Question

Statistics need help, please
If \(Y_1,Y_2,\cdots ,Y_n\)~N\((\mu, \sigma^2)\), then, \(\dfrac{\overline Y -\mu}{\dfrac{\sigma}{\sqrt n}}=Z\)~N\((0,1)\)

Answer 1

And then, on Sample mean distribution, I have \(Z=\dfrac{\overline Y -\mu}{\sigma}\)~N(0,1)
My question is : where is \(\sqrt n\)?

Answer 2

We will use the second Z to make a chi-square distribution. But it is unclear to me. Please, explain

Answer 3

@Zarkon Please, explain me

Answer 4

@Loser66 , so if you have a normal distribution with mean mu and variance sigma squared, you can transfer your distribution to standard normal but caculating Z score which is (X-mu) over sigma, where X is original distribution data

Answer 5

If you have sampling distribution of sample mean, with know population variance sigma squared, then according to CLT, you have the (X-mu)/standard error is normally distributed with mean 0 and variance 1

Answer 6

so basicallyZ score is only used to transfer a given normal distribution to standard normal and the formula with sigma oveer sqrt(n) is to be used if you have a sapling distribution of sample mean

Answer 7

Thanks for the explanation, but my question is about \(\sqrt n\)
For both, they are almost the same but \(\sqrt n\), both are standard normal, right? but they are NOT equal, right? I meant \(Z=\dfrac{\overline Y-\mu}{\sigma /\sqrt n}\neq \color {red}{Z}=\dfrac{\overline Y-\mu}{\sigma}\)
That is only one way made it clear to me.
If they are equal, I ...... don't understand

Answer 8

this \[\dfrac{\overline Y-\mu}{\sigma}\]

is not standard normal (unless n=1)

but \[\dfrac{Y_i-\mu}{\sigma}\]
is standard normal

Answer 9

Yes, I see the different between them, the first one is \(\sqrt n(\overline Y-\mu)/\sigma \)
and the second one is \((Y_i-\mu)/\sigma\)
How do they relate to each other?

Answer 10

only that they are both standard normal (provided your \(Y_i\) are independent)

Answer 11

But they are different, right?

Answer 12

What do you mean by different? They are both random variables that have the same distribution

Answer 13

do they take the same value for each realization...no, but that really doesn't matter

Answer 14

Like this
\(Y_1, Y_2,\cdots ,Y_n\)~N(0,1)
but they are different since they are independent even though all are standard normal distribution.
Same as those Zs

Answer 15

like I said they will have different realizations

Answer 16

I'm not sure what you are trying to get at?

Answer 17

If they use \(Z_1=\sqrt n\dfrac{\overline Y -\mu}{\sigma}\) ~ N (0,1)

and \(Z_2=\dfrac{Y_i -\mu}{\sigma}\)~ N (0,1)
then I am ok. I understand they are 2 different standard normal distributions.
But they use the same Z. It makes me think they are only 1 normal distribution but represented in 2 forms.

Answer 18

they are different

\[P(Z_1=Z_2)=0\]

Answer 19

again, provided n>1

Answer 20

One more question :)
I don't get the last result from the solution \(m_z (t) = e^t^2/2 from this

Answer 21

about the mean and variance being 0 and 1 respectively?

Answer 22

\(m_{(Y-\mu)}(t/\sigma)\) = ?

Answer 23

you don't understand why it is equal to

\[e^{(t/\sigma)^2(\sigma^2/2)}\]?

Answer 24

yes, please

Answer 25

you understand why \[m_{(Y-\mu)}(t)=e^{(t^2)(\sigma^2/2)}\]?

Answer 26

oh, just replace t by t/sigma, right?

Answer 27

yes

Answer 28

ha!! :)
Thanks a lot. Ah!! I am struggling with statistics.

Answer 29

what book are you using?

Answer 30

Mathematical Statistics with Applications 7th Edition

Answer 31

Mendenhall

Answer 32

Yes, Dennis D. Wackerly , William Mendenhall III and Richard L. Scheaffer

Answer 33

I'm teaching out of that book this semester ;)

Answer 34

we are on section 6.3...starting 6.4 tomorrow

Answer 35

Really? I stuck everywhere because I know nothing about statistics.
My Prof will teach me 7.2 or 7.3 next week but I have to study the parts I don't get.

Answer 36

ic

Answer 37

Thank you so much. :)

Answer 38

no problem

Answer 39

I am sorry for bothering you again. I would like to know how to find cdf /pdf of chi-square on Ti 83. My Prof said something about the table, but I know there is a way we can find it out by calculator, just not know how to yet. :)
Please, teach me.

Answer 40

hit 2nd - vars

the \(\chi^2\) are numbers 7 and 8

Answer 41

I use Ti 83 plus, so that I don't have upper /lower and df
But I guess!! :)
2nd +Vars
then (0, 5, 7)
where 0 is lower, 5 is upper limit and 7 is degree of freedom, right?

Answer 42

\(Y_1\)~N (0,1)
\(Y_2\)~N (0,1)
\((Y_1-Y_2)\)~ N( 0,2)
and then \(\dfrac{Y_1-Y_2}{\sqrt 2}\) ~ N (0,1)
why?

Answer 43

I understand why Y1-Y2 ~ N (0,2), just the last one.

Answer 44

if you have a random variable X and you divide by its standard deviation then you get a random variable with s.d. 1

2 is the variance and \(\sqrt{2}\) is the standard deivation

Answer 45

Perfect. Thanks a ton