Probability in Gaussian Distriubution Problem Please show al

Probability in Gaussian Distriubution Problem

Please show all steps. Thank you!

Probability in Gaussian Distribution problem Given a Population described by a Gaussian distribution with mean overline mu = 5 , and standard deviation sigma = 0.4. Determine the probability that mu is less than or equal to 5. Determine the probability that mu is greater than or equal to 3. determine the probability that mu is between or equal to 3 and 5. Please show all steps. Thank you!

Solution

Mean ( u ) =5
Standard Deviation ( sd )=0.4
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X > 5) = (5-5)/0.4
= 0/0.4 = 0
= P ( Z >0) From Standard Normal Table
= 0.5                  
P(X > = 5) = (1 - P(X < 5)
= 1 - 0.5 = 0.5                  
b)
P(X < 3) = (3-5)/0.4
= -2/0.4= -5
= P ( Z <-5) From Standard Normal Table
= 0                  
P(X > = 3) = (1 - P(X < 3)
= 1 - 0 = 1                  
c)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 3) = (3-5)/0.4
= -2/0.4 = -5
= P ( Z <-5) From Standard Normal Table
= 0
P(X < 5) = (5-5)/0.4
= 0/0.4 = 0
= P ( Z <0) From Standard Normal Table
= 0.5
P(3 < X < 5) = 0.5-0 = 0.5