I am figuring out a method to calculate the population mean via sample info. Is that possible?
For example, sample info sample size n=100, ̅x=53, σ=7.7. The population size is 6700. Can I calculate the population mean when P(x ̅≥53 hours)=0.5,
Can anyone give me hints and tell me is there any trading between population and sample mean? Thanks a lot
With a sample of size $n=100,$ the sample mean $\bar X$ will tend to be very close to the population mean $\mu.$
Furthermore, by the Central Limit Theorem, $\bar X$ will have very nearly the distribution $\mathsf{Norm}(\mu, \sigma/\sqrt{n} = 7.7/10 = 0.77).$
So you can say $$P(\bar X \ge 53) \approx P\left(Z = \frac{\bar X - \mu}{\sigma/\sqrt{10}} \ge \frac{53-\mu}{0.77} = 0\right) = 0.5,$$ where $Z$ is a standard normal random variable. Solve for $\mu.$