Find standard deviation and probability

Question

The results of a test given to a group of students are normally distributed with a mean of 71. In the test, 85% of students scored at less than 80. To pass the test a student must score at least 65. 

a) find the standard deviation of the score 

b) find the probability that a student chosen at random passes the test 

Answer 1

There seems to be a typo in the problem. I am assuming the problem should say "85% of the students scored less than 80%".

Recall the formula:

z = (x-µ)/σ

We are given x=80% and mu = 71. If we can find z, we can solve for sigma.

We are told that 85% of the students scored less than an 80%. Sketch a bell curve. What is the z-score that separates the bottom 85% from the top 15%?

Use this z-score, the given value for mean, and the value x=80

So...

1.04 = (80- 71) / σ

σ = (80-71)/1.04 = 8.65

To calculate the probability of passing, we need to find the z-score for x=65

z= (65 - 71)/8.65 = -.69

P(X>65) = P( Z > -.69) = 0.7549 ≈ 75.5%