Question

In a survey of a group of men, the heights in the 20-29 age group were normally distributed, with a mean 69.1 inches and a standard deviation of 4.0 inches. A study participant is randomly selected. Find the probability that a study participant has a height less than 68.

Answer 1

`P(X<68)=P((X-\mu)/(\sigma)<(68-\mu)/(\sigma))=P(Z<(68-69.1)/(4.0))=P(z<-0.275)`

And we can use the normal standard distribution table or excel to find the probability of interest:

`P(z<-0.275)=0.392`

So then we can conclude that the probability that a study participant has a height less than 68 is approximately 0.392 or 39.2%

Explanation:

We can define X as the random variable that represent the heights of a population desired, and for this case we know the distribution for X is given by:

`X \sim N(69.1,4.0)`

Where
`\mu=69.1` and
`\sigma=4.0`

We want to find the following probability:

`P(X<68)`

And we can use the z score formula given by:

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

Using this z score formula we have this:

`P(X<68)=P((X-\mu)/(\sigma)<(68-\mu)/(\sigma))=P(Z<(68-69.1)/(4.0))=P(z<-0.275)`

And we can use the normal standard distribution table or excel to find the probability of interest:

`P(z<-0.275)=0.392`

So then we can conclude that the probability that a study participant has a height less than 68 is approximately 0.392 or 39.2%