I'm fairly comfortable calculating the confidence interval. But now I'm seeing a problem where I'm giving a confidence interval $CI(27.6621, 30.3379)$ and I'm requested to calculate the confidence level. I'm also given the sample size $n=85$ and a standard deviation $\sigma=7.5$. I can't find a formula for that, I feel like it's something simple I'm not seeing. Any help would be appreciated. Thanks in advance!
Calculate confidence level from a given confidence interval
confidence intervalstandard deviationstatistics
Best Answer
Assuming that the data comes from $\mathcal{N}(\mu,\sigma^2)$ with known $\sigma^2$, we have that $$ \dfrac{\sqrt{n}(\bar{X}_n-\mu)}{\sigma}\sim\mathcal{N}(0,1). $$ If $z_{1-\alpha}$ is such that $P(\mathcal{N}(0,1)\geq z_{\alpha/2})=\alpha/2$, then the CI for $\mu$ is $$\left(\bar{X}_n-\dfrac{\sigma}{\sqrt{n}}z_{\alpha/2},\bar{X}_n+\dfrac{\sigma}{\sqrt{n}}z_{\alpha/2}\right).$$ Substract both sides to get that $$\dfrac{2\sigma}{\sqrt{n}}z_{\alpha/2}=c,$$ where $c$ is known. In a normal table find the $\alpha$ such that $z_{\alpha/2}=c\sqrt{n}/(2\sigma)$.