It is estimated that times to completion for major league ba

It is estimated that times to completion for major league baseball games follow a normal distribution with a mean of 132 minutes and a variance of 144 minutes.

a. What is the probability that a game lasts between 120 and 150 minutes?

b. 33% of all games last longer than how many minutes?

c. What is the probability that a game lasts fewer than 120 minutes?

d. If 100 games are chosen at random, what is the probability that at least 25 of these games last fewer than 120 minutes?

Solution

Mean ( u ) =132
Standard Deviation ( sd )=12
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 120) = (120-132)/12
= -12/12 = -1
= P ( Z <-1) From Standard Normal Table
= 0.15866
P(X < 150) = (150-132)/12
= 18/12 = 1.5
= P ( Z <1.5) From Standard Normal Table
= 0.93319
P(120 < X < 150) = 0.93319-0.15866 = 0.7745                  
b)
P ( Z > x ) = 0.33
Value of z to the cumulative probability of 0.33 from normal table is 0.44
P( x-u/ (s.d) > x - 132/12) = 0.33
That is, ( x - 132/12) = 0.44
--> x = 0.44 * 12+132 = 137.28  
c)
P(X < 120) = (120-132)/12
= -12/12= -1
= P ( Z <-1) From Standard Normal Table
= 0.1587