Still having trouble with confidence Intervals, different info this time.?

Question

The score on the entrance test for a well known law school has a mean score of 200 points and a standard deviation of 50 points. What value should the lowest passing score be set if the school wishes only 2.5% of those taking the entrance test to pass?

Answer 1

Assuming the scores are spread over a normal distribution

If school only want top 2.5% to pass then they want everything outside the top bracket of a 95% confidence interval

upper 95% confidence limit = mean + z(1-alpha/2) x s.d

z(1-alpha/2) = quantile of the normal range. We have alpha=0.95 and z(0.975)=1.96

So upper confidence limit = 200+1.96 x 50 = 298

So bottom pass mark for acceptance should be 299.

Answer 2

If you have a TI-83 Plus or TI-84 calculator, do this:
a) Press 2nd, DISTR, 3 to select invNorm(.
b) Enter parameters so that your display looks like this: invNormal(.975, 200 , 50. You need not
close the parentheses, but it’s okay if you do.
c) Press ENTER and your answer will be 297.998, which you can round off appropriately.

If you don’t have a calculator, do this:
1) Let’s start by finding the area to the right of that score beginning at the mean:
.5000 - .025 = .475
2) In tables, find the value closest to .475. That’s l.96. Since this is a standard number, you probably already know this without looking it up in tables.
3) Let’s calculate the x value corresponding to z = 1.96.
4) From z = (x- u)/sigma
1.96 = (x – 200)/50
1.96(50) +200 = x
298 = x
The calculator probably has a little more accuracy.
Incidentally, if you ask questions such as this, some time could be saved if you'd inidicate whether you have a graphing calculator, and, if so, what model it is and if you are allowed to use the statistics functions.

Hope this helps.
FE

Answer 3

you need to estimate the 97.5% of the distribution. Assuming it to be normal, let
z = (x - 200)/50 is normalized variable with mean 0, s.d. 1
then
z =1.96 for it to be on the 97.5 percentile
(x - 200)/50 = 1.96
x - 200 = 1.96*50 = 196/2 = 98
x = 298
298 is the lowest passing score.