To study a certain characteristic about a population of people we take a sample of $100$ individuals. The $80$ percent confidence interval for the mean is $(0.9,1.1)$.
Part I: Find the sample mean and standard deviation (easy). $\bar x = 1$ and $\hat \sigma = 0.7752$
Part II: What is the probability that $(0.9,1.1)$ doesn't contain the mean.
What about Part II? Is it just $1 - 0.8 = 0.2$, or is it a trick question?
Part 2 isn't a trick question. The answer is 0.2. The trick question which tends to be asked in these situations is "what is the probability $\mu$ lies in the confidence interval?" Then the answer is "either $1$ or $0$, depending on whether it does or does not."