The question states "Times spent studying by students the week before exams follows a normal distribution with standard deviation of 8 hours. A random sample of four students was taken in order to estimate the man study time for the population of all students. What is the probability the sample mean exceeds the population mean by more than 2 hours?"
I'm not given any means to work with, so I can't use Z-Score as far as I can tell. I'm clearly not seeing something. Is there a way to back solve for the answer? Is it something along the lines of just 2/standard error and finding Z greater than that?
Best Answer
The sample mean $\bar{X}$ has normal distribution, mean the population mean $\mu$, and standard deviation $\tau=\frac{8}{\sqrt{4}}$. We want $\Pr(\bar{X}-\mu\gt 2)$, which is $\Pr(Z\gt \frac{2}{\tau})$, where $Z$ is standard normal. This looks like precisely the approach you suggested.