7.1 The Central Limit Theorem for Sample Means Averages

in this lesson we are going to talk

about what's known as the central limit

theorem for sample means now there are

different central limit theorems but

we're going to only focus on the central

limit theorem for sample means so before

we get into the actual theorem i want to

give you some background understanding

we need to talk about what's known as a

sampling distribution

so

i want you to imagine

that i find

a sample i pick a random sample

i'll call it sample one

okay and then i find the mean of that

sample so i'll label that x bar and i'll

put a little one down here to remind you

that it's coming from sample one

then let's say i take another sample

we'll call it sample

two and so i find the mean of that

sample

and i'll call that x bar two so that we

know it comes from sample two

then i take another sample

we'll call it sample three so i'll

find the mean of that sample and i'll

call it x bar three

now if i continue to do this

a bunch of times

and all my samples

have size n

okay when i take all of those means out

so i take this mean

i take this mean i take this mean i take

all the means of all of the samples that

i find

and i create a new distribution where my

variable x

is all of those means

when i put all those means together and

i

assign it to this variable x we have

what's known as the sampling

distribution

so now that we understand what a

sampling distribution is

that's

a new variable that we define

as the means of each of these random

samples that we're taking of the same

size n

now we can get into now that we

understand what that means

we can get into what the central limit

theorem is

now the central limit theorem for a

sample

if you're talking about specifically the

mean if you draw repeated samples over

and over and over again of the same size

then

the sampling distribution

is going to be

approximately

normal

and as the sample size increases

we'll call it n generically as the

sample size n increases

the closer

the sampling distribution

gets

to a normal distribution

okay so what this means is if i had all

these samples that i talked about

previously sample one sample two sample

three if originally i took sample sizes

of say 20.

if i repeat that process

and i go ahead and take sample sizes of

30 or 40

and then i repeat it again and then i

start taking sample sizes that are 50.

as those sample sizes that i take are

bigger and bigger and bigger

this sampling distribution gets closer

and closer and closer to that normal

shape where you have the one mound

and then it's symmetric

so let's take a look at our new notation

okay so if we have some random variable

with the distribution

we'll call it just x

we'll call it any distribution

and the mean of that distribution is mu

x

and the standard deviation let me see

and the standard deviation of that

distribution is sigma with your little

sub x

then as the sample sizes that you take

from that distribution over and over

again if that size

n

increases

then the random

variable

x bar

be normally distributed

and

it will have this notation here

it's normal and it's going to have the

same mean

as the distribution that we're talking

about

however the standard deviation is going

to change you're going to take whatever

the standard deviation of the

distribution that we're talking about

and you divide it by the square root of

n

that will give you the standard

deviation of the sampling distribution

so that's this here

so this is the standard deviation of the

sampling distribution

and then of course if you want to

standardize it normally before when we

were talking about in chapter 6 we would

say

z is equal to

your data value

minus the mean over your standard

deviation it's basically the same thing

but the notation is changing a little

bit because your data value in this case

your value of x is going to be a mean of

one of those samples

your mu is going to be a me the mean of

your distribution that you're talking

about and then the standard deviation

is going to be the same as the

distribution x but for the sampling

distribution you have to divide it by

the square root of n

so it's almost the same thing with the

few minor tweaks to it

okay so now let's go down and walk

through an example

because this isn't going to start to

click until you actually work through

some examples okay so we have

um let's see an unknown distribution

we'll call it x

and it has a mean of 90. so this is the

mean of our distribution and the

standard deviation is 15.

we're taking samples of size n is 25

and they're drawn randomly from this

population

so if we take samples from this

distribution we know that it's going to

be approximately normal

and that the mean is going to be the

same

but the standard deviation is because

we're talking about a sampling

distribution is going to be 15 divided

by the square root of n which in this

case is 25.

we don't know if the distribution x is

normal or not but we do know when we do

the sampling distribution based on the

central limit theorem that it should be

approximately normal and it gets closer

and closer to being exactly normal

when the sample sizes get bigger and

bigger and bigger

so we can make this approximation now it

says in part a it says what is the

probability that the sample mean

is between 85 and 92

so here are the key words here to let us

know that we're looking

at this distribution we're looking at

the samples not the x's

we're looking at the means of the

samples

okay so previously

we would do this we would say the

probability that x

is between 85 and 92 but this says we're

looking for the probability that the

sample mean is between 85 and 92 so just

by that one word sample that changes the

whole problem and instead of putting x

is between 85 and 92 i'm going to make

it x bar is between 85 and 92.

now because we know this is going to be