# 6.2 The central limit theorem for sample means (averages)

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Suppose $X$ is a random variable with a distribution that may be known or unknown (it can be any distribution). Using a subscript that matches the random variable, suppose:

• ${\mu }_{X}$ = the mean of $X$
• ${\sigma }_{X}$ = the standard deviation of $X$
If you draw random samples of size $n$ , then as $n$ increases, the random variable $\overline{X}$ which consists of sample means, tends to be normally distributed and

$\overline{X}$ ~ $N\left({\mu }_{X},\frac{{\sigma }_{X}}{\sqrt{n}}\right)$

The Central Limit Theorem for Sample Means says that if you keep drawing larger and larger samples (like rolling 1, 2, 5, and, finally, 10 dice) and calculating their means the sample means form their own normal distribution (the sampling distribution). The normal distribution has the same mean as the original distribution and a variance that equals the original variance divided by $n$ , the sample size. $n$ is the number of values that are averaged together not the number of times the experiment is done.

To put it more formally, if you draw random samples of size $n$ ,the distribution of the random variable $\overline{X}$ , which consists of sample means, is called the sampling distribution of the mean . The sampling distribution of the mean approaches a normal distribution as $n$ , the sample size, increases.

The random variable $\overline{X}$ has a different z-score associated with it than the random variable $X$ . $\overline{x}$ is the value of $\overline{X}$ in one sample.

$z=\frac{\overline{x}-{\mu }_{X}}{\left(\frac{{\sigma }_{X}}{\sqrt{n}}\right)}$

${\mu }_{X}$ is both the average of $X$ and of $\overline{X}$ .

${\sigma }_{\overline{X}}=\frac{{\sigma }_{X}}{\sqrt{n}}=$ standard deviation of $\overline{X}$ and is called the standard error of the mean.

An unknown distribution has a mean of 90 and a standard deviation of 15. Samples of size $n$ = 25 are drawn randomly from the population.

Find the probability that the sample mean is between 85 and 92.

Let $X$ = one value from the original unknown population. The probability question asks you to find a probability for the sample mean .

Let $\overline{X}=$ the mean of a sample of size 25. Since ${\mu }_{X}=90$ , ${\sigma }_{X}=15$ , and ${n}_{}=25$ ;

then $\overline{X}$ ~ $N\left(90,\frac{15}{\sqrt{25}}\right)$

Find $((P\left(85, \overline{x}), 92\right))\phantom{\rule{20pt}{0ex}}$ Draw a graph.

$((P\left(85, \overline{x}), 92\right))=0.6997$

The probability that the sample mean is between 85 and 92 is 0.6997.

TI-83 or 84: normalcdf (lower value, upper value, mean, standard error of the mean)

The parameter list is abbreviated (lower value, upper value, $\mu$ , $\frac{\sigma }{\sqrt{n}}$ )

normalcdf $\mathrm{\left(85,92,90,}\frac{15}{\sqrt{25}}\mathrm{\right) = 0.6997}$

Find the value that is 2 standard deviations above the expected value (it is 90) of the sample mean.

To find the value that is 2 standard deviations above the expected value 90, use the formula

value = ${\mu }_{X}+\text{(#ofSTDEVs)}\left(\frac{{\sigma }_{X}}{\sqrt{n}}\right)$

value = $90+2\cdot \frac{15}{\sqrt{25}}=96$

So, the value that is 2 standard deviations above the expected value is 96.

The length of time, in hours, it takes an "over 40" group of people to play one soccer match is normally distributed with a mean of 2 hours and a standard deviation of 0.5 hours . A sample of size $n$ = 50 is drawn randomly from the population.

Find the probability that the sample mean is between 1.8 hours and 2.3 hours.

Let $X$ = the time, in hours, it takes to play one soccer match.

The probability question asks you to find a probability for the sample mean time, in hours , it takes to play one soccer match.

Let $\overline{X}$ = the mean time, in hours, it takes to play one soccer match.

If ${\mu }_{X}=$ _________, ${\sigma }_{X}=$ __________, and $n=$ ___________, then $\overline{X}~N\text{(______, ______)}$ by the Central Limit Theorem for Means.

${\mu }_{X}=$ 2 , ${\sigma }_{X}=$ 0.5 , $n=$ 50 , and $X~N\left(2,\frac{0.5}{\sqrt{50}}\right)$

Find $((P\left(1.8, \overline{x}), 2.3\right))$ . $\phantom{\rule{20pt}{0ex}}$ Draw a graph.

$((P\left(1.8, \overline{x}), 2.3\right))=0.9977$

normalcdf $\mathrm{\left(1.8,2.3,2,}\frac{.5}{\sqrt{50}}\mathrm{\right) = 0.9977}$

The probability that the mean time is between 1.8 hours and 2.3 hours is ______.

What is randomization
definition of stimulus
clinical studies are people who evaluate behavior, medical, and surgical intervention
Connie
clinical studies are people who evaluate behavior, medical, and surgical intervention
Connie
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The ability to learn is one of the most outstanding human characteristics. Learning occurs continuously throughout a person’s lifetime. To understand how people learn, it is necessary to understand what happens to the individual during the process. In spite of numerous theories and contrasting views
MUBARAK
Psychologists generally agree there are many characteristics of learning.
MUBARAK
Learning is the process by which one acquires, ingests, and stores or accepts information. The main characteristic of learning that; it is a process of obtaining knowledge to change human behavior through interaction, practice, and experience.
MUBARAK
Our experiences with learned information compose our bodies of knowledge.
MUBARAK
Is there not one universal understanding or relatable emotion whether physical or communicated verbally they could Trigger empathy?
Tory
what is immune system
a complex network of cells that protects the body against infection
Stephen
Sorry. Cells and proteins
Stephen
what is perspectives
someones point of view the way ur brain sees the way situations unfold
Echo
perspective is your view on topics
Trish
Perspective is your opinion on certain situations
Connie
your perspective is your interpretation of what is being said and done around you and how you hear and view them .
Jim
is this in reference to any particular use of the word? Because there are also the 7 major "perspectives" in psychology: psychodynamic, cognitive, behavioral, biological, cross-cultural, evolutionary, & humanistic
Alex
aren't they like schemas of the world, the future and yourself
naina
Perspective is your opinion on things that you feel, think or hear.
hiba
I'm trying to write a paper about video game and violence and suggestions or researches would help with this
rebal
do you also have to write about aggression and how its linked to video games and violence
naina
because for alevel psychology we could have a 16 marker essay for how media influences aggression which basically includes video games and violence on tv shows
naina
yes basically I'm arguing that video games cause violence not necessarily in a direct way but it plays apart. I'm trying to oppose the popular opinion of video games doesn't cause violence
rebal
It's not connected that much actually tbh
Kira
the way we think
according to my textbook rebal
naina
there is a lab study by Craig an Anderson- computer games mortal combat
naina
Matt delisi et al did a correlation study
naina
Lindsay Robertson et al - longitudinal study
naina
Craig Anderson also did a meta analysis of 136 studies
naina
I will check them thanks naina
rebal
Yes, that would be my recommendation.
merits and demerits of observation as a method of studying human behavior in education psychology
what is psychology
psychology: scientific study of behaviour and mental processes.
Ahmed
UMESH
UMESH
energy and thought will give mind spirit and proper exercises , flexibility and making mind readind easier
Ahmed
Edg
scope of educationnalpsycology
USMAN
seriously i will pay someone to do an essay for me my god i need help so much 😪😪😪
dearbhlagh
what do you need help with ?
Keiko
The better question is how much?
Pixeled
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Jacob
hey all I'm Beth I just started psych 101 have 1st test any suggestions I should memorize ?
memorize dememorizing
devesh
how the blinds person his dream
Ahmed
Hi! I started as well
La
what is prototype
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Arnav
Event schema is how you deal with situations is this installation is good you gonna handle a great but if it's badd you have to be strong I handle it the best way you can and stay positive
Connie
why do we adapt to negative events more quickly as compared to the positive ones?
Fareha
The negative ones are easier to adapt because of the people the we surround ourselves with if we surround ourselves with negative people who go to see things negatively but if we surround ourselves with positive people we are going to see things more positive
Connie
oh ok thanks!
Fareha
Connie
what are the remedy for ADHD
methylphenidate
counter conditioning
it is d conditioning were u add an unpleasant stimulus ad a pleasant stimulus to give a good response. eg a girl that hate or fears snake u add her mom to d picture because she loves her mother she gradually tends to like snake
dalusi
can I get an update on the discussion at hand
Segun
I'd like one as well, please
Trish
I third this.
Redacted
you guys could refer to a research by Mary Cover Jones. she did a study on counter conditioning.
Shubhra
i too would like to get an update on the dicussion please
Connie
so basically we are discussing about counter conditioning. if you all know about classical conditioning which was done by pavlov, later a similar thing was done by JB Watson, but on a child. this child was made to learn a phobia.
Shubhra
after his unethical experiment, his student mary cover jones also did an experiment where she proved, if a fear can be learnt, it can also be unlearn. hence, counter conditioning. where a negative stimulus (any fearful object) is followed by a pleasant stimulus (eg food).
Shubhra
After several pairings, the fear is neutralized.
Shubhra
hey, i wanted to know in positive counter conditioning a several trials are done to make the person unlearn their phobias but in aversive (-ve) counter conditioning,after just 1 trial a person learns that behaviour, why the negative behaviour is learnt in just 1 try?
Fareha
Joseph
hi, may I have many MCQ?
hi
Teri
what would you like to know
Connie
We want to know everything
victor
what's MCQ?
Ronah
Multiple Chouce Questiom
Connie
Who knows something about Multiple Personality Disorder ?
victor
Victor what you want to know in it
Brindhu
victor
Teri
@ Victor. Dissociative identity disorder or multiple personality disorder has two or more distinct personality states.they may be disconnected among thoughts, identity, consciousness and memory .this could be happened when to trauma of childhood incident or any other impact in life
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how do you identify these altars ? and how do you easily identify MPD patient ?
victor
what is experimental bias
Teri
experimental bias is when you experiment with something and you like it and tell someone else about it and you tell them ira good even though they don't have the same reacton to it that you did
Connie
define Experiment
Teri
Experiment is trying something new you don't know how it's gonna work out or even if it will work or if it won't so you try it anyway it's sometimes it works and sometimes it doesn't
Connie
what is the difference between CBT and REBT
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it's called bystander effect when people reject you
Jason
hey Jason
Bonsa
There are 2 meanings of surviving danger one is to be stronger than you were when you went in and the 2nd is a to have your wits about you and to keep yourself from getting into danger in the 1st place but if you have to be in danger Use your best common knowledge to get out of it
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how do I overcome my fears of public speaking
Bonsa
There's many theories to that as well many people say the picture the audience in their underwear but that does not always work so what I do is I forget about the people in the audience and pretend like I'm by myself and you gonna find yourself about more comfortable and a lot more at ease
Connie
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