A university expects to receive, for the next academic year, 16000 applications for admission to the bachelor’s degree program. The SAT score obtained by the applicants is modeled as a normal random variable with mean μ = 950 and standard deviation σ = 100. If the university accepts only applicants among the top 25%, what is the minimum SAT score necessary to get accepted into the program?
I'm trying to find the z score for the SAT score, but I don't know how to decide on a proper sample value. Please help!
In your edit, you wrote that you were looking for a "$z$ score for the SAT score". Note that the question asks for the SAT score itself. If what you're seeking is a $z$ score, more often than not, the answer is going to be something like $-5 < z < 5$. (That's probably a pretty wide interval, but statistics classes like to have students look at outliers; it's probably more, uh, exciting that way :).) If what you're looking for is an SAT score, though, that's going to be closer to $1000$, especially since we want a good SAT score, which is going to be higher than $\mu = 950$.
That's a hint that we don't want the $z$ score; that is, we already know the $z$ score. If we look up $1 - 0.25 = 0.75$ on a $z$ table, we'll get a $z$ score of $0.67$. (We have to look up the probability in the middle of the table itself; once we find $0.75$, or something close to it [on my chart it's $0.7486$], we see what number leads off the row — in this case $0.6$ — and then what number heads the column — in this case $.07$. Putting them together we get $0.67$.) We look for $0.75$ because we want three quarters of the area under the curve to be to the left of the $z$ score. Likewise, if on your TI-83, you press DISTR, select invNorm(, and enter $0.75$, you'll get the same value, just with more accuracy, approximately $0.6745$.
Now you have a $z$ score. You have $\mu$. You have $\sigma$. What don't you have? (And what is the real-world meaning of that thing you don't have?) Can you set it up and solve it from here?