Question

The quarterly returns for a group of 53 mutual funds with a mean of 2.1​% and a standard deviation of 5.1​% can be modeled by a Normal model. Based on the model ​N(0.021​,0.051​), what are the cutoff values for the ​

a) highest 10​% of these​ funds? ​
b) lowest 20​%?
​c) middle 40​%?
​d) highest 80​%?

Answer 1

here mean=0.021 and standard deviation=0.051

we use stadard normal variate z=(x-mean)/sd

(a) answer is 0.0864

here we want to find the value of x such that P(X>x)=0.1

first we find z such that P(Z>z)=0.1

or P(Z<z)=1-P(Z>z)=1-0.1=0.9

and z=1.28 ( using ms-excel=normsinv(1.28))

corresponding x=mean+z*sd=0.021+1.28*0.051=0.0864

(b) answer is -0.0219

we want to find x such that P(X<x)=0.2

first we find z such that P(Z<z)=0.2 and z=-0.8416 and

x=mean+z*sd=0.021-0.8416*0.051=-0.0219

(c) here we find x 1 and x2 such that P(x1<X<x2)=0.4

coresponding z1 and z2 would be P(z1<Z<z2)=0.4

P(Z<z2)-P(Z<z1)=0.4 , since Z is symmetrical so z1=-z and z2=z

now P(Z<z)-P(Z<-z)=0.4

or 2*P(Z<-z)=0.2

or P(Z<-z)=0.1 and z=1.2816

and x1=mean-z*sd=0.021-1.28*0.051=-0.0443

x2=mean+z*sd=0.021+1.28*0.051=0.0864

(d) P(X<x)=0.8

and z=0.8416 for P(Z<z)=0.8

and corresponding x==mean+z*sd=0.021+0.8416*0.051=0.0639