17 Find the following probabilities based on a standard norm

17)

Find the following probabilities based on a standard normal variable Z. Use Table 1. (Round your answers to 4 decimal places.)

Solution

Normal Distribution
Mean ( u ) =0
Standard Deviation ( sd )=1
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X > 0.65) = (0.65-0)/1
= 0.65/1 = 0.65
= P ( Z >0.65) From Standard Normal Table
= 0.2578  
b)
P(X <= -1.64) = (-1.64-0)/1
= -1.64/1= -1.64
= P ( Z <-1.64) From Standard Normal Table
= 0.0505                  
      
c)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 0.06) = (0.06-0)/1
= 0.06/1 = 0.06
= P ( Z <0.06) From Standard Normal Table
= 0.52392
P(X < 1.59) = (1.59-0)/1
= 1.59/1 = 1.59
= P ( Z <1.59) From Standard Normal Table
= 0.94408
P(0.06 < X < 1.59) = 0.94408-0.52392 = 0.4202                  

d)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < -0.86) = (-0.86-0)/1
= -0.86/1 = -0.86
= P ( Z <-0.86) From Standard Normal Table
= 0.19489
P(X < 2.48) = (2.48-0)/1
= 2.48/1 = 2.48
= P ( Z <2.48) From Standard Normal Table
= 0.99343
P(-0.86 < X < 2.48) = 0.99343-0.19489 = 0.7985