Solving this problem using the statistical program R on the

Solving this problem using the statistical program R on the computer:
Let Z N(0, 1). Find a constant c for which (
a) P(Z c) = 0.1587
(b) P(c Z 0) = 0.4772
(c) P(c Z c) = 0.8664
(d) P(0 Z c) = 0.2967
(e) P(|Z| c) = 0.1470

Solution

here Z~N(0,1)

the R code is given as---

### Z~N(0,1)
##a)
qnorm(0.1587,0,1,lower.tail=FALSE)
##b)
qnorm(0.5-0.4772,0,1)
##c)
qnorm((0.8664+1)/2,0,1)
##d)
qnorm(0.5+0.2967,0,1)
##e)
qnorm((2-0.1470)/2,0,1)

description

a) it is directly from the R code. lower.tail=FALSE means P[X>x] type of probability is calculated.

b) P[c<=Z<=0]=0.4772 => P[Z<=0]-P[Z<=c]=0.4772 =>0.5-P[Z<=c]=0.4772 =>P[Z<=c]=0.5-0.4772

c) P[-c<=Z<=c]=0.8664 => P[Z<=c]-P[Z<=-c]=0.8664 => P[Z<=c]-[1-P[Z<=c]]=0.8664 =>P[Z<=c]=(0.8664+1)/2

d) P[0<=Z<=c]=0.2967 => P[Z<=c]-P[Z<=0]=0.2967 =>P[Z<=c]=0.2967+0.5

e) P[|Z|>=c]=0.1470 => P[Z>=c]+P[Z<=-c]=0.1470 => 1-P[Z<=c]+1-P[Z<=c]=0.1470 =>P[Z<=c]=(2-0.1470)/2

answer

a)0.9998151

b)-1.999077

c)1.500056

d)0.8298917

e)1.45021