If my data $X_i\sim N(\mu,σ^2)$ is iid with a sample $n=35$
If my 99% confidence interval is $[.56,1.86]$ where $P(t>2.7238)=.005$ for a $t(34)$ random variable $t$
How would I derive $\bar{X}$ and $S^2$ and test
$H_0:\mu=1.5$ against $H_1:\mu\neq1.5$ at the 1% level? I am really stuck, I appreciate any help!
Your confidence is of the form $\bar x \pm t^\ast SE$, where $t^\ast = 2.7238$ and $SE = \sqrt{S^2/n}$. This should allow you to find $\bar x$ and $S^2$, and you can take it form there.