Math 3470 Introductory Probability and Statistics Suppose th

Suppose that Z has the standard normal distribution Then answer the following questions. Find P(|Z| 2). Find the value z such that P(Z > x) = 0.05. Let X be a normal random variable with mu = 5 and sigma = 10. (a) Find P(-20

Solution

a) P(|z|<2)

= P(-2<z<2)

= P(z<2) - P(z<-2)

Using z table from the below link, we can plug in the values

http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf

= 0.9772 - 0.0228

= 0.9544

b) P(|z|>2)

= 1 - P(-2<z<2)

= 1 - (P(z<2) - P(z<-2))

Using z table from the below link, we can plug in the values

http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf

= 1 - (0.9772 - 0.0228)

= 0.04566

c) P(Z > x) = 0.05

1 - P(Z<x) = 0.05

P(Z<x) = 0.95

Using z table from the below link, we can plug in the values

http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf

x = 1.65

Question 2

a) Mean = 5, Deviation = 10

Z1 = (X1 - mean)/sigma = (-20-5)/10 = -2.5

Z2 = (X2 - mean)/sigma = (15-5)/10 = 1

P(Z1<x<Z2) = P(-2.5<x<1) = P(x<1) - P(x<-2.5)

Using z table from the below link, we can plug in the values

http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf

=> 0.8413 - 0.062

=> 0.7793

b) P(-12.4 < X-5 < 12.4)

adding 5 both left and right sides we get

P(-7.4<X<17.4)

Z1 = (X1 - mean)/sigma = (-7.4-5)/10 = -1.24

Z2 = (X2 - mean)/sigma = (17.4-5)/10 = 1.24

P(Z1<x<Z2) = P(-1.24<x<1.24) = P(x<1.24) - P(x<-1.24)

Using z table from the below link, we can plug in the values

http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf

=> 0.895 - 0.013

=> 0.882

=> 0.7793