2. Your professor gives one of two types of exams, easy tests and hard tests. 75

Question

2. Your professor gives one of two types of exams, easy tests and hard tests. 75% of the time he gives hard tests, 25% of the time he gives easy tests Easy tests have an average score of 85 and a standard deviation of » Hard tests have an average score of 70 and a standard deviation of 10 . 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 90 on a test given by the professor? (c) What is the probability of getting a score below 80?

Explanation / Answer

Ans:

a)Average score of all tests=0.75*70+0.25*85=52.5+21.25=73.75

standard deviation of all tests=sqrt(0.752*100+0.252*25)=sqrt(57.8125)=7.6

b)P(X>90)

z=(90-73.75)/7.6=2.14

P(z>2.14)=1-P(z<=2.14)=1-0.9838=0.0162

c)P(X<80)

z=(80-73.75)/7.6=0.82

P(z<0.82)=0.7939

d)P(>90/easy test)=0.1587 (as z=(90-85)/5=1,P(z>1)=0.1587)

P(>90/hard test)=0.02275 (as z=(90-70)/10;P(z>2)=0.02275)

i)P(easy test/>90)=0.1587*0.25/[0.1587*0.25+0.02275*0.75]

=0.039675/0.05674

=0.699

ii)P(hard test/>90)=0.02275*0.75/[0.1587*0.25+0.02275*0.75]

=0.01706/0.05674

=0.301

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