# Statistics: detectibility of two objects, with 95% confidence

1. Jul 14, 2014

### lavster

1. The problem statement, all variables and given/known data

Apologies if this is not meant to go in to homework area; this is not a homework or coursework question, but more to convince myself that what I am reading is correct

I have attached the diagram I am pondering over... How can I prove that if I want to differentiate between two objects at the 95% level then the difference between the mean of the two distributoins must be 3.29σ (*assume the standard deviation is the same for both distribution*)

I.e. i want to prove the distance in the attached image is 3.29σ

2. Relevant equations

3. The attempt at a solution

95% (two tailed distribution is normally +/-1.96*σ). Therefore I naively thought that the distance would be two times this ie 3.92σ. This has the same numbers but in the wrong order!!! Im sure im missing something obvious but as I am not clued up in my statistics Im finding it difficult to see what is wrong with my approach. Can anyone shed some light?!

thanks

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2. Jul 14, 2014

### LCKurtz

If $X\sim n(\mu_1,\sigma^2)$ and $Y\sim n(\mu_2,\sigma^2)$ then $W = X-Y \sim n(\mu_1-\mu_2,2\sigma^2)$. If your hypothesis is that the means are different you want to reject $H_0:\mu_1=\mu_2$ with a two tailed test at 95% confidence you need$$|Z| =\left|\frac{W-0}{\sqrt 2\sigma} \right| > 1.96$$ That gives $|W|>1.96\sqrt 2\sigma =2.77\sigma$. Perhaps I misunderstand the question.