Question

The price-earnings ratios of a sample of stocks have a mean value of 13.5 and a standard deviation of 2. If the ratios have a bell-shaped distribution, what can be said about the proportion of ratios that fall between 11.5 and 15.5

Answer 1

`P(11.5<X<15.5)=P((11.5-\mu)/(\sigma)<(X-\mu)/(\sigma)<(13.5-\mu)/(\sigma))=P((11.5-13.5)/(2)<Z<(15.5-13.5)/(2))=P(-1<z<1)`

And we can find the probability with this difference

`P(-1<z<1)=P(z<1)-P(z<-1)`

And we can use the normal standard distribution or excel and we got:

`P(-1<z<1)=P(z<1)-P(z<-1)=0.841-0.159=0.682`

So then we expect a proportion of 0.682 between 11.5 and 13.5

Explanation:

Let X the random variable that represent the price earning ratios of a population, and for this case we know the distribution for X is given by:

`X \sim N(13.5,2)`

Where
`\mu=13.5` and
`\sigma=2`

We want to find the following probability

`P(11.5<X<15.5)`

And we can use the z score formula given by:

`z=(x-\mu)/(\sigma)`

Using this formula we got:

`P(11.5<X<15.5)=P((11.5-\mu)/(\sigma)<(X-\mu)/(\sigma)<(13.5-\mu)/(\sigma))=P((11.5-13.5)/(2)<Z<(15.5-13.5)/(2))=P(-1<z<1)`

And we can find the probability with this difference

`P(-1<z<1)=P(z<1)-P(z<-1)`

And we can use the normal standard distribution or excel and we got:

`P(-1<z<1)=P(z<1)-P(z<-1)=0.841-0.159=0.682`

So then we expect a proportion of 0.682 between 11.5 and 13.5