Scores for an entry exam to a university...?

Question

Scores for an entry exam to a university are normally distributed with a mean of 75 and a standard deviation of 6.5:

a) If the top 5% are admitted with a full scholarship, what is the lowest score a student can obtain to earn a full scholarship?

b) The university decides that students who scores below or at the 10th percentile must take remedial courses. If a student scores 63 on the entry exam, will he/she be taking remedial courses? Explain why.

c) What is the probability that a randomly selected student will score more than 90?

Answer 1

The key here is remembering that a given normal distribution's mean and standard deviation, if not 0 and 1, respectively, requires converting the x values into Z values, to be able to use a Z table to look up the correct probabilities.

You also have to determine if the value you looked up matches the area under the curve you seek.  If your table only goes from Z = 0 to, say, 3.5, and you have negative values of Z, you'll have to use the fact that the Z curve is symmetric about Z =0, such that the probability of P(0 < Z < z) is the same as P(-z < Z < 0).  Some tables may run from -3.5 to 3.5 which simplifies things, but remember the probability you look up off this table is for P(Z < z), so if you need P(Z > z) you'll need to subtract the area you look up from 1, i.e., P(Z > z) = 1 - P(Z < z).

Answer 2

z = (x-μ)/σ

a) Using invNorm (.95, 0, 1) = 1.645,

1.645 = (x-75)/6.5

Solve for x,

x = 85.7 <==the lowest score for a full scholarship.

 

b) z = (63-75)/6.5

P(z < (63-75)/6.5) = 3.24% < 10%

So, the student has to take remedial courses.

 

c) P (z > (90-75)/6.5) = 1.05% <==Answer