Question

Please put Rcode and Routput. Thank you. 1. Use R to: (a)Find P(-1.45 < Z < 2.72) b) Find P(Z < -2.24) (c) Find P( -1.34 < Z < - 0.74) d) Find P( 1.53 < Z) e) Find P( Z > 1.31) 2. Use R to: (a)Find z0 when P( - Z0 < Z < Z0)= 0.82 (b) Find z0 when P(Z > Z0)= 0.32 (c) Find z0 when P(Z < Z0)= 0.28 3. Use R: Suppose yearly income of people in Minnesota is normally distributed with mean 45700 and standard deviation 2830. Find probability that a randomly selected person has income between 44000 and 48000. 4. Use R: Suppose yearly income of people in Minnesota is normally distributed with mean 45,700 and standard deviation 2830. Find probability that 10 randomly selected people have mean income between 44000 and 48000. 5. Use R: Suppose yearly income of people in Minnesota is normally distributed with mean 45,700 and standard deviation 2830. Find probability that 10 randomly selected people have mean income greater than 47,000.

Answer 1

Here we need to find probabilities coming from the normal distribution.

Let Z~N(0,1)

then

i) P(a<Z<b)=P(Z<b)-P(Z<a)

ii) P(Z<-a)=1-P(Z<a)

iii) P(-a<Z<a)=

that is P(Z<a)-P(Z<-a)=

that is P(Z<a)-(1-P(Z<a))=

that is 2*P(Z<a)=+1

that is P(Z<a)=(+1)/2

the value of a will be found by inverse.

In R for cumulative probability, we use "pnorm" function and for inverse probability, we use "qnorm" function.

The R codes and output is

############################################--R Code-##################################

# number 1
#part a
pnorm(2.72)-pnorm(-1.45)

#part b
pnorm(-2.24)

#part c
pnorm(-.74)-pnorm(-1.34)

#part d
pnorm(1.53,lower.tail = F)

#part e
pnorm(1.31,lower.tail = F)

# number 2
#part a
qnorm((1+0.82)/2)

#part b
qnorm(0.32,lower.tail = F)

#part c
qnorm(0.28)

#number 3
# here mean=M=45700, standard deviation=S=2830
M=45700
S=2830
pnorm(48000,M,S)-pnorm(44000,M,S)

#number 4
# here sample of 10 has mean=M=45700, standard deviation=S=2830/sqrt(10)
M=45700
S=2830/sqrt(10)
pnorm(48000,M,S)-pnorm(44000,M,S)

#number 5
# here sample of 10 has mean=M=45700, standard deviation=S=2830/sqrt(10)
M=45700
S=2830/sqrt(10)
pnorm(47000,M,S,lower.tail = F)

###########################-R script-#####################################################