Question

Let z denote a random variable having a normal distribution with ? = 0 and ? = 1. Determine each of the probabilities below. (Round all answers to four decimal places.)

(a) P(z < 0.3) =  

(b) P(z < -0.3) =  

(c) P(0.40 < z < 0.85) =  

(d) P(-0.85 < z < -0.40) =  

(e) P(-0.40 < z < 0.85) =  

(f) P(z > -1.26) =  

(g) P(z < -1.5 or z > 2.50) =

Answer 1

given that

population mean

Population Standard daviation

hense X=Z

a)now we have to fin P(Z<0.3)

So the veriable lie on the right side to the mean

Now for Z=0.3 the probablity value from the standard table of probablity distribution we have

P=0.6179

As we see the bell shape curve we will find that this is the area left to the point or Z of it is the probablity of less than Z<0.3

P(Z<0.3)=0.6179

b)now we have to fin P(Z< - 0.3)

So the veriable lie on the left side to the mean

Now for Z= - 0.3 the probablity value from the standard table of probablity distribution we have

P=0.3821

As we see the bell shape curve we will find that this is the area left to the point of it is the probablity of less than Z< - 0.3

P(Z< - 0.3)=0.3821

c) now we have to find

P(0.4<Z< 0.85)

Now for Z=0.40 the probablity value from the standard table of probablity distribution we have

P=0.6554 (this is the area left to the Z score so it is the P(z<0.40))

Now for Z=0.85 the probablity value from the standard table of probablity distribution we have

P=0.8023 (this is the area left to the Z score so it is the P(z<0.85))

now the probablity between these two point willl probablity between thes two points

P(0.4<Z<0.85)=0.8023-0.6554 =0.1469

d) now we have to find

P( - 0.85<Z < - 0.40)

Now for Z= - 0.85 the probablity value from the standard table of probablity distribution we have

P=0.1977 (this is the area left to the Z score so it is the P(z<- 0.85))

Now for Z= - 0.40 the probablity value from the standard table of probablity distribution we have

P=0.3446 (this is the area left to the Z score so it is the P(z< -0.40))

now the probablity between these two point willl probablity between thes two points

P( - 0.85<Z< - 0.40)=0.3446-0.1977 =0.1469

e)now we have to find

P( - 0.40<Z <   0.85)

so for Z= - 0.40 the probablity value from the standard table of probablity distribution we have

P=0.3446(this is the area left to the Z score so it is the P(z<- 0.40))

Now for Z= 0.85 the probablity value from the standard table of probablity distribution we have

P=0.8023 (this is the area left to the Z score so it is the P(z< 0.85))

now the probablity between these two point willl probablity between thes two points

P( - 0.40<Z<   0.85)=0.8023-0.3446=0.4577

f) now we have to find P(Z> -1.26)

So the veriable lie on the left side to the mean

Now for Z= - 1.26 the probablity value from the standard table of probablity distribution we have

P=0.1038

As we see the bell shape curve we will find that this is the area left to the point of it is the probablity of less than Z< - 1.26

and total probablity will alweys =1

so P(Z> - 1.26)=1-P(Z< -1.26) =1-0.1038=0.8962

c) now we have to find

P(Z< - 1.5 or Z> 2.50)

Now for Z= - 1.5 the probablity value from the standard table of probablity distribution we have

P=0.0668   (this is the area left to the Z score so it is the P(z< -1.5))

Now for Z=2.50 the probablity value from the standard table of probablity distribution we have

P=0.9938 (this is the area left to the Z score so it is the P(z<2.50))

and total probablity will always be=1

so P(Z>2.50)=1-P(z<2.50)=1- 0.9938=0.0062

now we konw that

P(A or B)=P(A)+P(B)

so

P(Z< - 1.50 or Z> 2.50)=P(Z< - 1.50) +P(Z>2.50)=0.0668 + 0.0062= 0.0730