0
$\begingroup$

For $n=20$ samples of normal distribution with variance $10$, what should be $\alpha$, so the length of the confidence interval will be exactly $2$?

So the calculation is:

$$2=2z_{1-\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}}=2z_{1-\frac{\alpha}{2}}\frac{\sqrt{10}}{\sqrt{20}}=2z_{1-\frac{\alpha}{2}}\sqrt{\frac{1}{2}}\\z_{1-\frac{\alpha}{2}}=\sqrt{2} $$

BUT! There's a problem - $\sqrt 2 > 1$

Why is it happening and how to fix it?

  • 0
    I could be wrong here, but I think you may be confusing $z$-values with probabilities. What's the problem with having it $>1$? – 2017-01-19

1 Answers 1

2

Your $z$-value can be greater than $1$ - you're confusing them with probabilities. You need to look in the $z$-table and find a corresponding choice of $\alpha$ for $\sqrt{2} \approx 1.41$ and set that to $1-\frac{\alpha}{2}$.

  • 0
    but $z_{1-\alpha/2}$ represents an area under the distribution graph, so how can it be equal to $\sqrt 2$? – 2017-01-19
  • 0
    Incorrect, I'm afraid - you need to revise $z$-scores: https://en.wikipedia.org/wiki/Standard_score – 2017-01-19
  • 0
    Okay... got it! ty – 2017-01-19