Question

The quarterly returns for a group of 52 mutual funds with a mean of 3.2% and a standard deviation of 5,8% can be modeled by a Normal model. Based on the model N(0.032,0058)what are the cutoff values for the a) highest 40% of these funds? b) lowest 20%? c) middle 80%? d) highest 80%? a) Select the correct choice and fill in any answer boxes in your choice below. (Hound to two decimal places as needed.) A > O c. x

Answer 1

Let X denote the quarterly returns. Then $X\sim N(0.032,0.058)$

a)

We want to find 'x' such that $P(X>x)=0.40$

P(Z > 2 – 0.032 0.058) = 0.40 * 1-0.032 = 0.2533 = x=0.0467 0.058

{Using probability tables}

So, the answer is x>4.67%.

b)

We want to find 'x' such that $P(X<x)=0.20$

$P(Z<\frac{x-0.032}{0.058})=0.20\Rightarrow \frac{x-0.032}{0.058}=-0.8416\Rightarrow x=-0.0168$

{Using probability tables}

So, the answer is x<-1.68%.

c)

We want to find 'x' such that $P(-x<X<x)=0.80$

$P(X<x)-P(X<-x)=0.80\Rightarrow P(X<x)-P(X>x)=0.80$

$\Rightarrow P(X<x)-[1-P(X<x)]=0.80 \Rightarrow 2P(X<x)-1=0.80$

$\Rightarrow P(X<x)=0.90$

$\Rightarrow P(Z<\frac{x-0.032}{0.058})=0.90\Rightarrow \frac{x-0.032}{0.058}=1.2816\Rightarrow x=0.1063$

{Using probability tables}

So, the answer is -10.63%<x<10.63%.

d)

We want to find 'x' such that $P(X>x)=0.80$

$P(Z>\frac{x-0.032}{0.058})=0.80\Rightarrow \frac{x-0.032}{0.058}=-0.8416\Rightarrow x=-0.0168$

{Using probability tables}

So, the answer is x>-1.68%.