taken before and after the event i.e OBT

One tail ¢ test is used to test the above hypothesis at 1%

significance level

4 There is no effect of Deep Breathing Technique on Systolic There is a positive effect of DBT on SBP

Blood Pressure (Experimental Group) The readings are taken before and after the event i.e OBT

One tail f test is used to test the above hypothesis at 1% significance level

Following the first step of setting the hypothesis in the above section is the second stage where all the data is analyzed using descriptive statistics Using the ¢ test formula above (Tstas), the mean of the differences as well as standard deviation can be calculated and illustrated in Table 2 As listed in Table 2 (for control group), that the average DBP is

87.75 before the OBT drill and 87.27 after the OBT drill, with a variance of 7.54 and 9.34

respectively Based on the ¢ test, it is calculated that the P-value (= 0.089 and 0.274) > a

= 0.01, and [Tsa| < teauca (1.757 < 2.763 and 1.114 < 2.763) Therefore, it is concluded

that both Ho: and Ho2 are not rejected at 1% level of significance

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Table 2 t test (Control Group)

evident, the P-value is less that o at 0.01 (P-value = 0.000 < o = 0.01); also, [Tstats] > teritical

(13.722 > 2.467 and 10.459 > 2.467) And the delivered conclusion is “Ho3 is rejected”, and we can say that there is a significant positive effect of DBT on DBP

Table 3 t test (Experimental Group)

Table 3

t test (Experimentation Group)

t-Test: Paired Two Sample for Means DIASTOLIC B.P SYSTOLIC B.P

Before After Before After Mean 87.275 79.896 128.827 121.034

On the other hand, the SBP was 128.82 before the DBT drill and 121.03 after the drill

again indicating a significant positive effect as the desired level is 120 References including the P-value being 0.000 (lower than « at 0.01) in the ¢ test confirms this (Ho4 is rejected)

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Table 4 shows the status of hypothesis for control and experimental groups after testing and statistical analysis

Table 4 Hypothesis status

Table 4 Hypothesis status

S No Null Hypothesis Status of Null Hypothesis

1 There is no effect of Ordinary Breathing Technique on Diastolic Hp; Accepted

Blood Pressure (Control Group)

2: There is no effect of Ordinary Breathing Technique on Systolic Ho2 Accepted

Blood Pressure (Control Group)

3 There is no effect of Deep Breathing Technique on Diastolic Blood Ho3 Rejected

Pressure (Experimental Group)

4 There is no effect of Deep Breathing Technique on Systolic Blood Hos Rejected

Pressure (Experimental Group)

The conclusion

Based on the above analysis it can be concluded that Deep Breathing Technique has a significant positive effect on the Systolic Blood Pressure (SBP) as well on Diastolic Blood Pressure (DBP) According to the above findings, it is proposed that techniques like Deep Breathing can be made compulsory in the educational institutes to contribute to

a healthier student life The time spent in college also prepares students to meet uncertainties, which they are going to face in the future, especially in their career Therefore, it is vital that students not only grow academically, but remain in a healthy state of mind

Overall, P-value is so far a ‘convenient’ way serving researching on macro-scale and it is believed to have proven itself to be practically useful in many fields Prominent researchers believe that the p-value still fills an important function in statistical analysis

of such data, even though they emphasize the importance of confidence intervals and remain skeptical of an absolute rule of 0.05 A note-worthy point to be stated here is that specifically in the field of Clinical and epidemiological, researchers have been highly critical of the use of hypothesis testing and p-values in biological and medical studying It

is frequently brought up that reporting of descriptive statistics and effect measurements with confidence intervals are essential and required in addition to the P-value, including

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in the recommendations for use of statistics in the Journal of the Norwegian Medical

of economics Decisions can be made by thinking as if a hypothesis is being tested, even though the manager is not aware of it Learning the formal details of hypothesis testing will help managers make better decisions and better understand the decisions made by

others

Referring to P-values, it can be judged whether a value is greater or less than a previously specified limit This allows a rapid decision as to whether a value is statistically significant or not It provides a measure of the statistical plausibility of a result With a defined level of significance, p-values allow a decision about the rejection or

maintenance of null hypotheses made by managers P-value hypothesis testing offers a direct way to compare the relative confidence that the managers can have when identifying different types of factors that affect the company's performance; therefore, making optimal decisions to boost employees productivity

2.3 OTHER RESEARCHES AND TECHNIQUES

2.3.1 RELIABILITY ANALYSIS ON THE LIFE OF THE LIGHT BULBS Isadora V Sousa and R Radharamanan

The case

The performance of products over their lifetime is a concern for both manufacturers and consumers This article analyzes the reliability of light bulbs as the straightforward

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question is, “What is the life of a light bulb?” The problem in Lighting Systems Corporation was selected as a case study for a graduate course in reliability engineering The purpose

This information on the reliability of light bulbs is extremely valuable for a manufacturer

in developing marketing material and establishing a warranty policy, as well as a consumer, would like to ensure that the light bulb will last a specified period and thus receive a good value for his purchase

The method

With a new production process in Lighting Systems Corporation, the obvious solution to establishing the life of the light bulbs is hypothesis testing with one-tailed and two-tailed tests

Step 1: Set up hypotheses and select the level of significance a

Ho: Null hypothesis (no change, no difference)

Ha: Research hypothesis (investigator's belief), « =?

Step 2: Calculate test statistics using the following formula: