Chi Square Distribution Problem

[Nexttime35]

Homework Statement

Suppose that X has normal distribution. Find the distribution of n(Sample Mean - μ)²/σ².

The Attempt at a Solution

I honestly have no idea where to begin with this problem. Any ideas?

 

[Greg Bernhardt]

Tell us what you know about this problem. It must be homework for something you've been learning in class.

 

[Nexttime35]

Well I know that the Sample mean has a normal distribution ~ N(μ,σ²/n), which I think is useful to solve this problem. Now, I am confused about how to use this normal distribution for the sample mean to solve the problem. Any thoughts, using this idea?

 

[LCKurtz]

If $\bar X \sim N(\mu, \frac {\sigma^2}{n})$ what is the distribution of$$
Z=\frac{\bar X -\mu}{\frac\sigma {\sqrt n}}$$Once you answer that, you need to work out the distribution of $Z^2$.

 

[Nexttime35]

Hi LCKurtz,

Thanks for the help. I am wondering where you got this equation from? I know it's the z-score, converting the sample mean into the z-score, but how did you come up with the equation? Thanks.

 

[Nexttime35]

Actually I believe I know where this comes from.

the Sum from i=1 to n of Zi² = the Sum from i=1 to n of ((Xi-μ)/σ)² = chi square distribution with n degrees of freedom.

 

[Nexttime35]

So, would the distribution of this be N(0,n)?

 

[LCKurtz]

Homework Statement

Suppose that X has normal distribution. Find the distribution of n(Sample Mean - μ)²/σ².

Hi LCKurtz,

Thanks for the help. I am wondering where you got this equation from? I know it's the z-score, converting the sample mean into the z-score, but how did you come up with the equation? Thanks.

Now you are confusing me. It is your question asking for the distribution of$$
\frac {n(\bar X - \mu)^2}{\sigma^2} =\frac{(\bar X - \mu)^2}{(\frac \sigma {\sqrt n})^2}$$
So if $Z = \frac{(\bar X - \mu)}{\frac \sigma {\sqrt n}}$, you know $Z\sim n(0,1)$ and your problem is to find the distribution of $Z^2$.