# Introduction

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This module provides an introduction to Chi-Square Distribution as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

## Student learning outcomes

By the end of this chapter, the student should be able to:

• Interpret the chi-square probability distribution as the sample size changes.
• Conduct and interpret chi-square goodness-of-fit hypothesis tests.
• Conduct and interpret chi-square test of independence hypothesis tests.
• Conduct and interpret chi-square homogeneity hypothesis tests.
• Conduct and interpret chi-square single variance hypothesis tests.

## Introduction

Have you ever wondered if lottery numbers were evenly distributed or if some numbers occurred with a greater frequency? How about if the types of movies people preferredwere different across different age groups? What about if a coffee machine was dispensing approximately the same amount of coffee each time? You could answer thesequestions by conducting a hypothesis test.

You will now study a new distribution, one that is used to determine the answers to the above examples. This distribution is called the Chi-square distribution.

In this chapter, you will learn the three major applications of the Chi-square distribution:

• The goodness-of-fit test, which determines if data fit a particular distribution, such as with the lottery example
• The test of independence, which determines if events are independent, such as with the movie example
• The test of a single variance, which tests variability, such as with the coffee example

Though the Chi-square calculations depend on calculators or computers for most of the calculations, there is a table available (see the Table of Contents 15. Tables ). TI-83+ and TI-84 calculator instructions are included in the text.

## Optional collaborative classroom activity

Look in the sports section of a newspaper or on the Internet for some sports data (baseball averages, basketball scores, golf tournament scores, football odds, swimmingtimes, etc.). Plot a histogram and a boxplot using your data. See if you can determine a probability distribution that your data fits. Have a discussion with the class about yourchoice.

#### Questions & Answers

What you see is what you believe? or what you believe is what you see?
what you believe is what you see.
Shalomkegeta
what you think what you believe leads to things you see.
sushant
It is all in one's conscious mind.
Dark
everyone has their own interpretation of what they see and everyone can see differently.
Helena
I think they endlessly interact based on one's mental schemas. That's how it's easy to spiral up or down, like in depression.
Emily
Dark Dream tell me why you were asking about my age ..By the way your are my little brother Dreamo...
Dark Dream you are my little brother .....By the way tell Me why you were asking about my age?
Just to distance my self and my respect, maybe? And actually I'm a sister.
Dark
And maybe you could tell your age now?
Dark
I' m Around 🐒 ... I forgot..can u guess
Milky
I'd be careful. These messages are public for all to see
Miky
I think around 20s or 30s?
Dark
Yeah. But you didn't put your name. Identity still secured.
Dark
Excuse me for imposing, but why are you so amid to know their age?
Miky
No, not really. Never mind that. Just skip the question. It's all fine when you tell me you're older. That's fine, really.
Dark
I Haven't mind it my little broo
Milky
Miky !!! It's seems like my name..Isn't it?
Milky
Hm?
Dark
good night guys!!!! sweet dreams .....will talk tomorrow
Milky
And dark dream May u have ' Light Dreams'
Milky
sorry Dark ....Got seriously ? was just joking my little bro
Milky
Never mind that.
Dark
It's nothing.
Dark
you said Dark you are reading books I really appreciate that .....Reading makes a man perfect said Bacon
Milky
Just working with some stuffs but thanks. I'm a sister, actually.
Dark
OOhh sorry!!! my little sister ,,,,,Sending best wishes for whatever you are doing ....gud night
Milky
you R most welcome my Bro!!! Dark dream first tell me your age the i will tell whether I m ur Elder or young brother....
I'm 15.
Dark
hypnosis is an artificially induced state in which suppressed memories and feelings are revealed....
Is There Any One ? Why there is calm and silence😷😝
Milky
Because there are tensions and noises.
Ira
Absences build the world- the absence of light creates darkness like the absence of calmness creates tension.
Ira
it creates endings and beginnings
Sarah
God save us from tensions and noises ira😝
Milky
Dunno, if that's what you believe in. Sorry, my name is not Ira, it's my sister's. Just can't log in with my account.
Ira
okay,,,Not big deal my sister Biscocho😃😃
Milky
okay,,,Not big deal my sister Biscocho😃😃
Milky
No, of course.
Ira
have u heard of the breaking news my brothers and sisters?
Milky
Can I ask? Why I can't log in with my real account? I just changed my birthday there because I thought that was the problem. Whenever I'm sending a message with that account in here it just says like there has a problem then try it again later but nothing good really happen.
Ira
You are a religious person, aren't you?
Ira
I Think first of all u should uninstall the app and the installed the app and sign it with your Google account or make another account.....
Milky
hahahha Biscocho my sister how you made that opinion about me ,,,First tell me about that ,,,,a psychological approach...
Milky
Iras' sister why I made that opinion about me.....
Milky
Milky
A psychological approach? Maybe? Like earlier you told me or us the God save us thing. Maybe not really a psychological approach but more likely to be an impression.
Ira
Sorry for the question. HIHI
Ira
But still, I can't logged in with my account. It still recognizes the past setting of my account but don't really mind it. HIHI I'm just a bothersome.
Ira
Maybe this one will do.
Dark
It's Okay.....there is nothing like bothersome in it........
Milky
we are here to help each other ..........
Milky
can't get you Light Dream!!!!!
Milky
Yeah, thanks. Can you tell me your age? It's fine not the specific age like middle age or teenager or whatsoever age.
Dark
You are Most welcome my Bro!!!Dark Dream My brother first u tell m about your age then I will tell u whether I am your elder or younger brother...
Milky
Like I'm really young and just fond of reading books or textbooks that does not fit on my age and I'd crossed my path with this one and got to talk to someone I don't really know and I just feel bad like I feel I'm kind a rude and I'm answering or questioning you rudely or maybe just an overthinking
Dark
Debate question: the social engineering is the dark side of the psychology? what do you guys think
What was social gathering like in ancient times?
no
abdul
how do i know if i have a personality disorder? and no, i dont have insurance
i can't see ur whole line
tap 'more" to see the whole paragraph
Milky
how to join conversation
A square is given at the end ,,Tap the Square and start Conversation....
Milky
what is the difference between place theory and time theory
you
abdul
place theory is a theory of hearing that sates that our perception of sound depends upon from where each component frequency produces vibration...
Milky
place theory is when sound wave of different frequency stimulate different area of the cochlea (inner)
Zainab
what is saturation?
unable To hold or contain any more e.g The syrup is saturated you can't add further sugar to it...
Milky
what is Conscious ,Subconscious and Unconscious?
Milky
conscious is the state of being aware of mental activities while unconscious is d opposite
Zainab
what is Biopsychology
?
tom
yes
Shalomkegeta
hey
N_Young
I'm so sorry I'm late I've been really busy. okay let's get started what are you all having trouble with in this class
Reyna
Study of the body, how it works and how why...?
tom
as far I know , psychobiology or biopsychology is the study of the behavior using biological basis... slightly different from neuroscience... remark: I'm not a professional, its just my opinion according my studies
Josue
Hey, i just got transferred to psychology and i am a bit lost because i resumed late. i need help on how to catch up and also how i do i read and understand and also pass my exam. Exam is in 45 days.
Venturah
hey. can you tell more about psychology studies? and math in psychology?
Gabriel
hi
Mercy
hello
Lieutenant
Hello Mercy
Syed
is there anyone?
Lieutenant
hi
James
hi!! How are?
Milky
Fine
Syed
anyone?
Lieutenant
anyonnne
Lieutenant
lots of good people here are...
Milky
yes
Syed
its science
abdul
it's Science) what this suppose to mean?
Milky
its meaning also all covered
abdul
okay.......
Milky
IS THIS A CHAT ROOM FOR ONLINE CONSULTING
For everything relating to psychology I think. Is it?
Daniela
Related*
Daniela
yes
abdul
i need finance
abdul
Abdul Latheff people are here to learn sth from each other ,,,Its like learning Institution Okay please Behave like an educated man...
Milky
I gave my opinion yesterday... I don't know if it helped
Josue
my definition about biopsychology...
Josue
what is mind?
Human body is an organisation which is owned by you. Mind is CEO of that organisation which is responsible for running all the affairs as per your desire and directive.
SS
Mind is how you control it
SS
OK,Thanks for sharing your ideas...
Milky
Mind is divided into three elements what are these three?
Milky
Conscious, unconscious and sub conscious
tom
will define it ,Tom?
Milky
@ Tom Thornton, I agree with conscious and subconscious as elements of mind, but unconscious is a state of brain as an organ, where mind has no power over it, while conscious and subconscious are both controlled by it. excuse me if I am wrong.
Bora
Conscious is the everyday stuff like brushing your teeth, getting dressed, knowing how to do things without needing to be shown
tom
Unconscious is the stuff you don't know yet, the unknown side of the brain
tom
Subconscious is the stuff you know but don't know you know
tom
Hey I'm not knocking it, this is just my input wrong or right, it's just as I understand it?
tom
@ bora brigjaj how very interesting approach. Where or how did you get it? Any book or article you can recommend to dig more into it?
Daniela
okay !!! thanks Tom for sharing your ideas with your little brother....
Milky
according with the neuroscience the unconscious is the part which ocurr automatically (like brushing our teeth -we don't pay attention to every little movement, we just do it-) its like an automatic pilot. on the other side the conscious is when we are aware about the facts (e.g., we are going to..
Josue
thanks Josue.....
Milky
haha Subconscious remains .....
Milky
...learn how to drive, we are paying attention to every detail..the first times is difficult, but when we practice the unconscious take charge and afterwards we do it automatically). there are another definition about this topics (the definition given by Sigmund Freud , I think there is no longer...
Josue
...used, don't know for sure) in these days there are a lot of controversy in the psychology about this... I prefer the neuroscience definition... what do you guys think (I have a lot of literature about that..if you guys want). give me your feedback please ;)
Josue
jaja you're right milky... it remains
Josue
sending lot of prayers to my Elder Brother who shares their invaluable ideas with their little brothers ,so that can understand and relate examples from every day life...
Milky
It will be honour for U Josue to share your ideas with persons who don't really know and because of u and persons like u in this group they will learn sth.. Great achievement Brothers😃😎
Milky
in the new age studies the term unconscious is used (no longer subconscious) , on older ages (Freud, there are three definitions)
Josue
Just a final words: I think milky with this question have opened a door...in the new age studies the mind no longer exists (we all know the history of this duality opened by René Descartes), nowadays the neuroscience is giving a lot of scientific information
Josue
the mind is just an illusion (there is no ghost or something related, they stated there is no mind ..just processes on the brain. there is no duality) ...then what happen with psychology?
Josue
Hhmmm .....Josue have u done any course in psychology ,,,I mean whats your educational background bro?
Milky
if there is no mind..just the brain and inner workings , the psychology is redundant? ... perhaps the answer is no, the psychology and the neuroscience can coexist..even be complementary fields, there is a lot of reasons but I don't want to bore everyone... but this is a exceptional question...
Josue
okay!!! got the idea....Bro Josue
Milky
I am studying psychology at the university, but a lot of my knowledge I am doing it by my own research... my principal background is the body language.. afterwards I began to study the branches of the body language... and here I am..
Josue
that's Good and we are lucky to have a Psychologist around us😃😃
Milky
het
gloria
wat is psychology
gloria
psychology is relationship between mental processed and it's effects on behaviour...
Milky
* Processes
Milky
what are the methods of psychology
gloria
Gloria there are a lot of methods here, example: case studies, interviews, observational study...and the list keep going on...I am not really sure if this was the answer you were looking for..
Josue
there are experimental methods i.e lab experiments and Non Experimental methods as Josue Defined earlier...
Milky
Gloria!!!! Are u getting?
Milky
good Night ,,,,,Guys ,,, sweet dreams😴😴
Milky
same to u
gloria
sweet REM-NREM sleep jajaja
Josue
sorry Rosue but I cant Get the meaning of JaJa ...OOhh can u speak Spanish also?
Milky
Si exactamente
Josue
Bueno ,,,Josue
Milky
guys me I just understand English plz wants is this language
gloria
Gloria you from Kenya ?
samuel
no Zambia kitwe
gloria
n
abdul
wat do u mean I understand English and bemba
gloria
Gloria inbox me
samuel
okay Gloria will talk in English Onward in conversation....
Milky
me I want to know more about English
aimie
what is understanding self mean
aimie
to gain something personal to fulfill personal desire
Sarah
what is betrayal then
Sarah
how do you distinguish between being mean and being honest
Sarah
Hi my friends.. I think this is a psychology forum? or can we ask about another topics? I also study mathematics, pharmaceutical chemistry, hypnosis, persuasion, social engineering... we surely have a great time..
Josue
being honest is totally different from being mean. being honest may express being mean. but being mean may use the methods of ditching, flattering, saying that m being honest and many others.
Deepak
ohk...
pooja
Pooja how are you?
James
hi.
Lima
hi
Jayson
Am OK and you
Lima
hi
Roushan
hii
anyone can tell about hypnosis
Roushan
I have played with a few parts of sleep hypnosis. It's very complex until you get the hang of it.
Sarah
can you tell about it
Roushan
@ Gloria they are naturalistic observation case study survey correlational study
Zainab
what is the psychology-specific word for a disorder that has one or more additional disorders attached to it? for example: anorexia nervosa is typically accompanied with anxiety, depression, personality disorders, etc. I know their is a specific word that describes the multitude of disorders.
Comorbidity? ***en.m.wikipedia.org/wiki/Comorbidity
Angel
nooo 😭 it's close though. thanks :) I remember hearing it in an article last year and it's been killing me since
Eduardo
actually, it is the exact word now that i saw the exact meaning on google 😁😁😁 thank you so much
Eduardo
I would say it is "syndrome" and it is defined as a number of medical signs and symptoms that are correlated to each other. I know that depression is an illness rather that has many signs and symptoms.
Bora
Some disorders are comorbid with certain Conditions... Not necessarily a syndrome, just comorbidity
George
comorbid condition
Kayla
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