A SAT score is designed to have a normal distribution with mean 400 and standard deviation 200. If we take 5 independent SAT scores, what is the probability that the mean of them is greater than 500?
I'm just making sure I am thinking about this right. So I am trying to find $P(X > 500)$ so I would be using something like:
$$ P(X > 500) = 1 - \phi \frac{500-400}{200}$$
$\phi$ would be the normal integration of this number in some cases this fraction is denoted as Z = $\frac {X- \mu}{\sigma}$
I know I should alter this since I'm given that I'm using 5 independent trials and that I'm finding the probability of their mean rather than a certain score.
Hint:
Yes, you are on the right track. You should "alter" it. Notice we take a sample, $X_1,\dotsc, X_5$. Then they are asking about the mean of the five, $$\bar X = \frac{X_1+\dotsb+X_5}{5}.$$ Specifically, they are asking for $$P(\bar X >500).$$
Now can you proceed?