Question

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

(a)

P(z < 0.1) =

(b)

P(z < −0.1) =

(c)

P(0.40 < z < 0.85) =

(d)

P(−0.85 < z < −0.40) =

(e)

P(−0.40 < z < 0.85) =

(f)

P(z > −1.25) =

(g)

P(z < −1.5 or z > 2.50) =

Let z denote a variable that has a standard normal distribution. Determine the value z* to satisfy the following conditions. (Round all answers to two decimal places.)

(a) P(z < z*) = 0.0256
z* =

(b) P(z < z*) = 0.0098
z* =

(c) P(z < z*) = 0.0507
z* =

(d) P(z > z*) = 0.0198
z* =

(e) P(z > z*) = 0.0098
z* =

(f) P(z > z* or z < −z*) = 0.2009
z* =

Answer 1

Ans.

We find all answers using R:-

1 (a)  P(z < 0.1)

> pnorm(0.1,0,1)
[1] 0.5398278

P(z < 0.1) = 0.5398

(b) P(z < −0.1)

> pnorm(-0.1,0,1)
[1] 0.4601722

P(z < −0.1) = 0.4602

(c). P(0.40 < z < 0.85) = P(z<0.85) - P(z<0.40)

> pnorm(0.85,0,1)- pnorm(0.40,0,1)
[1] 0.1469157

P(0.40 < z < 0.85) = 0.1469

(d). P(−0.85 < z < −0.40) = P(z<-0.40) - P(z<-0.85)

> pnorm(-0.40,0,1)- pnorm(-0.85,0,1)
[1] 0.1469157

P(−0.85 < z < −0.40) = 0.1469

(e). P(−0.40 < z < 0.85) = P(z<0.85) - P(z<-0.40)

> pnorm(0.85,0,1)- pnorm(-0.40,0,1)
[1] 0.4577592

P(−0.40 < z < 0.85) = 0.4578

(f). P(z > −1.25) = P(z<1.25) = 1-P(z<-1.25)

>1- pnorm(-1.25,0,1)
[1] 0.8943502

P(z > −1.25) = 0.8944

(g). P(z < −1.5 or z > 1.50) = P(z<-1.5) or P(z>1.50) = P(z<-1.5) = P(z>1.50)

P(z<-1.5)

> pnorm(-1.5,0,1)
[1] 0.0668072

P(z<-1.5) = 0.0668

P(z>1.5) = 1 - P(z<1.5)

> 1-pnorm(1.5,0,1)
[1] 0.0668072

P(z>1.5) = 1 - P(z<1.5) = 0.0668

P(z < −1.5 or z > 1.50) = 0.0668

2 (a). P(z < z*) = 0.0256

> qnorm(0.0256,0,1)
[1] -1.9498

z* = -1.95
(b). P(z < z*) = 0.0098

> qnorm(0.0098,0,1)
[1] -2.333918

z* = -2.33
(c) P(z < z*) = 0.0507

> qnorm(0.0507,0,1)
[1] -1.638104

z* = -1.64

(d). P(z > z*) = 0.0198

> qnorm(0.0198,0,1,lower.tail=F)
[1] 2.057897

z* = 2.06

(e). P(z > z*) = 0.0098

> qnorm(0.0098,0,1,lower.tail=F)
[1] 2.333918

z* = 2.33
(f). P(z>z* or z<-z*) = 0.2009

P(z>z*) = P(z<-z*)

> qnorm(0.2009,0,1,lower.tail=F)
[1] 0.8384108

z* = 0.84