Question

Your professor gives one of two types of exams, hard tests and INSANELY hard tests.

• 60% of the time he gives insanely hard tests, 40% of the time he gives hard tests

• Hard tests have an average score of 70 and a standard deviation of 10.

• Insanely hard tests have an average score of 50 and a standard deviation of 20.

• Both types of tests produce normally distributed scores

(a) What is the average score on all tests given by the professor?

(b) What is the probability of getting a score above a 70 on a test given by the professor?

(c) A randomly chosen student gets above a 70 on a test given by the professor, what is the probability it was an insanely hard test?

Answer 1

a)

average score on all tests given by the professor = 0.60*70 + 0.40*50 = 62

b)

P(X>70 on hard test)

Z =(X - µ ) / σ = (70.00-70) / 10=0.000

  

P(X ≥70.000) = P(Z ≥0.000) =P ( Z <0.000) = 0.50

excel formula for probability from z score is =NORMSDIST(Z)

----------------------

P(X>70 on insanely hard test)

Z =(X - µ ) / σ = (70.00-50) / 20=1.0

  

P(X ≥70) = P(Z ≥1.0) =P ( Z <-1.000) = 0.1587

excel formula for probability from z score is =NORMSDIST(Z)

so, probability of getting score 70 in any test = 0.60 * 0.50 + 0.40 * 0.1587 = 0.3635

c)

P(score above 70 on hard test) = 0.50

P(score above 70 on insanely hard test) = 0.1587

P(score above 70) = 0.3635

P(insanely hard test | score above 70) = 0.40 * 0.1587 / 0.3635 = 0.1746