Two-Tailed Test for Hypothesis Testing in Statistics, Research and Lean Six Sigma

Hello everyone,
In this blog, we'll study a two-tailed test in the context of hypothesis testing. So far with respect to hypothesis testing, we have studied,

• The concept of hypothesis testing, 
• The test statistic, level of significance and level of confidence, 
• Concept of Type 1 and Type 2 errors, and 
• Risk of rejecting a null hypothesis in case it is true

Testing a Hypothesis

For testing a hypothesis we have three different situations

Case 1: H0: μ=μ0 and Ha: μ not equal μ0

When the null hypothesis stated as the population mean equals to the hypothetical population mean and alternative hypothesis stated as the population mean is not equal to the hypothetical population mean. It implies the population mean is either lower than or higher than the hypothetical population mean

Case2: H0: μ=μ0 and Ha: μ < μ0

When the null hypothesis stated as the population mean equals to the hypothetical population mean and alternative hypothesis stated as the population mean less than the hypothetical population mean

Case3: H0: μ=μ0 and Ha: μ>μ0

When the null hypothesis stated as the population mean equals to the hypothetical population mean and alternative hypothesis stated as the population mean greater than the hypothetical population mean

Now In case 1, we use two-tailed test to test the null hypothesis and in two other situations we use a one-tailed test such as left-tailed test and right-tailed test for case 2 and 3 respectively

Rejection Regions, Acceptance Region, and Critical Limits

In a two-tailed test, there are two rejections regions also known as critical regions, one on each tail of the curve. For a 5% significance level, the value of alpha (α) is 0.05. it defines the probability of the rejection area for the null hypothesis when it is true. And if we apply the two-tailed test, then it equally splits on both sides of the curve such as (α/2)=0.025

Now the acceptance region is 95% or 0.95, here in this region the null hypothesis will be accepted. The acceptance region is the area under the normal curve between the critical limits defined by Z=+/-1.96 at a 5% significance level. You may name these limits as lower critical limit and the upper critical limit for suitability

Now the area under the left half of the curve up to the lower critical limit Z= -1.96 is 0.475 and the area under the right half of the curve up to the upper critical limit  Z=1.96  is also 0.475 and both taken together equals to 0.95 or 95% area of the curve.

Now to find the area under the normal curve for Z=1.96, you may refer the table “Area under the standard normal distribution”  in any book of business statistics

Here 1.96 is the critical value, and if the value of test statistic like ‘Z’ test comes out less than  or equal to1.96  then the null hypothesis will remain in acceptance region and if the value of test statistic comes out above 1.96 at 5% significance level then the null hypothesis will rest in the rejection region.

Hence mathematically Acceptance Region is defined by;

A: |Z|<=1.96

And Rejection Region is defined by;

R: |Z|>1.96

Now to better understand the concept of the two-tailed test, let's take an example

Example Problem

A sample of 400 male students is found to have a mean height of 67.47 inches. And the mean height of a large population of male students is 67.39 inches and the standard deviation is 1.30 inches. Conclude that at a 5% significance level, whether the sample is taken from the population having a population mean 67.39inches.
Now to conclude that whether the sample is taken from this population at a 5% significance level, we first state the null and alternative hypothesis…..

Solution

So the null and alternative hypothesis for the given problem can be stated as
H0: μ=67.39 and Ha: μ not equal 67.39
Here alternative hypothesis is two-sided hence we use two-tailed test, also it is given that

• The sample mean (xbar)=67.47inches,  
• The hypothetical population mean (μ0)=67.39 inches, and 
• Population standard deviation (Std. Dev.p)=1.30 inches
• And Number of students in sample (n) = 400

Now let's process the z test by replacing the value of  x bar, mu not, sigma p, and N in the z statistic, assuming the population to be normal

Z= (xbar-μ0)/(Std. Dev.p /sqrt(n)

Z= (67.47-67.39)/(1.30/sqrt(400) = 0.080/0.065=1.231

The calculated value of ‘Z’ is 1.231 less than the critical value of Z=1.96 at a 5% significance level under a two-tailed condition hence it will lie in the acceptance region and therefore null hypothesis is accepted.
And we may conclude that the given sample with mean height of 67.47 Inches drawn from the population with a mean height of 67.39 inches and a standard deviation of 1.30 inches at a 5% significance level.

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So this was the concept of a two-tailed test with respect to hypothesis testing.

• If you have any questions or suggestion then please do write your feedback in the comment box below and hit the like button if you liked this blog and share it among your friends and colleagues.
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Thanks for reading, see you in my next blog with a new topic.

Narender Sharma

Prevention of the Community Transmission of COVID-19 using 5S Methodology

Hello everyone,
In this blog, we are going to study, how to prevent the community spread of COVID-19 using 5S. But first, let me remind you the 5S, The 5S are Sort, Systemize, Sanitize, Standardization and Sustain

1. Sorting

Now let's begin with Sort. Sort implies an organization of necessary and unnecessary things for us. So here in the case of COVID-19, we’ll study what are necessary for us and what we must avoid stopping the transmission of COVID-19.

Dos

First talk about the necessary things to do

• Stay at home and work from home
• Follow the social distancing
• Wash hands with soap and water for 20 seconds
• Cover your nose and mouth while sneezing or coughing with tissue paper or your own elbow
• See a doctor if you feel fever, dry cough, and breathing problem, and also isolate and quarantine yourself for 14 days as a preventive measure to stop the transmission of COVID-19
• Filter the information coming to your phone in the form of video, text or graphics
• Share only, if the information is valid and the source is confirmed

Don'ts

Now things we must avoid to do, to prevent the transmission of COVID-19 are

• Don’t be the part of a gathering
• Don’t spit in public
• Don’t touch your nose, mouth, and eyes
• Don’t ignore the symptoms like fever, dry cough, and breathing problem
• Don’t leave home to go outside in case you are sick
• Don’t handshake with anyone, especially if you are experiencing cold and fever
• And Don’t believe in rumors

2. Systemization

Now come to 2nd ‘S’ the Systemization or ‘Set the things in an Order’

• Keep the soap and sanitizer in the washroom at an optimum level to wash the hands frequently for 20 seconds daily.
• Keep the mask or handkerchief at a place where you can get it easily
• Keep the tissue paper in your pocket and throw it in a covered dustbin after use...
• Make the list of people u are meeting daily to track your 14 days history until we overcome this life-threatening challenge of COVID-19
• Stay at a distance of 1-2 meters while gathering at the workplace if not possible to avoid.

3. Sanitization

Now come to 3rd ‘S’ the Sanitization’, to prevent the transmission of COVID-19 the role of sanitization is utmost important and it includes;

• Washing hands with soap and water for 20 seconds or sanitize with an alcohol-based sanitizer
• Sanitize the items at the workplace like table-tops, laptop, paper tray, pen holder, etc.
• Sanitize the toys of the children and avoid kissing them.
• Sanitize the things which are being commonly used or touched frequently by the people at the workplace like meeting rooms, desktops, door handles, water taps, bathrooms, etc.
• Sanitize your mobile phones and avoid sharing it with other people.
• And the most important, sanitize the information to avoid rumors by deleting videos, text, and graphics which are coming from untrusted and unknown sources and which you can’t verify personally

4. Standardization

Now come to 4th  ‘S’ the Standardization,  The standardization ensures the standard practice of  first 3S and visual management and it includes in case of prevention of covid19 transmission

• Place a chart in washroom or workplace to monitor the handwashing frequency in a day
• Place a chart to track the body temperature in the morning.
• Place a chart at the place of workout to track your daily exercise
• Place a chart in the kitchen to track the balanced diet for breakfast, lunch, and dinner to improve immunity
• Place the chairs at a 1-meter distance in the drawing-room, balcony, garden, and on the terrace to practice social distancing at home and workplace
• Place a chart to track the 14 days history that showcases, the place and person you met with
• Visit only WHO website, who.int and the website of the ministry of health and family welfare, mohfw.gov.in as the standard source of information with respect to the COVID19 pandemics.

5. Sustain

Now come to 5th  ‘S’ the sustain, and it means habit formation and it includes;

• Do the daily exercise for 30 minutes like yoga, aerobics, jogging, and dancing.
• Make it a habit to wash hands for 20 seconds with soap and water before every meal like breakfast, lunch, and dinner
• Make it a habit to use the tissue paper or your own elbow while sneezing or coughing
• Practice to avoid touching your mouth, nose, and eyes as much as possible
• Practice safe greetings like Namaste, wave, nod, or a bow
• Practice to talk with the people at a distance of 1 meter 
• Practice avoiding to visit in crowded places.
• Develop creative skills like writing, painting, cooking, stitching, etc. while staying at home.
• Develop the habit of reading the books rather read the long messages in WhatsApp and see the meaningless videos
• Learn from past, stay in present and plan for the future to make yourself ready to face this kind of life-threatening challenge.
• And Practice spreading the love…. not the COVID-19 and hatred

Like Comment and Share

I hope the things will come in control very soon. If you have any questions or suggestion then please do write your feedback in the comment box below. Share your quick reactions and share this blog with your friends and colleagues.

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Thanks for reading!

Narender Sharma

COVID-19 Community Spread: Failure Modes, Potential Effects, Occurrence, Detection and Control

As all of you know we are at the doorstep of a life-threatening challenge. Yes, I am talking about the community spread of CORONA virus or COVID-19 in India, the challenge becomes more critical because we know the cause but still don't know how to eliminate that cause. As of now, no vaccine is ready to kill the coronavirus, research is going on across the world by the medical scientists and doctors.
Many countries across the world are already entered into this stage and trying to control the mortality rate, the average mortality rate across the world is 4.5%.  Hope is there, but it'll take at least one-year and before that, we have to break the transmission of COVID-19 as early as possible so that it couldn't outbreak in India.
I don't know.. what would be the number... if it outbreaks..because till the date 979 confirmed cases of COVID-19 and 25 deaths have been confirmed and what could be the actual number of infected people in 1.4 billion population in that situation...don't know... but definitely it would be an alarming figure.
The only matter of some relief is that 87 cases recovered successfully from this dangerous disease because of extraordinary efforts by the medical and paramedical staff in this tough time, I salute them from my bottom of heart. So we are at the threshold of an outbreak of COVID-19, but before we move on and study the risk of community spread using FMEA we have to understand, how we have arrived at this stage!

The Beginning

At the first stage, there was not a single case in India the only threat was infected cases in affected countries like China, Italy, Iran, etc. etc. At the second stage, there were isolated imported cases in India, these cases couldn't come into detection in thermal scanning and symptomatic testing because the symptoms of COVID-19 appear in 2 to 14 days after exposure. Therefore, imported cases were advised by the authorities for self-isolation in their homes for 14 days but the infection has passed on to isolated people, isolated places, and isolated items from the imported infected cases. And after that we entered into the third stage, the isolated infected cases are being reported from all parts of India with a few death cases. here we are at the doorstep of community spread and therefore I thought to use FMEA to study the risk of community spread and its subsequent effects as a real-life case study.
Now if you want to know what is an FMEA and how to use an FMEA for risk analysis then please click here

Failure Modes and Potential Effects (Severity) of Community Spread of COVID-19

In this case, the failure is community spread and the modes of community spread are imported cases, local infected cases, infected items and isolated infected places in different parts of India and if we talk about the potential effects of the community spread then the effects will be deadly, the infected cases will increase at an exponential rate and so the mortality rate, like in Italy even after that availability of best medical facilities the current mortality rate is about 10% with 8165 death cases out of 80,539 infected cases but here in India the medical facilities are limited against 1.4 billion population and because of that the mortality rate could go higher than 10% and will become out of control and as a subsequent effect. The GDP growth rate dipped to less than 2% with the shattering world economy, so I ranked the severity of the community spread as 10.

Causes/Mechanism (Chances of Occurrence) of Community Spread of COVID-19

Now let's talk about the occurrence of infection in case of community spread of COVID-19. As we know that coronavirus transmits from an infected person to healthy person through water droplets while sneezing coughing and kissing, it also transmits through handshaking and hugging with the infected person and also through an infected item or surface when used or touched by the healthy person, because it stays alive on the surfaces for a few hours to 9 days depending on the surface. Therefore there are very high chances of occurrence of infection in case of community spread which means on every 3 test cases one will be an infected case this implies about 33% population will be infected from COVID-19 which is an alarming figure. Therefore here I rank the occurrence of infection in case of community spread as 9.

Detection Controls (Ability of Detection of Infected Cases) of Community Spread of COVID-19

Now we discuss the detection, right now the current controls for detection of the actual number of infected cases are;

• Thermal scanning and symptomatic testing of the suspected cases.
• Isolation and quarantine of imported cases, 
• Isolation, and quarantine of confirmed cases of COVID-19,
• Tracking the 14 days history of the confirmed cases

But the biggest problem is, the symptoms appear in 2-14 days in an infected case after the exposure with COVID-19….And therefore till the symptoms will appear, an infected case would have transmitted the infection to 3 other healthy persons unknowingly. And this creates the gap between the number of confirmed cases of COVID-19 and the actual number of infected cases of COVID-19. Hence the chances of detection of the actual number of infected cases are very low with current detection controls. And Therefore I ranked the detection of infected cases with current controls as 7, Now the resultant value of RPN, means risk priority number for community spread will be 630.
It implies the risk of community spread is very high in the context of India. Now before the current situation outbreaks and turned into community spread, it's an urgent requirement to break the transmission chain of COVID-19. And for that, there must be a proper action plan including containment actions, corrective actions, and preventive actions.

Containment Actions, Corrective Actions, and Preventive Actions

Right now the govt. announced nationwide lockdown for 21 days and asking people to remain in their homes and follow the social distancing.

• All international and domestic flights, rail and road transportation will remain canceled during the lockdown period. 
• All imported cases are in isolation and quarantined in their homes, close surveillance imposed on them. 
• All confirmed cases are also in isolation and quarantined for medical treatment. Thermal scanning and symptomatic testing are used if a person reports symptoms like fever, dry cough, and breathing problems.

All these actions are considered as the containment actions that possibly hold the number of infected cases for the given period of time. But according to the director-general of WHO,  Dr. Tedros Adhanom Ghebreyesus, “Social distancing and asking people to stay at home are the defensive measures, that will not help us to win…to win we need to attack the virus with aggressive and targeted tactics, tested every suspected case…isolating and caring for every confirmed case, and tracing and quarantining every close contact.”

Now some of the infected cases recovered successfully because of isolation and proper medical care, therefore these cases are considered as a correction. So to increase the number of recovered cases, the first and foremost requirement is to build a large number of hospitals with a good capacity of beds, ventilators, doctors, medical and paramedical staff. The biggest problem is the detection failure of the infected cases and the root cause of this problem is, the symptoms appear in 2-14 days after the exposure.

The Need of the Hour

Therefore the need for the hour is the formation of the task forces at the district level for asymptomatic testing at a large level and provide the medical care to the infected cases to kill the virus aggressively so that infected cases could turn into recovered cases. Like Germany tested 300,000  to 500,000 cases per week and recovered most of the infected cases caring for them in isolation, and control the mortality rate effectively at 0.6%.
The govt. of India along with all state govt. is doing well to control things as much as possible.

• Providing personal protective equipment (PPE) like masks, medical suits, gloves, testing kits to the doctors, medical and paramedical staff. 
• Using the train coaches for medical care for COVID-19 infected cases is the ultimate example of resource utilization. 
• Apart from this sanitizing the cities, villages, malls, cinema halls, railway stations, bus stand, airports, seaports, religious places, apartments, community halls, hotels, marriage halls, markets, and other public gathering places.

Stop Community Spread of COVID-19

The center and all state govt. are continuously providing the advisory to aware the people how to stop the community spread, through print, electronic and social media such as;

• Stay at home during the lockdown and practice social distancing
• wash hands with soap for 20 seconds frequently or use an alcohol-based sanitizer
• Avoid touching your mouth, nose, and eyes,
• Use your elbow to cover the mouth while sneezing and coughing

Apart from this all state govt. are tracing and isolating the imported cases as early as possible before they transmit COVID-19 to others. Providing medical help to the suspected cases in case of fever, dry cough, and breathing problem, and test for COVID-19.

Conclusion

I have a firm belief that following these actions with public support the mortality rate will remain in control and further reduced to 0. And this will turn the ranking of severity from 10 to 1. And also it will reduce the infected cases out of test cases possibly to 5% and it will turn the occurrence ranking from 9 to 7 and then 1. It also improves the detection ability which is the main problem, with asymptomatic testing to a large number of people per week as early as possible. And so the chances of detection of infected cases improve from low to very high and turn the detection ranking from 7 to 2 and then 1.
Hence by early detection of the infected cases and controlling them less than 5% and recovered them with proper medical care and hence control the mortality rate as low as possible and possibly make it zero, we will definitely win the situation of community spread.

Video Tutorial: Process Capability | Process Improvement using Cp and Cpk Analysis

This video includes;

1. Meaning of process capability
2. Control limits and specification limits
3. Relationship of limits, specifications, and distributions
4. Process width and specification width
5. Process capability w.r.t. specification limits (Cp)
6. Process capability w.r.t. target (Cpk)
7. Calculation of Cp and Cpk
8. Analysing different cases of process for varying Cp and Cpk
9. Process improvement

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Video Tutorial: Measures of Central Tendency | Calculation of Median and Mode

This video includes;

1. Calculation of median for individual series
2. Calculation of median for discrete series
3. Calculation of median for continuous series
4. Graphical location of median
5. Calculation of mode for individual series
6. Calculation of mode for discrete series
7. Calculation of mode for continuous series
8. Graphical location of mode
9. Empirical relation between mean, median and mode

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