The total maximum score on a calculus exam was 100 points Th

The total maximum score on a calculus exam was 100 points. The mean score was 74 and the standard deviation was 11. Assume that the scores are normally distributed.

(a) Suppose that 180 students took the exam, and their scores were all independent. What is the probability that exactly 5 students scored below 52, exactly 100 students scored between 52 and 85, and exactly 75 students scored above 85?

Solution

Mean ( u ) =74
Standard Deviation ( sd )=11
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X < 52) = (52-74)/11
= -22/11= -2
= P ( Z <-2) From Standard Normal Table
= 0.0228                  

Probability of scoring below 52 is 0.0228
it follows Binomial , n=180, p = 0.0228
P( X = 5 ) = ( 180 5 ) * ( 0.0228^5) * ( 1 - 0.0228 )^175
= 0.162
b)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 52) = (52-74)/11
= -22/11 = -2
= P ( Z <-2) From Standard Normal Table
= 0.02275
P(X < 85) = (85-74)/11
= 11/11 = 1
= P ( Z <1) From Standard Normal Table
= 0.84134
P(52 < X < 85) = 0.84134-0.02275 = 0.8186                  
Probability of scoring between 52 to 82 is 0.8186
it follows Binomial , n=180, p = 0.8186

P( X = 100 ) = ( 180 100 ) * ( 0.8186^100) * ( 1 - 0.8186 )^80
= 0
c)
P(X > 85) = (85-74)/11
= 11/11 = 1
= P ( Z >1) From Standard Normal Table
= 0.1587                  

Probability of scoring above 85 is 0.1587
it follows Binomial , n=180, p = 0.1587

P( X = 75 ) = ( 180 75 ) * ( 0.1587^75) * ( 1 - 0.1587 )^105
= 0