Please translate the following into standard normal probabil

Please translate the following into standard normal probability distribution (transform x to z) with Mean = 30 and standard deviation = 4. Find P(x < 40) ; P(x > 21) ; P(30 < x < 35)

Solution

X follows a normal distribution with mean =E[X]=30 and standard deviation=S(X)= 4.

and Z follows a standard normal distribution that is mean of Z is zero and standard deviation is 1

so Z=(X-30)/4 as E[Z]=(E[X]-30)/4=(30-30)/4=0

and S(Z)=S(X)/4=4/4=1

so the transformation can be done by subtracting 30 from the given value and then dividing it by 4.

so

1. P[X<40]=P[(X-30)/4<(40-30)/4]=P[Z<2.5]   [answer]

and the probability is P[Z<2.5]=0.993790 [using minitab]

2. P[X>21]=P[(X-30)/4>(21-30)/4]=P[Z>-2.25] [answer]

and the probability is P[Z>-2.25]=1-P[Z<-2.25]=0.987776 [using minitab]

3. P[30<X<35]=P[(30-30)/4<(X-30)/4<(35-30)/4]=P[0<Z<1.25]   [answer]

and the probability is P[0<Z<1.25]=P[Z<1.25]-P[Z<0]=0.894350-0.5=0.394350 [using minitab]