# 10.3 Hypothesis testing: two population means and two population

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## Assumptions and conditions: two sample proportions test

When constructing a two sample proportions hypothesis test the assumptions and conditions must be met in order to use the normal model.

• Randomization Condition: The data must be sampled randomly. Is one of the good sampling methodologies discussed in the Sampling and Data chapter being used?
• Independence Assumption: The sample values must be independent of each other. This means that the occurrence of one event has no influence on the next event. Usually, if we know that people or items were selected randomly we can assume that the independence assumption is met.
• 10% Condition: When the sample is drawn without replacement (usually the case), the sample size, n, should be no more than 10% of the population. This must be true for both groups.
• Success/ failure Condition: must be met for both groups. When working with proportions we need to be especially concerned about sample size when the proportion is close to zero or one. To check that the sample size is large enough calculate the successes by multiplying the sample percentages by the sample sizes and calculate failure by multiplying one minus the sample percentages by the sample sizes. If all of these products are larger than ten then the condition is met.
• ${\stackrel{̂}{p}}_{1}{\left(n}_{1}\right)$ >10 and ${\left(1-\stackrel{̂}{p}}_{1}\right){\left(n}_{1}\right)$ >10 and
• ${\stackrel{̂}{p}}_{2}{\left(n}_{2}\right)$ >10 and ${\left(1-\stackrel{̂}{p}}_{2}\right){\left(n}_{2}\right)$ >10
• Independent Groups: The two groups you are working with must be independent of one another. Is there reason to believe that one group would have influence on the other group? A hypothesis test that is comparing a group of respondent’s pre-test and post-test scores would not be independent. Likewise, groups made by splitting married couples into husband and wife groups would not be independent.

Comparing two proportions, like comparing two means, is common. If two estimated proportions are different, it may be due to a difference in the populationsor it may be due to chance. A hypothesis test can help determine if a difference in the estimated proportions $\left({\stackrel{̂}{p}}_{A}-{\stackrel{̂}{p}}_{B}\right)$ reflects a difference in the population proportions.

The difference of two proportions follows an approximate normal distribution. Generally, the null hypothesis states that the two proportions are the same. That is, ${H}_{o}:{p}_{A}={p}_{B}$ . To conduct the test, we use a pooled proportion, ${\stackrel{̂}{p}}_{\mathrm{pooled}}$ .

## The pooled proportion is calculated as follows:

${\stackrel{̂}{p}}_{\mathrm{pooled}}=\frac{{x}_{A}+{x}_{B}}{{n}_{A}+{n}_{B}}$

## The distribution for the differences is:

${\stackrel{̂}{p}}_{A}-{\stackrel{̂}{p}}_{B}~N\left[0,\sqrt{{\stackrel{̂}{p}}_{\mathrm{pooled}}·\left(1-{\stackrel{̂}{p}}_{\mathrm{pooled}}\right)·\left(\frac{1}{{n}_{A}}+\frac{1}{{n}_{B}}\right)}\right]$

## The test statistic (z-score) is:

$z=\frac{\left({\stackrel{̂}{p}}_{A}-{\stackrel{̂}{p}}_{A}\right)-\left({p}_{A}-{p}_{B}\right)}{\sqrt{{\stackrel{̂}{p}}_{\mathrm{pooled}}·\left(1-{\stackrel{̂}{p}}_{\mathrm{pooled}}\right)·\left(\frac{1}{{n}_{A}}+\frac{1}{{n}_{B}}\right)}}$

## Two population proportions

Two types of medication for hives are being tested to determine if there is a medication A is more effective than medication B. Twenty out of a random sample of 200 adults given medication A still had hives 30 minutes after taking the medication. Twelve out of another random sample of 200 adults given medication B still had hives 30 minutes after taking the medication. Test at a 1% level of significance.

## Determining the solution

This is a test of 2 population proportions.

How do you know?

The problem asks for a difference in proportions.

Let $A$ and $B$ be the subscripts for medication A and medication B. Then ${p}_{A}$ and ${p}_{B}$ are the desired population proportions.

## Random variable:

${\stackrel{̂}{p}}_{A}-{\stackrel{̂}{p}}_{B}=$ difference in the proportions of adult patients who did not react after 30 minutes to medication A and medication B.

${H}_{o}:{p}_{A}={p}_{B}\phantom{\rule{25pt}{0ex}}$ or $\phantom{\rule{25pt}{0ex}}{p}_{A}-{p}_{B}=0$

${H}_{a}:{p}_{A}>{p}_{B}\phantom{\rule{25pt}{0ex}}$ or $\phantom{\rule{25pt}{0ex}}{p}_{A}-{p}_{B}>0$

The words "is more effective" tell you the test is one-tailed.

Distribution for the test: Since this is a test of two binomial population proportions, the distribution is normal: ${p}_{c}={\stackrel{̂}{p}}_{\mathrm{pooled}}$

${\stackrel{̂}{p}}_{\mathrm{pooled}}=\frac{{x}_{A}+{x}_{B}}{{n}_{A}+{n}_{B}}=\frac{20+12}{200+200}=0.08\phantom{\rule{12pt}{0ex}}1-{\stackrel{̂}{p}}_{\mathrm{pooled}}=0.92$

Therefore, $\phantom{\rule{10pt}{0ex}}{\stackrel{̂}{p}}_{A}-{\stackrel{̂}{p}}_{B}~N\left[0,\sqrt{\left(0.08\right)\cdot \left(0.92\right)\cdot \left(\frac{1}{200}+\frac{1}{200}\right)}\right]$

${\stackrel{̂}{p}}_{A}-{\stackrel{̂}{p}}_{B}$ follows an approximate normal distribution.

Calculate the p-value using the normal distribution: p-value = 0.1404.

Estimated proportion for group A: $\phantom{\rule{12pt}{0ex}}{\stackrel{̂}{p}}_{A}=\frac{{x}_{A}}{{n}_{A}}=\frac{20}{200}=0.1$

Estimated proportion for group B: $\phantom{\rule{12pt}{0ex}}{\stackrel{̂}{p}}_{B}=\frac{{x}_{B}}{{n}_{B}}=\frac{12}{200}=0.06$

## Graph:

${\stackrel{̂}{p}}_{A}-{\stackrel{̂}{p}}_{B}=0.1-0.06=0.04$ .

The p-value is above 0.1404.

Compare $\alpha$ and the p-value: $\alpha =0.01$ and the $\text{p-value}=0.1404$ . $\alpha <$ p-value.

Make a decision: Since $\alpha <\text{p-value}$ , do not reject ${H}_{o}$ .

Conclusion: At a 1% level of significance, from the sample data, there is not sufficient evidence to conclude that there is a difference in the proportions of adultpatients who did not react after 30 minutes to medication A and medication B.

what's a psy.d degree
Psychology Doctorate. The main difference between PsyD and PhD Psychology is that PsyD mainly focuses on clinical sessions and PhD focuses on research.
Shrajan
so most of the phd holders become professors teaching what they have learnt
Bridget
Hi, I have a question. What factors influence social facilitation and social inhibition in groups?
it really depends on that person on how the can task in life either they can work better with other people or you do worse working with people.
Lametra
Lametra
Lametra
Do you know of any studies done to show social facilitation and social inhibition
Amritpal
social facilitation is often influenced by the person's perception of the situation and their appraisal of the task. for example, a track athlete who views their competition as a challenge rather than a threat, will tend to run the fastest they have ever ran under a large crowd.
Another condition is if the person has mastered the ability to perform that task. A track athlete who trains everyday, will perform above average under a large crowd, however, someone who does not run, will tend to do worse than their actual ability under a large crowd
social inhibition can be explained through evolutionary biology or attachment personality theory. some people develop an avoidance personality trait, which when they feel under pressure, they tend to leave the group or avoid a gathering due to high lev of stress
Does anyone know how I can transfer my higher national diploma credits in usa?
Oh wow. I never realised there was a chat on this app!
Then welcome dearest
Bridget
Thank you! ❤️
Amritpal
thank you
Ana
Hello Ana, good afternoon from here
hello im new student here.
Rai
hello
Rai
I m also student.
Rai
hi
Kavya
what is easier option to psychology?
Dhanaraj
I'm new too
Deniz
As in the the specialisation or ?
Bridget
good morning
Ana
understanding human nature is what we are going to talk about.
Ana
Philip
“ Every child is left to evaluate his experiences for himself, and to take care of his own personal development outside the classroom. There is no tradition for the acquisition of a true knowledge of the human psyche. The science of human nature thus finds itself today in the position that chemis
Ana
well, I think every individual is different and to a great extent we can't say confidently that we understand the human nature in its entirety
Philip
lack determines what we will become in life?.
Ana
Not 100% but it plays a huge role
Philip
Hi! I wasn't aware there was a chat feature on this app 😊
Ophelia
👍Philip Anyaegbu
Ana
cool
Ana
👍
Ana
yes Ana I'm here
Philip
alternatively we are different, we have basic similar qualities
Bridget
Yes true Bridget.
Philip
hello there
Ciel
Hi
Philip
hi
Ej
hi
KUNDAN
hi
Ana
who we are how we think what we do insight
Ana
@ bridget. yes that's true
Ana
Ana ioanidis? do you speak Greek?
Qwanta
Qwanta no I dont speak Greek my husband does
Ana
are you a psychologist? or your husband?
Qwanta
I need some reliable informations about my studies...
Qwanta
not a psychologist
Ana
Ana
i wanna take psychology for college any advice?
am
so what is the cause why a person experiences a psychological disorder if it not of their own?
it's partly genetics,culture, environment and substance abuse. generally these are the common factors which can lead to most of the disorders
war
waoo
Rai
r u proficcer
Rai
u r psychology subject
Rai
not a professor but a clinical psychologist
war
can i ask you, i keep getting headache, and had a history of hypothyroid ans diagnosed as having mild anxiety disorder. i want to finish. my undergraduate psychology thesis.. but always unable to have good ambition and energy to focus on writing the case and the theories... whats the best remedy
Astaroxche
Ana
297 according to the DSM 5
how many disorders are there
interesting question
Mahmoud
it is😅
Nelly
I'm guessing it's in the tens of thousands, maybe hundreds if you include counter interactions and stuff like it. although I'm sure it's almost impossible to know for sure, unless you're very rich and connected to the right people. but as I said: guessing.
Beenie
There are more than 200 classified forms of mental illness. Some of the more common disorders are: clinical depression, bipolar disorder, dementia, schizophrenia and anxiety disorders. Symptoms may include changes in mood, personality, personal habits and/or social withdrawal. that is what I think
Mahmoud
too difficult to number. diagnosing a disorder is just checking off boxes on a compilation of symptoms that might match any particular condition on the DSM
what is psychology of the guest
I don't understand the question can you elaborate?
Edgar
the study of philosophy gives to the sociologist
why women are viewed as far more emotional than men?
because they actually are ,I guess.
Collins
May be they are biologically milder than man. It does not mean they are not equal with man. Man can also be emotional and can are oppressed with traditional norms. For example, man are not to be cry, in actual man are also emotional being and they cannot have the right to show their sentiment.
shine
sollungal bro sollungal...
Balaji
because of she has hormonal fluctuations than men..
bavi
usually women are more emotional than men and they are multi talented people
lavanya
Both men and women are capable of expressing emotions. But women has the highest percentage of doing that. Because our society is conditioned by nature in such a way that men are expected to suppress their emotions and motivate them through their acts or thoughts, which has it's side effects of....
Santos
denial of any emotion which they feel that useless at that point. Women in the other hand were encouraged/not controlled to suppress their emotions and let them out what they feel about it. I feel that's the wonderful superpower of the women.
Santos
y
Mahmoud
Men's emotion comes mostly with memories triggered by senses. That means, they thoughts have the power to decide whether to let go of emotions or not.
Santos
Q
Mark
because women tend to be more agreeable than men.
Edgar
women at a young age are conditioned to be more in tune with emotion than man should be less
David
The question is "Why women Viewed are as far more emotional than men ?" it's not a question whether women are more emotional than men. This is more an issue about the point of view from the observer, his/her assumption what emotional behavior is or what emotional behavior is.
Mehmet
The anwers are answers more to the question " Are women more emotional than men?"
Mehmet
i think both genders are very emotional but on the other hand males tend to try not to express their feelings maybe they're scared of being judged "men dont cry"
am
that's because,it's actually been proven that they are
victor
I'd say there are two flips. 1. The point of view of an observer; not everybody who doesn't shed tears every now and then is unemotional. It's just that they try to pretend (fear revealing their vulnerable side before others) 2. It's an individual case. Many humans, emotional, act strong.
Riya
that's how we have been cultured. Those who talk emotional and cry all the time, are weak is a common perception.
Riya
being a man or a woman can be a certification to follow some certain of behaviours in different cultures. like being emotional for women is generally accepted. it's a cultural certification to allow this behaviour. I think this is the general concept of this matter.
Hassan
also I think the fact of milking babies made women to stay at home more common. and they hade to interact with emotional babies. they learn how to cooperate this situation and it became a general culture for longtime.
Hassan
Mehmet, thank you for trying to get this conversation on track with clarifying the question.
Rebecca
I think the different emotions by women or men that are expressed by each are viewed by society in a different manner. example.... male politicians who yell and get angry aren't viewed as emotional, which in this world is often seen as weak, they are viewed as strong leaders who don't back down.
Rebecca
Women who yell in politics are viewed as shrill, emotional, and weak. I think this discrepancy is due to the majorly patriarchal/male dominated world. This is a way to keep women from leadership in any form. It is oppression of the matriarchy.
Rebecca
Women would not put up with the destruction of their children, their mates, the world. Women are primarily care takers. They have a different brain structure, than men. They have different hormones to help them bond with others, such as their children. That has a lot to do with why we procreate.
Rebecca
The views on why women are viewed as more emotional are more than one reason. We have a different make up of hormones and the different emotions we do express are viewed differently in society. This benefits the domination that men have in this world.
Rebecca
That women are MORE emotional is a made up construct to keep men in power.
Rebecca
I hope the two explanations I give, that basically say the same thing in a little different ways, makes sense. I would like to hear what you have say on all that! PEACE
Rebecca
hi are you on what'sap
no not really a fan of social laziness
Robert
huh
Parmizan
is there any way for a Btech E.C.E. graduate to take on MSC. PSYCHOLOGY
devesh
ya you can just by cracking entrance , if in India
Mansi
Any stream in UG can go for M.Sc (Psychology)
classification of traits and how they are measured
what was Freud's first name?
Sigmund
Dentriodite
sigmund
Margie
Sigismund Schlomo Freud
Sigmund Freud
Talal
sigmund
Ankita
Leon Vygotsky and Sergei Rubenstein please tell me contributions of these personalities in in 4 lines
Uzma
sigmund Freud
Harpreet
Lev Vygotsky was the founder of socio-cultural theory
Jo
So psychology was based off of the Greek gods I that right or no?
psychology is the study. anything -ology is the study of a certain field. Psyche is a mortal woman who becomes divine in Greek mythology. The etymology of the word "psyche" in Greek means "spirit" or "soul"
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