Thread: Solve for B

1. Solve for B

This is driving me nuts!

-Z < (XBar - B) / (B/sqrt(n)) < Z

I want to solve this for B (and keep it in the center). I've burned through an eraser.....can anyone show me the steps?

Thanks!

2. Originally Posted by B_Miner
This is driving me nuts!

-Z < (XBar - B) / (B/sqrt(n)) < Z

I want to solve this for B (and keep it in the center). I've burned through an eraser.....can anyone show me the steps?

Thanks!
Do you mean $\displaystyle -Z < \frac{(\bar{x} - B)} {\frac{B}{\sqrt{n}}} < Z$?

We can split that problem into two inequalities:

1) $\displaystyle -Z < \frac{(\bar{x} - B)} {\frac{B}{\sqrt{n}}}$ and 2) $\displaystyle \frac{(\bar{x} - B)} {\frac{B}{\sqrt{n}}} < Z$

For the first one, we can first multiply both sides of the inequality by $\displaystyle \frac{B}{\sqrt{n}}$,
and we arrive at $\displaystyle \frac{-Z(B)}{\sqrt{n}} = \bar{x} - B$.
Then, add $\displaystyle B$ to both sides:
$\displaystyle \frac{-Z(B)}{\sqrt{n}} + B = \bar{x}$
We can factor out $\displaystyle B$:
$\displaystyle B(\frac{-Z}{\sqrt{n}}+1) < \bar{x}$
and then divide both sides by $\displaystyle \frac{-Z}{\sqrt{n}}+1$, so the final inequality is:
$\displaystyle B < \frac{\bar{x}}{\frac{-z}{\sqrt{n}}+1}$.
It's the same process for the second inequality. Once you have both inequalities, you can mesh them together to form one big inequality with $\displaystyle B$ in the center.

3. Originally Posted by lisczz
Do you mean $\displaystyle -Z < \frac{(\bar{x} - B)} {\frac{B}{\sqrt{n}}} < Z$?

We can split that problem into two inequalities:

1) $\displaystyle -Z < \frac{(\bar{x} - B)} {\frac{B}{\sqrt{n}}}$ and 2) $\displaystyle \frac{(\bar{x} - B)} {\frac{B}{\sqrt{n}}} < Z$

For the first one, we can first multiply both sides of the inequality by $\displaystyle \frac{B}{\sqrt{n}}$,
and we arrive at $\displaystyle \frac{-Z(B)}{\sqrt{n}} = \bar{x} - B$.
Then, add $\displaystyle B$ to both sides:
$\displaystyle \frac{-Z(B)}{\sqrt{n}} + B = \bar{x}$
We can factor out $\displaystyle B$:
$\displaystyle B(\frac{-Z}{\sqrt{n}}+1) < \bar{x}$
and then divide both sides by $\displaystyle \frac{-Z}{\sqrt{n}}+1$, so the final inequality is:
$\displaystyle B < \frac{\bar{x}}{\frac{-z}{\sqrt{n}}+1}$.
It's the same process for the second inequality. Once you have both inequalities, you can mesh them together to form one big inequality with $\displaystyle B$ in the center.
Just a note that when getting the inequalities you should consider if it matters whether B is greater than 0 or less than zero ....

4. B is >0 (this is using the CLT to estimate a confidence interval of the parameter from the exponential distribution (aka theta)

Thanks so much for the detailed response lisczz!

You guys are great for doing this!!

5. I actually do have a question. This is what confuses me about inequalities. You have to assume that

-Z / sqrt(N) +1 is >0 so that the sign of the inequality does not flip when you divide by it? Is that correct?