Question

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

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?

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

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